Vehicle Accident Analysis and Reconstruction Methods, Second Edition. [2nd ed.] 9780768088281, 0768088283

Citation preview

Accident Analysis Reconstruction Methods Second Edition Raymond M. Brach and R. Matthew Brach

ERRATA Vehicle Accident Analysis and Reconstruction Methods, Second Edition (R-397) Corrections to equations: Chapter 3

y   b  (hC  R) 

(3.44)

y( )  c1  e1  1  c2  e1  1  c3  e21  1  c4  e21  1

(3.50)

Chapter 4

( x i  a ) 2  ( y i  b ) 2  R 2 , i  1, 2 , 3

(4.1)

Chapter 9

K2  L C1  2(C2  C3  C4  C5 )  C6  /10

(9.4)

Cavg  C1  2  C2  C3  C4  C5   C6  /10

(9.15)

Vehicle Accident Analysis and Reconstruction Methods

Other related resources from SAE International: Tire Forensic Investigation By Thomas Giapponi (Product Code: R-387) Automotive Safety Handbook, Second Edition By Ulrich W. Seiffert and Lothar Wech (Product Code: R-377) An Engineer in the Courtroom By William J. Lux (Product Code: R-155) Crash Reconstruction Research Edited by Michael S. Varat (Product Code: PT-138)

For more information or to order a book, contact SAE International at 400 Commonwealth Drive, Warrendale, PA 15096-0001 USA; phone 877-606-7323 (U.S. and Canada only) or 724-776-4970 (outside U.S. and Canada); fax (724) 776-0790; e-mail [email protected]; website http://books.sae.org.

Vehicle Accident Analysis and Reconstruction Methods

Raymond M. Brach and R. Matthew Brach

Warrendale, Pennsylvania, USA

Copyright © 2011 SAE International

eISBN: 978-0-7680-5740-9

400 Commonwealth Drive Warrendale, PA 15096-0001 USA E-mail: [email protected] Phone: 877-606-7323 (inside USA and Canada) 724-776-4970 (outside USA) Fax: 724-776-1615 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: [email protected]; phone: 724-772-4028; fax: 724-772-9765. ISBN 978-0-7680-3437-0 SAE Order No. R-397 DOI 10.4271/R-397 Library of Congress Cataloging-in-Publication Data Brach, Raymond M. Vehicle accident analysis and reconstruction methods / Raymond M. Brach and R. Matthew Brach. p. cm. Includes bibliographical references and index. ISBN 978-0-7680-3437-0 1. Traffic accident investigation. 2. Traffic accidents—Simulation methods. 3. Traffic accidents— Mathematical models. 4. Automobiles—Dynamics. I. Brach, R. Matthew. II. Title. HV8079.55.B723 2011 363.12’565--dc22 2010053248 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: [email protected] Phone: 877-606-7323 (inside USA and Canada) 724-776-4970 (outside USA) Fax: 724-776-1615 Visit the SAE Bookstore at http://store.sae.org

Dedication

A great deal of support was provided by our families during the preparation of this work. Our wives, Carol Brach and Paula Brach, encouraged this endeavor and shared in the effort required to complete the task. Matt’s children, Elizabeth, Olivia, and Daniel, sacrificed family time to permit writing to be done. This book is dedicated to them.

v

Contents

Foreword_____________________________________________________ xv Preface to Second Edition_______________________________________ xvii Preface to First Edition_ ________________________________________ xix Acknowledgments____________________________________________ xxiii Chapter 1

Uncertainty in Measurements and Calculations___________________ 1

1.1 1.2 1.3 1.4 1.5

Introduction Upper and Lower Bounds Differential Variations Statistics of Related Variables 1.4.1 Linear Functions 1.4.2 Arbitrary Functions (Approximate Method) Application Issues 1.5.1 Other Methods of Evaluating Uncertainty

1 4 5 8 8 9 11 12

Chapter 2

Tire Forces_______________________________________________ 15

2.1 Introduction 2.2 Rolling Resistance 2.3 Slip, Longitudinal Force, and Lateral Force 2.3.1 Longitudinal Slip 2.3.2 Comments, the Coefficient of Friction, and the Frictional Drag Coefficient

15 17 17 19 22

vii

Contents

2.3.3 Longitudinal Tire Force 2.3.4 Lateral Tire Force 2.4 Friction Circle and Friction Ellipse 2.4.1 Idealized Friction Circle and Idealized Friction Ellipse 2.4.2 Actual Friction Circle and Actual Friction Ellipse 2.5 Modeling Combined Steering and Braking Tire Forces 2.5.1 The Bakker-Nyborg-Pacejka Model for Lateral and Longitudinal Tire Forces 2.5.2 Modified Nicholas-Comstock Combined Tire Force Model 2.6 Application Issues 2.6.1 Antilock Braking Systems (ABS) 2.6.2 Light Vehicle Frictional Drag Coefficients 2.6.3 Frictional Drag Coefficients for Heavy Trucks 2.6.4 Hydroplaning

23 25 27 28 30 31 31 33 37 39 40 41 46

Chapter 3

Straight-Line Motion_______________________________________ 51

3.1 Uniform Acceleration and Braking Motion 3.1.1 Equations of Constant Acceleration 3.1.2 Grade and Equivalent Drag Coefficients 3.2 Vehicle Forward-Motion Performance Equations 3.3 Stopping Distance 3.3.1 Distance from Speed 3.3.2 Speed from Distance 3.4 Application Issues 3.4.1 Stopping Distance 3.4.2 Motion Around Curves 3.5 Vehicle Fall Equations 3.5.1 Equations of Motion of a Vehicle Leading to a Fall

51 51 54 54 59 60 60 62 62 63 64

64

Chapter 4

Critical Speed from Tire Yaw Marks___________________________ 69

4.1 4.2 4.3

viii

Estimation of Speed from Yaw Marks Yaw Marks 4.2.1 Radius from Yaw Marks Critical Speed 4.3.1 Critical Speed Formula 4.3.2 Roadway with Superelevation

69 71 73 73 73 74

Contents

4.4 Application Issues 4.4.1 Tire Marks in Practice 4.4.2 Other Curved Tire Marks 4.4.3 Coefficient of Friction, f 4.4.4 Driver Control Modes 4.4.5 Tire Forces in a Severe Yaw 4.4.6 The Critical Speed Formula and Edge Drop Off (Road Edge Reentry) 4.5 Uncertainty of Critical Speed Calculations 4.5.1 Estimation of Uncertainty by Differential Variations 4.5.2 Accuracy of the Critical Speed Method 4.5.3 Statistical Variations

76 76 76 76 77 77 79 79 79 80 81

Chapter 5

Reconstruction of Vehicular Rollover Accidents__________________ 85

5.1 Introduction 5.2 Rollover Test Methods 5.3 Documentation of the Accident Site 5.4 Pretrip Phase 5.5 Trip Phase 5.5.1 Analysis of Vehicle Trip 5.5.2 Complex Vehicle Trip Models 5.5.3 Reconstruction of the Trip Phase 5.5.4 Rim Contact 5.6 Roll Phase 5.6.1 Speed Analysis 5.6.2 Analysis of the Rolling Vehicle 5.6.3 Information about the Accident Scene and Site 5.6.4 Information about the Accident Vehicle 5.6.5 Rollover Reconstruction Tools 5.7 Example Rollover Reconstruction 5.7.1 Speed Analysis 5.7.2 Detailed Roll Analysis 5.8 Vehicle Roll Rate During Rollover 5.9 Further Considerations

85 86 88 89 95 96 101 102 105 106 107 109

110 111 115 117 117 120 126 128

Chapter 6

Analysis of Collisions, Impulse-Momentum Theory______________ 129

6.1 6.2 6.3

Introduction Quantitative Concepts Point-Mass Impulse-Momentum Collision Theory

129 131

132

ix

Contents

6.3.1 Coefficient of Restitution, Frictionless Point-Mass Collisions 6.3.2 Collisions Where Sliding Ends; the Critical Impulse Ratio, μ0 6.3.3 Sideswipe Collisions and Common-Velocity Conditions 6.4 Controlled Collisions 6.4.1 Coefficients of Restitution 6.4.1.1 Stiffness Equivalent Collision Coefficient of Restitution 6.4.1.2 Mass Equivalent Collision Coefficient of Restitution 6.5 Planar Impact Mechanics 6.5.1 Overview of Planar Impact Mechanics Model 6.5.2 Application Issues: Coefficients, Dimensions, and Angles 6.5.2.1 Coefficient of Restitution and Impulse Ratio 6.5.2.2 Distances, Angles, and Point C 6.5.3 Work of Impulses and Energy Loss (Crush Energy) 6.6 RICSAC Collisions

136 136 137 139 141 142 143 144 150 153 153 155 155 158

Chapter 7

Reconstruction Applications, Impulse-Momentum Theory________ 163

7.1 7.2 7.3 7.4 7.5 7.6 7.7

Introduction Point-Mass Collision Applications Rigid Body, Planar Impact Mechanics Applications; Vehicle Collisions with Rotation Collision Reconstruction Using a Solution of the Planar Impact Equations Reconstructions Using a Spreadsheet Solution of the Planar Impact Equations Low-Speed In-Line (Central) Collisions Airbags, Event Data Recorders (EDR), and ΔV 7.7.1 Crash Data 7.7.2 Precrash Data

163 164

168 171 172 186 189 191 192

Chapter 8

Collisions of Articulated Vehicles, Impulse-Momentum Theory_ ___ 193

8.1 Introduction 8.2 Assumptions for Application of Planar Impact Mechanics to Articulated Vehicles

x

193

195

Contents

8.3 8.4 8.5

Articulated Vehicle Impact Equations Validation of the Articulated Vehicle Impact Equations Using Experimental Data Appendix: Data Sheets for Example 8.4

198 206 222

Chapter 9

Crush Energy and ΔV_____________________________________ 225

9.1 Introduction 225 9.2 Crush Stiffness Coefficients Based on Average Crush from Rigid Barrier Tests 236 9.3 Application Issues 241 9.3.1 Crush Stiffness Coefficients from Vehicle-to-Vehicle Collisions 241 9.3.2 Damage to One Vehicle Unknown 243 9.3.3 Side Crush Stiffness Coefficients, Two-Vehicle, Front-to-Side Crash Tests 243 9.3.4 Nonlinear Models of Crush 243 9.3.5 Arbitrary Number of Crush Measurements 243 Chapter 10

Frontal Vehicle-Pedestrian Collisions_________________________ 245

10.1 10.2 10.3 10.4 10.5 10.6 10.7

Introduction General and Supplementary Information Hybrid Wrap Model Forward Projection Model Analysis Model 10.5.1 Pedestrian Motion 10.5.2 Vehicle Motion Values of Physical Variables 10.6.1 Pedestrian-Ground Drag Coefficient Reconstruction Model

245 248 248 248 249 249 252 252 253 254

Chapter 11

Photogrammetry for Accident Reconstruction__________________ 259

11.1 Introduction 11.2 Reverse Projection Photogrammetry 11.2.1 Overview and Requirements of the Reverse Projection Process 11.2.2 Reverse Projection Procedure 11.2.3 Summary of the Main Steps in the Reverse Projection Photogrammetry Process

259 260 261 263 268

xi

Contents

11.3 Planar Photogrammetry 11.4 Three-Dimensional Photogrammetry 11.4.1 The Fundamental Information Related to Three-Dimensional Photogrammetry 11.4.2 Mathematical Basis of Three-Dimensional Photogrammetry 11.4.3 Projection Equations 11.4.4 Collinearity Equations 11.4.5 Coplanarity Equations 11.4.6 Multiple Image Considerations 11.4.7 Considerations of the Use of Three-Dimensional Photogrammetry in Practice 11.5 Appendix: Projective Relation for Planar Photogrammetry

274 281 283 284 285 287 287 287 288 297

Chapter 12

Railroad Grade Crossing and Road Intersection Conflicts_________ 299

12.1 Introduction 12.2 Clearing a Crossing or Intersection Using a Sight Triangle 12.3 Sight Distance for Stopping Before a Crossing or Intersection 12.4 FHWA Grade Crossing Equations 12.4.1 Stopping Distance 12.4.2 Stopping Sight Distance 12.4.3 Clearing Sight Distance 12.5 Locomotive Horn Sound Levels at Railroad Grade Crossings 12.5.1 Computation of Horn Sound Levels at a Distance from a Point Source 12.5.2 Insertion Loss of Light Vehicles

299

300 304 309 309 311 311 313 313 316

Chapter 13

Vehicle Dynamic Simulation________________________________ 321

13.1 13.2 13.3 13.4 13.5

xii

Introduction Planar Vehicle Dynamics Simulation Tire Side-Force Stiffness Coefficients 13.3.1 Light-Vehicle Side-Force Coefficients 13.3.2 Heavy-Vehicle Side-Force Coefficients Examples Appendix: Differential Equations of Planar Vehicular Motion Variables Notation

321 322 325 325 326 326

338 339 339

Contents

Appendix A Units and Numbers_______________________________ 341 A.1 Use of SI (Metric Units of Measure in SAE Technical Papers) A.2 Numbers, Significant Figures, and Rounding A.2.1 Significant Figures A.2.2 Rounding of Numbers A.2.3 Consistency of Significant Figures When Adding and Subtracting A.2.4 Consistency of Significant Figures When Multiplying and Dividing A.2.5 Other Forms of Number Manipulation A.3 Unit Conversions for Common Units

341 342 342 343 344 344 345 346

Appendix B Glossary of Common Terms and Acronyms in Accident Reconstruction______________________________________ 355 References___________________________________________________ 383

Bibliography of Vehicle Dynamics Books___________________________ 407 Index_______________________________________________________ 409

About the Authors_____________________________________________ 417

xiii

Foreword

Vehicle accident reconstruction is a discipline which has developed and grown enormously over the (nearly) forty years that it has been my good fortune to be involved in it. In the early 1970s the methods were crude, being largely confined to simple calculations on skidding and stopping distances and on how far this vehicle or that person would move in a given time. It was as well that the calculations were simple, since slide rules were the only practicable portable calculator, and personal computers were an improbable dream. In those days the ordinary practitioner knew very little about such things as tire dynamics, road surface properties, vehicle crush behavior, or what happens when a pedestrian is hit by a car. But gradually accident reconstructionists picked up knowledge on these matters from various fields of learning—vehicle and highway engineering, safety research, driver psychology, trauma medicine—and at the same time the means of handling it, in the shape of calculators, computers, and eventually the internet came into being. A good example is the CRASH program, developed for NHTSA as a road safety research tool. Although by around 1980 it was being recognised as something that reconstructionists could use, it required a mainframe computer, and even then there were no graphics, so the scope for misusing it was considerable. Yet by 1990 a number of versions of it, complete with graphics and much greater flexibility than the original, were available for use in portable personal computers. But a problem throughout this period was how to get a deep but practical knowledge of what was an ever-widening field of study. The admirable manuals published by Northwestern University both then and now covered a great deal of the field at a practical but relatively shallow level, being primarily directed at police officers rather than engineers. Anyone wishing to delve deeper was confronted by daunting volumes on the rigorous analysis of tire and vehicle dynamics, or the construction of highways, or

xv

Foreword

the medical details of occupant and pedestrian trauma. One response was the advent of accident reconstruction societies and conferences for the dissemination of information and pieces of research: as far as I can ascertain, the first society with a wide reach was IAARS, founded in 1980, while notable among conferences has been the annual Accident Reconstruction Session at the SAE World Congress, which began in 1987. But despite all this, there was still the lack of a scholarly and mathematically rigorous text for the reconstructionists. This is where Ray and Matt Brach come in. I first met Ray Brach in the late 1980s, when he visited me in London. I immediately recognised him as a practical as well as academic engineer, with a deep interest in solving everyday problems, notably vehicle collisions. I am pleased to say that the acquaintance has grown over the years, with getting to know his son Matt as well, and more recently through their management of the SAE Congress sessions. A book by them was therefore something to look forward to, and the first edition, published in 2005, did not disappoint. But a considerable bonus with the book was a training course, and I had the pleasure of hosting Ray and Matt when they presented it to a group of investigators (myself included) from TRL in 2006. A new edition is, of course, very welcome. The investigations which the authors have carried out into tire models, as used in the various simulation programs and now brought into Chapter 2, are particularly interesting to anyone hoping to simulate more extreme vehicle motions, while the new chapter on articulated vehicle collisions gives us a handle on a (for some of us) near intractable problem. Collisions at railroad grade crossings (the new Chapter 12) are less of an issue in this writer’s country, but then the treatment is equally valid for any vehicle-vehicle conflict at an intersection: and the unexpected and fascinating exposition on train horns is a fine example of how this field just gets wider and wider. This new edition is a most welcome addition to the accident reconstructionist’s library and will be a valuable source of guidance and information. Richard Lambourn Transport Research Laboratory www.trl.co.uk December 2010

xvi

Preface to Second Edition

This book remains unique as a presentation of methods of accident reconstruction — scientific, engineering, and mathematical methods. In contrast to many other books on this subject, the methods presented here are supported by references and/or data to establish their validity. It is not simply a second edition of an existing book, but is a revised and enhanced version. Typographical errors in the first edition have been corrected. Additional examples have been included in many of the chapters. The Glossary has been updated, particularly to include many new acronyms that have emerged with the use of event data recorders (EDRs). More noteworthy, however, is the improved coverage of tire forces; additional experimental data are presented and the topic of the tire ellipse/circle is taken out of the idealized realm, explained, and placed into a practical, realistic format based on experimental data. Tables and a nomograph for the use of frictional drag coefficients for sliding tires have been added. With appropriate modification, these can be used with applications involving antilock braking systems. Frictional drag coefficients for heavy trucks differ from those of light vehicles; this topic is now covered, with data and references included. Two new chapters have been added. Articulated vehicles are ever-present on our roads and are overly involved in serious crashes. The analysis of collisions of articulated vehicles requires special concepts and methods. The equations of the mechanics of such collisions are covered in a new chapter of this edition. The topic of the conflict of vehicles (road vehicles or rail trains) approaching each other at an intersection of roads or at a railroad grade crossing has received little or no coverage in other books. A new chapter in this edition presents methods for analyzing and reconstructing such events. In addition, this chapter covers the acoustics of train horns to allow estimation of sound levels from locomotives at grade crossings. Overall, the authors hope that this book will assist users of the methods presented here by placing their reconstructions of vehicle accidents on a firmer, more scientific foundation. As before, feedback is welcome, particularly if any errors are found.

xvii

Preface to First Edition

More than 19,000 registered junkyards (junkyards.com) operate in the United States. Each can have hundreds, or even thousands, of wrecked vehicles. Each vehicle sent to a junkyard following an accident represents a tragedy of some degree, with accompanying financial loss, human injury, and perhaps loss of life. One of the objectives of the reconstruction of vehicle accidents is to tell a vital part of that story. Automotive accident reconstruction is the process of determining what happened to the vehicles and persons involved in an accident and how it happened, using the information available after the accident occurred. This task must produce results that are reasonably accurate. Reconstruction is a procedure carried out with the specific purpose of estimating in both a qualitative and quantitative manner how an accident occurred using engineering, scientific, and mathematical principles based on evidence obtained through an accident investigation. The collection of facts associated with the circumstances of an accident is referred to as accident investigation. Determining what happened and how it happened is usually referred to as accident reconstruction. A third facet of postaccident analysis attempts to answer the question of why the accident occurred (causation or fault). This is almost always of interest to the various parties involved. Since vehicles are, or should be, controlled by humans, answering the why question more often than not involves motivation and human psychology. Legal issues can also arise in these incidents. These issues include the violation of criminal and traffic laws where the question of fault is placed before a jury. In this book, the topics of accident investigation, human factors, causation, and fault are not explicitly addressed. Often the effects of these topics are interrelated, and this interrelationship must be addressed to some extent. Indeed, it would be rare to be able to carry out a good reconstruction with little or no physical evidence and data. This book concentrates on reconstruction—the determination of how

xix

Preface to the First Edition

an accident happened. That said, it is sometimes necessary to combine investigation and reconstruction. Often a reconstructionist may recognize a need for information not gathered initially and must obtain it later. An accurate reconstruction cannot be carried out without a good investigation. A number of books already exist on the topic of accident reconstruction. With some notable exceptions, many of them are tomes devoted to how the authors and perhaps a few colleagues used intuition and insight to decide how they thought an accident happened. In a few cases, these books are collections of “war stories” or case histories, usually presentations of one view of the events. In contrast, this book is one of methods. The perspective taken here is that accident reconstruction is a field of applied science, namely an application of the principles of science, mathematics, and engineering; accident reconstruction is a quantitative endeavor. The same principles of mathematics, physics, and engineering that allow us to safely race vehicles more than 200 mph, build space stations, and navigate the depths of the oceans can be used to reconstruct vehicle accidents. This requires that reconstructions not only be based on the physical evidence and information gathered from an accident investigation but also be based on the laws of nature. A concerted goal of this book is to raise the analytical level of accident reconstruction practice such that commonly known scientific, engineering, and mathematical methods increasingly become a more common part of the field. During the preparation of this book, the authors have assumed that the readers and users of this book and accompanying software (available separately from the authors) have a proper educational background and experience to fully comprehend the material. The United States is not the only country with junkyards (repositories of disadvantaged vehicles, to be politically correct). Vehicle accidents, of course, happen all over the world. Fortunately for engineers, and everyone else for that matter, the laws of mathematics and physics are the same everywhere. Consequently, applications of the material contained herein are not limited by geography. Applications using international units do arise. Therefore, dual units, customary U.S. units (based on the units of foot, pound force, and second) and the metric system (based on the meter, Newton, and second) are used throughout the book. Certain items are avoided, such as the confusion between mass and weight. In customary U.S. units, mass sometimes is given the unit of slug. Its use is avoided here. One slug is equivalent to 1 lb-s2/ft; all appearances of the unit pound, abbreviated as lb, in this book refer to units of force. Similarly, in the metric system, a kilogram is considered to be a unit of mass and the corresponding unit of force is the Newton. The practice of stating weight (a force) in kilograms can cause confusion. This is avoided by always using units of Newtons for weight and kilograms for mass. The convention followed here is that weight is the measure of the force of gravity, and the metric unit of force is the Newton, abbreviated as N. A summary of the topical coverage of the book is not given here. The reader can simply look at the table of contents to see the list of topics. Two features of this book are unique, however, and deserve some mention. One is the coverage of the topic of uncertainty, and the other is the use of many examples throughout the book. Many analysis and reconstruction methods can be implemented using spreadsheet technology. This has been done by the authors, and solutions to the examples are in the form of input information and results

xx

Preface to the First Edition

printed directly from computer output. Tools contained in popular spreadsheets often allow analysis techniques (time forward computations) to be used for reconstructions (time reverse computations). A common omission made by all accident reconstructionists at one time or another is to measure something or make a calculation based on a measured or estimated parameter and come up with the answer, look at it, and if it seems to make sense, present it as a definite result or finding. For example, we say “that car was going 25 mph (36.7 ft/s, 11.2 m/s).” But could it have been 26.2 mph or 24.5 mph? How certain are we of the result? How certain can we be of the result? These questions refer to the uncertainty associated with a measurement or calculation based on a measurement, a group of measurements, or a group of calculations. A view is taken here that we actually are estimating values of dynamical variables. Some estimates have high accuracy and some not so high. The uncertainty associated with results based on these estimates of dynamic variables should always be considered. Determining uncertainty is not always easy to do—but difficulty is not a reason for omitting it. The topic of uncertainty is the first technical topic covered in this book. The uncertainty of a reconstruction may be difficult to calculate, but the authors hope that users of this book will appreciate that a reconstructed speed of a vehicle presented with five significant digits of precision has limited accuracy when the only available skid mark length used in the reconstruction calculation was measured by pacing off the distance. As already mentioned, a distinction is drawn here between accident reconstruction and accident investigation. The latter is considered to be the process of gathering physical and testimonial evidence from an accident scene, vehicles, and eyewitnesses. It is considered as a field of its own. Investigation is most often executed by police officers and sometimes by insurance investigators. As any other human endeavor, it can be done well and can be done poorly. Several institutes exist across the country, such as the Northwestern University Traffic Institute, Texas Transportation Institute, Institute of Police Technology and Management (University of North Florida), and others, for training investigators to standardize and improve investigation practices. Although there is a need for each to know what the other does and there is an overlap in knowledge and tasks, a trained accident investigator is not the same as an accident reconstructionist, just as an accident reconstructionist is not a trained accident investigator. Different aspects of an accident reconstruction frequently are segregated into the categories of human, vehicle, and environment. The study of human performance and behavior as it relates to vehicular accidents belongs to the study of human factors. This topic is not covered here. Another important aspect of accident reconstruction involving humans is that of occupant kinetics, kinematics, and biomechanics—the study of the motion of vehicle occupants and the physical interaction of a body with interior surfaces and restraints. These concepts are not covered. Environmental topics include such things as the design and performance of roadways, poles and barriers, signs, traffic signals, and their interaction with accidents and crashes. These topics are not covered. As in all professions, the work of accident reconstructionists involves communication and reporting of results. Though they can be extremely important, report writing and diagram preparation are not covered. Other

xxi

Preface to the First Edition

topics omitted include those of finite element analysis of vehicle crash deformation and dynamical crush simulation, such as Simulation Model of Automobile Collisions (SMAC), Simulation Model Nonlinear (SIMON), and others. Collectively, the authors of this book have over 45 years of experience in the practice of vehicle accident reconstruction as well as with the research associated with accident reconstruction methods. Based on this experience, the topics covered throughout the 11 chapters and appendices should provide the methods to quantitatively reconstruct the vast majority of vehicular accidents. Not all accidents involve a crash, or collision, of two vehicles, but most do. Planar impact mechanics (Chapters 6 and 7) is used extensively in the reconstruction of crashes, often combined with estimation of crush energy (Chapter 8). Evidence from the motion of vehicles before an impact or following an impact, or both, often supplies vital information to a reconstruction. Vehicle dynamics simulation (Chapter 11) is invaluable in modeling such motion. Simulation of vehicle dynamics requires the knowledge of how tire forces are generated (Chapter 2), a topic that all accident reconstructionists must thoroughly understand. Methods for the analysis of accidents involving a single vehicle, such as rollovers, pedestrians and bicycle riders hit by cars, or simply yaw marks made by a single vehicle during a sudden high-speed turn, are covered individually. Each accident reconstruction is unique as no two accidents are the same. Moreover, the reconstruction of these accidents can also require the use of different methodologies because of variations in physical evidence and investigative information. This leaves plenty of room for ingenuity and insight for the application of the methods presented in this book. The authors hope this book is useful to those who want to find out how accidents occurred.

xxii

Acknowledgments

With the second edition, as with the first, the authors benefited from many discussions with the members of the Accident Investigation and Reconstruction Practices (AIRP) Committee of the SAE and the attendees, authors, and presenters at the annual Accident Reconstruction sessions held at the SAE World Congress. The annual pilgrimage to Detroit continues to provide a fertile environment for improved understanding and advancement of the many technical subjects that form the basis of the methods used in the field of accident reconstruction. This enhanced second edition benefits from the numerous times that the authors presented material from the first edition while teaching the SAE Accident Reconstruction Methods seminar from 2004–2010. Many students who attended the class asked insightful questions and made numerous interesting suggestions regarding the material. These questions and suggestions led to many improvements in the material presented in the class. These improvements have been incorporated into this second edition. The efforts by Kevin Manogue in organizing the original manuscript and preparing examples, and by John McManus, Alan Asay, Don Parker, and Jim Sprague in reading some or all of the original manuscript were followed by Matt Londergan and Weimin Yue who each read part of this second edition. Special thanks goes to Linda Trego for her many helpful comments and changes that improved the manuscript. Raymond M. Brach R. Matthew Brach South Bend, Indiana

xxiii

CHAPTER

1 Uncertainty in Measurements and Calculations

1.1 Introduction Although there may be some cases when and/or where an accident can be reconstructed effectively without the use of calculations and without the use of investigative and experimental data, such cases are rare. And they are becoming more rare as the professional level of the field advances and as the demands for more professional and accurate reconstructions increase. Whenever measurements are made and whenever calculations are based on experimental data, a level of uncertainty exists. It is the purpose of this chapter to provide ways of quantifying uncertainty. This type of uncertainty is parametric uncertainty, and it relates to variations in the values of the input quantities (parameters) used in the calculations. Another type of uncertainty is modeling uncertainty. This is where two or more different methods (mathematical models) are used to calculate the same quantity. For example, Chapter 6 introduces the theory of impact using point mass concepts. (Point mass impact theory ignores rotation.) Point mass impact mechanics can be, and is, used to reconstruct vehicle collisions; its primary purpose, however, is to introduce the concepts of impact mechanics that lead to rigidbody impact—that is, planar impact mechanics. (Planar impact mechanics includes the effect of rigid body rotation during the impact process.) Planar impact mechanics is more rigorous and provides more accurate results. Except in some very special cases, using point mass impact mechanics and planar impact mechanics to reconstruct the same collision will produce different results. Such differences are an example of modeling uncertainty. Consider an example of parametric uncertainty involving measurements. Suppose a car with a conventional braking system is brought up to a speed, v, and its brakes are applied suddenly to a level where the wheels are locked. The tires leave visible tire marks

1

Chapter 1

on the road as the car skids to a stop. Suppose, further, that the speed of the car at the beginning of the skid marks is measured using a radar gun, and the length, d, of the skid marks is measured using a tape measure. A well-known equation from mechanics used to estimate the speed of a car leaving skid marks from an emergency stop [1.1] is

v = 2 fgd ,

(1.1)

where g is the gravitational constant. Because the speed and distance were measured, this equation can be used to compute the corresponding frictional drag coefficient, f. Although this is a common way of measuring f, it is just one way (for example, Goudie, et al. [1.2]). Solving Eq. 1.1 for f gives



v2 f = 2 gd

(1.2)

It is easy to see that if an error exists in the measurement of either v or d, there will be a corresponding error in f. As used here, the term error does not mean a “mistake” (for example, using the metric value of g instead of U.S. units). Rather, error is a difference from the “true” value of f because of measurement inaccuracy (for example, because the radar gun was held a few degrees off to the side or not directly behind the vehicle, or perhaps both), or even something else. The “something else” could be because the test conditions differed from a straight-line skid, such as if only three wheels locked and the car yawed before it stopped. Uncertainty can arise in other ways. Suppose each of two independent observers of the same friction experiment report the result as f1 = 0.45 and f2 = 0.454. One of the differences between f1 and f2 is that the latter has one more significant digit than the first1; can they be the same? Is one more correct than the other? Is one more accurate than the other? Sometimes these questions are difficult to answer. Suppose the above example is changed so that Eq. 1.1 is to be used to reconstruct the speed of a vehicle from the length, d, of a measured skidding distance and a value, f, the frictional drag coefficient. Suppose, further, that the values of f and d are not known exactly but may vary by some amount about a nominal value. For example, suppose the length, d, was measured several hours or more after an accident and the skid mark length may have changed due to the effects of weather and/or traffic. Suppose, further, that a value of f was measured at the accident site using a vehicle different from the stopping vehicle. Any variations in f and d are “propagated” through Eq. 1.1 and result in uncertainty in the speed. In the first example, variations due to measurements of v and d are propagated through Eq. 1.2 and result in uncertainty in the calculated value of 1. If the reader is not familiar with the concepts of significant digits and rounding, it is suggested that they read Appendix A.2, Numbers, Significant Figures, and Rounding.

2

Uncertainty in Measurements and Calculations

f. In this case variations in f and d cause uncertainty in the calculated value of v. Both of these examples have the common characteristic that variations in values placed into an equation lead to uncertainty in the result. This is the subject of this chapter. The terminology and notation used in this book are similar to other references but do have some specific differences. Although the word “error” is commonly used, particularly with respect to measurements (see [1.3, 1.4, 1.5]), its use is avoided here. This is to prevent any connotation that nonexact results are the result of some sort of mistake or blunder, and because error is defined differently by different authors. In general, the quantity to be calculated such as v and f on the left-hand sides of Eqs. 1.1 and 1.2, above, is given the symbol y. The quantity, y, is expressed as a sum of a reference value, Y, and a variation, δy. So the result of an operation (measurement or calculation of a quantity, y) results in y = Y ± δy. The closeness of the reference value to some “true” value often is referred to as the accuracy of y. In some circumstances, the difference y - Y is called a bias. The variation, δy, about the reference value often is referred to as the precision of y, or as Mandel [1.6] points out, the imprecision of y. After Mandel, the reference value, Y, can be placed into any of three categories based on how it is defined: a. A “true” value which generally is unknown. Any value used for Y is an estimate, and values are chosen using a method appropriate for the application. b. An “assigned” value that is agreed upon among experts in a field. An example of this is the value of the acceleration of gravity, g, at sea level, which is assigned the value g = 9.806650 m/s/s [1.7]. c. The mean of a randomly distributed population. With such a statistical definition, Y never is truly known but is estimated by the sample mean of a set of experiments. To generalize the above examples, the problem approached in this chapter is to determine Y and δy when they depend on variations of other variables, say x1, x2, . . . , xn. That is, suppose that

y = f ( x1 , x2 , . . . , xn )

(1.3)

Note that Eq. 1.3 can represent a complex sequence of more than one calculation carried out by a computer. In such cases, the variables x1, x2, . . . , xn could be functions of other variables, that is, x1 = x1 (u1, u2, . . . , un), x2 = x2( u1, u2, . . . , un), xn = xn( u1, u2, . . . , un), etc., where u1, u2, . . . , un are different independent variables. Three different approaches are covered. The first is to determine upper and lower bounds on y. Another is to use the analytical form of Eq. 1.3 and calculus to relate variations in the independent variables x1, x2, . . . , xn to Y and δy. Finally, a statistical approach is taken where the independent variables have known statistical properties. The meaning and interpretation of each of the components, Y and δy, differ in the three approaches and will be discussed in more detail.

3

Chapter 1

1.2 Upper and Lower Bounds One of the simplest ways of quantifying uncertainty is to establish upper and lower bounds on the dependent variable, y, caused by variations in all independent variables that possess significant variations. First, those quantities in the equation which possess a significant degree of variation are identified as variables of interest. Then a maximum range of each variable’s variation is determined. Finally, the lowest and highest values of the dependent variable are calculated for all possible combinations of the values of the independent variables. Values of all specific combinations of the independent variables are used in Eq. 1.3 in a way that produces the maximum and minimum values of y. From this δy and Y are assigned the values



1 2

1 2

δ y = ( ymax − ymin ) and Y = ( ymax + ymin )



(1.4)

Consider again the example given by Eq. 1.1. In this example, y is the speed, v, and so Y = V and δy = δv are sought. Under typical circumstances, f and d are known with less than perfect certainty. These are the independent variables. It is assumed that the acceleration of gravity, g, is known with sufficient accuracy and with negligible uncertainty and is a known constant. Suppose that d and f are unknown but can be bounded such that dmin ≤ d ≤ dmax and that fmin ≤ f ≤ fmax. Corresponding upper and lower bounds on the estimate of the initial speed, v, are



2 f min gd min ≤ v ≤ 2 f max gd max



(1.5)

The uncertainty δv is taken to be half of the difference between the upper and lower bounds,



1 2

δ v = ( 2 f max gd max − 2 f min gd min )



(1.6)

and the reference value, V, is the average of the upper and lower values of v; that is,



1 V = ( 2 f max gd max + 2 f min gd min ) 2

(1.7)

The result is that v = V ± δv. For example if fmin = 0.6, fmax = 0.8, dmin = 32.0 m and dmax = 34.0 m, then 19.4 m/s ≤ v ≤ 23.1 m/s. Then δf = 0.1, δd = 1 m and v = 21.3 ± 1.8 m/s. This example illustrates what is probably the simplest and most versatile method for determining uncertainty. It applies to any formula, no matter how complex, and is easy to execute. It even is possible to use this approach with computer simulations using multiple runs with different inputs. Care must be used when the formula for Y involves

4

Uncertainty in Measurements and Calculations

differences and division. For example, the lower limit of y = (x1 - x2)/x3 is obtained by using the lower limit of x1 and the upper limits of x2 and x3. Negative numbers also can be tricky. A drawback of this (and the next) method is that any likelihood of y to tend to be near the center of, or near either limit of, the range of Y ± δy cannot be assessed. Attributing the upper and lower bounds to a specific percentage of a population should not be done; statistical conclusions should follow the use of statistical methods and always be based on statistical data. Suppose a vehicle leaves skid marks of length, d, from an emergency stop with locked Example 1.1 wheels over a road surface with a frictional drag coefficient, f. The distance d is not known exactly, but is known to be greater than 32 m and less than 34 m. The frictional drag coefficient is known to be somewhere between 0.6 and 0.8. Determine the initial speed of the vehicle and associated bounds on the uncertainty. Solution  The maximum stopping distance is when d = dmax = 34 m and f = fmax = 0.8. From Eq. 1.1, vmax = 23.1 m/s (51.7 mph). For d = dmin = 32 m and f = fmin = 0.6, Eq. 1.1 gives vmin = 19.4 m/s (43.4 mph). Eqs. 1.6 and 1.7 give V = 21.3 m/s (47.6 mph) and δv = 1.8 m/s (4.1 mph). The end result is that the speed of the vehicle at the beginning of the skid marks is v = 21.3± 1.8 m/s (47.6 ± 4.1 mph).

1.3 Differential Variations Another common method of estimating uncertainty, often referred to as propagation of error, is covered in many laboratory courses taken in science and engineering (for example, see texts such as Beers [1.4] and Taylor [1.3]). As above, an equation or formula is being used in a reconstruction to calculate a physical quantity, y, representing a speed, time, distance, speed change, etc. The method uses differential calculus to relate y to the dependent variables x1, x2, . . . , xn. As before, the variables x1, x2, . . . , xn could be functions of other variables; that is, x1 = x1( u1, u2, . . . , un), x2 = x2( u1, u2, . . . , un), xn = xn( u1, u2, . . . , un), and so on. Using calculus, y = y(x) can be expressed in a Taylor series near x = a, as:

y ( x) = y (a) +

∂y 1 ∂ 2y ( x − a) + ( x − a)2 +  2 ∂x a 2! ∂ x a



(1.8)

If (x - a) is small, where a is a reference value of x, then δx = (x - a) is small and (x - a)2 and like terms of higher powers can be neglected. Let δy = y(x) - y(a), where y(a) = Y is the reference value of y. For a function of several independent variables, the nominal or reference values of x1, x2, . . . , xn are given by X1, X2, . . . , Xn. Under these conditions, a general formula for uncertainty can be found by replacing the variable differentials by variations, such that

5

Chapter 1

δy=

∂y ∂y ∂y δ x1 + δ x2 + ..... + δx ∂ x1 ∂ x2 ∂ xn n .

(1.9)

The derivatives in each of the terms in Eq. 1.9 often are referred to as sensitivity coefficients because their signs and magnitudes indicate how each of the variations, δxi, influences the uncertainty, δy. In applications, the absolute values of the sensitivity coefficients sometimes are used to prevent cancellation of the terms when using Eq. 1.9 to estimate uncertainty. Equation 1.9 is an approximation that amounts to a linearization of the function y(x1, x2, . . . , xn) around its reference value Y(X1, X2, . . . , Xn). Note that the derivatives are evaluated at the reference or nominal values. The relative uncertainty often is used and found by dividing Eq. 1.9 by Y, giving

δy

Y

=

X ∂ y δ xn X 1 ∂ y δ x1 X 2 ∂ y δ x2 + + ..... + n Y ∂ x1 X 1 Y ∂ x2 X 2 Y ∂ xn X n

(1.10)

Note that y = Y ± δy, and for relative uncertainty,

δy y = 1+ Y Y



(1.11)

The independent variables in the function y(x1, x2, . . . , xn) given by Eq. 1.3 often are thought to be representable in an n-dimensional vector space. In such circumstances it is common to define a norm2 of δy, where

δy= (

∂y 2 2 ∂y 2 2 ∂y 2 2 ) δ x1 + ( ) δ x2 +  + ( ) δ xn ∂ x1 ∂ x2 ∂ xn



(1.12)

Equations 1.9 and 1.12 are different expressions for the same quantity, and it is natural to ask which is correct, or at least, which is better? If the variations, δx1, δx2, . . . , δxn are viewed as orthogonal components of an n-dimensional vector, then it can be shown that the value from Eq. 1.12 will always be less than or equal to the value from Eq. 1.9, and so Eq. 1.9 gives a larger, or more conservative, value. However, the value from Eq. 1.12 generally provides a realistic estimate of uncertainty, and so it is commonly used. Another important difference is that each of the terms in Eq. 1.9 can have signs that depend on the form of the function y and its derivatives ∂y∕∂xi. This means that positive and negative variations can cancel each other. Although cancellations can occur, even to the extent where δy could be zero, this is not something that can be expected. Consequently, absolute value signs sometimes are used with each term of Eq. 1.9. This is not done here, but caution must be used when applying Eq. 1.9. The use of Eq. 1.12 avoids such problems and is recommended. 2. In mathematics, a norm is a measure of the size or length of a vector.

6

Uncertainty in Measurements and Calculations

In Example 1.1, y = v and there are two variables, x1 = f and x2 = d. After using Eq. 1.10, the relative uncertainty of v is given by

δv

1 δ f δd ) = ( + V 2 F D

(1.13)

where the quantities D and F are the reference values of the independent variables. Using the values from the example, it is clear that the variation in friction often has a considerably greater effect than the variation in the distance measurements. This is because, for values typically encountered in practice, δf∕F is larger than δd∕D. This is particularly true when the friction coefficient is small, such as under icy conditions. Recall that the derivation of Eq. 1.9 involves replacement of infinitesimal differentials by finite variations. Consequently, there is some degree of approximation involved when using differential variations. As an approximation, higher-order terms were dropped after expansion of y in a Taylor series about the reference values X1, X2, . . . , Xn. It is difficult to assess the error of the approximation because it depends on the functional form of the expression for y as well as the size of the variations in the dependent variables. The following example demonstrates that the agreement can be reasonably close. A vehicle skids to rest over a distance d = 33 m (108.3 ft) in a straight line with its wheels Example 1.2 locked. The measured value of the frictional drag coefficient is f = 0.70. Variations of the measured values of d and f are established as δd = ± 1m and δf = ± 0.10. Determine the initial speed and its uncertainty by computing a) upper and lower bounds and b) using differential variations. Solution  The reference, or nominal, value of the reconstructed speed can be obtained from Eq. 1.1 using D = 33 m and F = 0.70. This gives V = 21.3 m/s (69.8 ft/s). Using the same equation, computation of lower and upper bounds for (dmin, fmin) = (33, 0.60) and (dmax, fmax) = (34, 0.80) gives lower and upper bounds (vmin, vmax) = (19.4, 23.1) m/s Figure 1.1 (63.7, 75.8) ft/s, respectively. Distribution of a sample Using the method of differential variations, the uncertainty in the speed, δv, can be found using Eq. 1.13 after multiplying by the reference value of the speed. This gives δv = 1.8 m/s (6.0 ft/s). The reference value, V, is the same as before, so the reconstructed speed using differential variations is v = 21.3 ± 1.8 m/s (69.7 ± 6.0 ft/s). If Eq. 1.12 is used instead of Eq. 1.13, the uncertainty is δv = 1.6 m/s (5.1 ft/s) and the reconstructed speed is v = 21.3 ± 1.6 m/s (69.7 ± 5.1 ft/s). As mentioned above, the use of Eq. 1.13 gives

of skid numbers for 230 test sections of a two-lane country road with a wide range of average daily traffic [1.7]. The average is 0.392 and the standard deviation is 0.077.

7

Chapter 1

more conservative results than Eq. 1.12. To summarize, bounding gives the broadest uncertainty, δv = ± 1.8 m/s; the differential variations method gives δv = ± 1.6 m/s (or δv = ± 1.8 m/s from Eq. 1.13).

1.4 Statistics of Related Variables Sometimes the statistical distribution of a quantity or variable is known. For example, it is known that the height, h, of males in the United States is normally distributed with a certain mean, μh, and variance, σh,2; or that the distribution of skid numbers (see Appendix B, Glossary) measured at intervals along roads is, at least approximately, normally distributed (see Fig. 1.1). Furthermore, it often happens that the variable, say x, whose statistical properties are known, is related through a mathematical equation to another variable, say y. Let

y = f ( x)

(1.14)

be the mathematical equation where the function f(x) is given and the statistical distribution of x is known. In this context, x and y are referred to as random variables. The question often arises: what is the distribution of y? This is not an easy question to answer with mathematical rigor for an arbitrary function f(x), but there are some important cases where useful results can be found. In one case the function is linear; in another, an approximate method of answering the question is satisfactory, or it is the only means available. Each of the variables x and y has a certain statistical distribution (such as a normal distribution, a uniform distribution, a chi-square distribution, etc.). Each distribution has parameters (mean, median, variance, etc.). When random variables are related, such as by Eq. 1.14, it is common to find that knowing the statistical distribution of x doesn’t mean the distribution of y is known or can be found. In fact, relating distributions is a very difficult problem. The following section concentrates on finding the parameters μy and σy from the parameters μx and σx.

1.4.1 Linear Functions When the function in Eq. 1.14 is linear; that is,

y = f ( x) = ax + b

(1.15)

where a and b are constants, the statistical properties of y are well known (for example, see Guttman, et al. [1.8] and Montgomery and Runger [1.9]) and relatively simple. If x has any statistical distribution with mean, μx, and variance, σx2, then the mean of y is

8

µ y = aµ x + b



(1.16)

Uncertainty in Measurements and Calculations

and the variance of y is

σ y 2 = a 2σ x 2





(1.17)

For linear functions, Eq. 1.15, if x is normally distributed, then y will be normally distributed. Knowing this is helpful in relating uncertainty to variations through equations using statistics. Suppose that under certain circumstances, the perception-decision-reaction time, Example 1.3 tpdr, for drivers is known to be normally distributed with a mean, μt, of 1.75 s and a standard deviation of σt = 0.2 s. What are the mean and standard deviation of the distance, d, traveled by a vehicle with a uniform speed of 20 m/s (65.6 mph) under such circumstances? Solution  From mechanics, it is known that distance is the product of speed, v, and time, τ, so

d = vτ Here, v is a constant, and the relationship between d and τ is linear. If the time τ = τpdr, then d has a normal distribution, where the mean, μd, is

µd = vµt = 20.0(1.75) = 35.0 m (98.4 ft ) and the variance of d is

σ d 2 = v 2σ t 2 = 202 (0.2) 2 = 16.0 m 2 (172.2 ft 2 ) The standard deviation of d is σd = 4.0 m (13.1 ft). With this information and based on the properties of the normal distribution, it can be stated that under the given conditions, drivers will fully react in a distance of 35.0 ± 2(4.0) = (27.0, 43.0) m, (88.6, 141.1) ft, 95% of the time. Because of the bell shape of the normal probability curve, values near the center are more likely than those near the tails.

1.4.2 Arbitrary Functions (Approximate Method) For an arbitrary function, f(x), a way of relating the statistical parameters of x and y is to expand the function in a Taylor series about the mean of x, neglect higher-order terms (that is, linearize the expansion), and then use Eqs. 1.16 and 1.17. The Taylor series of y about the point x = μx is

y ( x) = f ( µ x ) +

∂f ∂x

( x − µx ) + µx

1∂2f 2 ∂ x2

( x − µ x )2 + . . . µx



(1.18)

9

Chapter 1

If terms of power 2 and higher are neglected, then this reduces to

y ( x) = f ( µ x ) +

∂f ∂x

( x − µx ) µx



(1.19)

Evaluating this at the mean of x, μx, gives the mean of y:

y(µx ) = f (µx ) = µ y





(1.20)

Substituting this into Eq. 1.19 and rearranging gives

y ( x) − µ y =

∂f ∂x

( x − µx ) µx



(1.21)

Viewing Eq. 1.21 as a linear relationship, such as Eq. 1.15, and using Eq. 1.17 gives the variance of y as 2

σy

2

 ∂ f   2 =    σx  ∂ x  µ x 

(1.22)

This process can be generalized to a function of many variables, y = f(x1, x2, . . . , xn), which gives

∂ f σ = ∑ i =1  ∂ xi 2 y



n

2

n n  2 ∂ f + σ 2  ∑∑ µ xi  xi i =1 j =1  ∂ xi 

µ xi

 ∂ f    ∂ xj

µ xj

  σ xi x j 

(1.23)

where the variables in all of the terms in Eq. 1.23 are evaluated at their mean values. The quantity is the covariance of xi and xj and is zero if variables xi and xj are statistically independent. When all of the variables xi and xj are statistically independent, then



σ y = σ x2 + σ x2 + ... + σ x2 1

2

n



(1.24)

This shows that the standard deviation, σy, of the result of a calculation involving many variables (n ≥2) grows as the root-mean-square of the variance of the input quantities, xi. Some words of caution: These expressions are approximate because the Taylor series expansion was linearized. In addition, although expressions for the mean and variance of y have been found, the statistical distribution of y generally is unknown, even when the distribution of x is known. Whereas the distribution of y is the same as the distribution

10

Uncertainty in Measurements and Calculations

of x when x is normally distributed and when x and y are related linearly, the same is not true when the function f(x) is nonlinear. The well-known formula from mechanics, Eq. 1.1,

Example 1.4

v = 2 fgd gives the initial velocity, v, for a vehicle that slows to rest uniformly with deceleration fg over a distance, d, where f is a constant frictional drag coefficient. Suppose that f is normally distributed with a mean, μf = 0.4, and a standard deviation, σ f = 0.08. What are the mean and standard deviation of the initial speed if the stopping distance is known to be exactly d = 20 m? Solution  From Eq. 1.20, the mean of the distribution of the initial speed is

mv = 2 mf g md = 2(0.4)9.81(20) = 12.53 m / s (41.1 ft / s ) and from Eq. 1.22 the standard deviation of v is

 gd s v =  2 fgd 

 s   f = mf

f

= 15.66(0.08) = 1.25 m / s (4.1 ft / s )

1.5 Application Issues Three methods were covered in this chapter for determining the uncertainty. Whenever more than one method is available for the solution of a problem, questions arise as to which is best and which should be used. Such an answer is dependent on the information available for the solution of the problem. But another response could be to use as many approaches as are available to solve the problem. If the answers agree, multiple results from independent approaches reinforce their validity and reduce the uncertainty in a subjective manner. If the answers differ significantly, this is an indication of a need to question the formulation of the problem or, at least, one of the solutions. Statisticians are careful to distinguish between a population and a sample. For example, the mean and standard deviation of a sample are designated as x and s, whereas the mean and standard deviation of a population are designated as μ and σ. When we make measurements, we are “sampling” from a population, and the statistics (x, s) are estimates of (μ, σ). How close the estimates are to the population parameters depends on many factors, such as how well the experiments are controlled and the number of times, n, we repeat the measurements (the sample size). In fact, most statistics books devote chapters, usually under the titles of point estimation, interval estimation, and/or confidence intervals, to how close samples estimate populations and how much confidence can be

11

Chapter 1

placed on the sample values. So, can (x, s) be used for (μ, σ)? The answer is found by using a statistical test. When sample sizes are large (see for example Fig. 1.1, where n = 230), transference of (x, s) to (μ, σ) has little risk of error, but if the sample size is small, then this should be done with caution and awareness. When dealing with a normal distribution, a rule of thumb is that for n ≥ 30, x ≈ μ and s ≈ σ. For small samples sizes, other methods can be used, such as in the following example. Example 1.5 Example 4.4 shows measurements that were made of the radius of a tire’s yaw mark from

seven critical speed tests. The radii are: 130, 137, 128, 126, 126, 115, and 132 ft.

Figure 1.2 Normal probability plot of the measured radii from yaw marks showing points that approximately form a straight line.

Develop a normal probability plot of A.  the radius data to examine if it follows a straight-line trend and thus indicates normality of the sample. B. Assuming the measurements are from a normal population, and using formulas from a statistics book or statistical software, determine a 95% confidence interval for the mean of the population. Solution A  A normal probability plot [1.8] of the data is shown in Fig. 1.2. The points follow a straight-line trend reasonably well (a criterion for normality), so the data can be considered to be from a normal population. Solution B The descriptive statistics of the radius data are given in Table 1.1. The statistics were found using a spreadsheet function that includes a 95% confidence interval for the population mean of the radius. It is 127.7143 ± 6.2888 = (121, 134) ft and (37.0, 40.8) m. Interpretation of the remaining descriptive statistics in the table is left to the reader.

1.5.1 Other Methods of Evaluating Uncertainty Methods of evaluating parametric uncertainty other than what are covered above exist. One method associated with statistical, or randomized, evaluation of formulas or functions is the use of Monte Carlo methods. Although the method is not explained in this book, a few examples of its use are presented in later sections of this book. Other applications of Monte Carlo methods are presented in a book on uncertainty [1.10]. Many direct applications to accident reconstruction have been published; for example, see references 1.12 through 1.22. These and other books and references on the subject should be consulted. Another method that can be used to study the uncertainty and sensitivity of calculations is the method commonly called design of experiments. It is an efficient method,

12

Uncertainty in Measurements and Calculations

particularly suitable when variations of a large number of variables is necessary. The method is covered in detail elsewhere [1.10] with an application to accident reconstruction in [1.23].

Descriptive Statistics Mean

127.7143

Standard Error

2.5701

Median

218.0000

Mode

126.0000

Table 1.1

Standard Deviation 6.7999 A final, and relatively new, method is one associated with (statistical) Sample Variance 46.2381 distribution-free methods. In some Kurtosis 2.0101 situations, specific values for part or all Skewness -0.8713 of the data used in a calculation, and the Range 22.0000 corresponding statistical distribution of Minimum 115.0000 the variations of the input, may not be Maximum 137.0000 known. Rather, only a range or interval Sum 894.0000 of possible values is known. This Count 7.0000 introduces another type of uncertainty, Confidence Interval, +/6.2888 that due to a lack of knowledge about (95.0%) the specific values of the data, known as epistemic uncertainty. When all other experimental uncertainties are removed, epistemic uncertainty can still remain and become the overall uncertainty. An approach different from what has been presented above, which assumes linearizably small and statistically distributed measurement uncertainties, is required. Ferson, et al. [1.24] present such an approach that is based upon interval statistics. The more usual methods assume that uncertainty results primarily from variability in the data that is caused by inherent randomness and/or finite sampling (termed aleatory uncertainty). Based on the aleatory (statistical distribution) approach, additional measurements will reduce the lower bound of the uncertainty estimate. However, when epistemic uncertainty is present, the situation can be different. The usual methods will underestimate the overall measurement uncertainty when epistemic uncertainty is present. Further, additional data will not necessarily lead to a reduced uncertainty. The reader is referred to the report by Ferson, et al. and the monograph by Salicone [1.25] for more detailed information. Treating epistemic uncertainty is associated with distribution-free methods such as those developed by Kolmogorov [1.26] and Smirnov [1.27]. Collectively, these are termed Kolmogorov-Smirnov (distribution-free) methods. Confidence limits from such methods do not depend upon the type of the distribution function, only on the fact that the distribution is continuous and that the samples have been drawn randomly and independently from the distribution. Such methods can produce very wide uncertainty limits.

13

CHAPTER

2 Tire Forces

2.1 Introduction Aside from gravitational forces, aerodynamic forces, and inter-vehicular forces developed during collisions, the forces and moments generated by the interaction between the tires of a vehicle and the road surface control the motion of a vehicle. Hence, vehicle dynamicists and vehicle accident reconstructionists require an understanding of these tire forces to effectively perform many of their tasks. Developing a means by which these forces can be accurately modeled mathematically is very important. This chapter presents the concepts underlying tire-road forces and moments. Concepts regarding forces developed by a single rolling tire are presented initially. This conceptual presentation is followed by the quantification of the individual lateral (steering) and longitudinal (traction: braking, acceleration) tire force components that lie in the plane of the road. Forces developed for combined lateral and longitudinal components (simultaneous braking and steering) are also covered. Only one (of many) different models for predicting lateral and longitudinal force components is presented. This is followed by a general method for combining lateral and longitudinal tire force components for conditions of combined braking and steering. To describe the forces and moments acting on a tire, it is necessary to establish a coordinate system that will serve as a reference with which tire system variables, component characteristics, forces, and moments can be described. One widely used coordinate system is the one adopted by the Society of Automotive Engineers [2.1], shown in Fig. 2.1. This figure shows three orthogonal force components, Fx, Fy, Fz and moment components Mx, My, and Mz associated with a rotating wheel, as well as two angular variables: α, the wheel slip angle, and γ, the camber angle.

15

Chapter 2

Figure 2.1 Tire coordinate system, force and moment components.

The force Fx is commonly referred to as the longitudinal force or traction force. It will be referred to here as the longitudinal force. It can be developed by engine torque as a driving, or accelerating, force or can be developed by brake application as a braking force. In various publications, Fy is referred to as the transverse, side, cornering, steering, or lateral force. It will be referred to here as the lateral, or steering, force. The lateral force component arises both as the result of an input to a steering wheel as well as a wheel slip angle developed on fixed, non-steerable axles. It is always perpendicular to the heading of the wheel. The vector resultant, F, of the lateral and longitudinal force components is in the plane of the contact patch, where F2 = Fx2 + Fy2; it is a type of friction force generated at the tire and road interface. The force, Fz, is the resultant normal force (perpendicular to the road surface) acting at the center of the tire patch. The moment about the z-axis is called the aligning moment. The moment about the x-axis is referred to as the overturning moment, while the moment about the y-axis is due to rolling resistance, brake torque, or drive torque. These three moments are not discussed in detail here, because the topic of interest for accident reconstructionists typically concerns only Fx, Fy, and Fz, individually and for combined braking and steering conditions. Moments are discussed elsewhere [2.2]. Analytical approaches have been developed that include all tire forces and moments [2.3, 2.4, 2.5, 2.6]. Although Fx and Fy are described and modeled here and elsewhere as forces, it must be kept in mind that they represent friction distributed over the contact patch surface. The normal force, Fz, is the effect of a pressure due to compression distributed over the contact patch.

16

Tire Forces

Two kinematic parameters are shown in Fig. 2.1: the camber angle, γ, and the wheel slip angle, α. The wheel slip angle is the angle formed between the direction of the wheel heading and the velocity of the wheel, or direction of travel of the wheel’s hub. The direction of the wheel heading is defined for convenience as the X-axis, as shown in Fig. 2.1. The camber angle represents wheel rotation about the X-axis as an angle between the wheel plane and the Z-axis. The area of mutual contact between the tire and the stationary surface over which it is moving is commonly referred to as the contact patch. Various factors influence the nature and shape of the static and dynamic contact patches between the tire and ground. These factors include the static vehicle weight supported by the tire; the tire’s inflation pressure, construction (bias ply vs. radial), profile, and velocity; suspension system characteristics; the presence of substances in the tire patch (water and contamination); road surface geometry (potholes, etc.); and the motion of the tire with respect to road contour. These dependencies are covered in detail elsewhere [2.7].

2.2 Rolling Resistance The rolling resistance of the tire is primarily a function of the hysteresis of the tire materials due to the deflection of the tire carcass that occurs as a portion of the tire passes through the contact patch while the wheel is rolling. Experimental data show that these deformations of the rolling tire account for approximately 87-94% of the rolling resistance losses. Friction at the patch accounts for an additional 5-10%, and 1-3% is attributed to drag due to friction of the wheel with the surrounding air [2.7]. It has been shown that for a radial truck tire, the energy losses can be further broken down to the various parts of the tire carcass as follows: tread region, 73%; sidewall, 13%; shoulder, 12%; and the beads, 2% [2.8]. At high vehicle speeds tires can develop circumferential standing waves that introduce additional carcass deflections. This topic and the dependence of the deflections on vehicle speed are presented elsewhere [2.7]. Numerous factors affect the value of the rolling resistance of a tire. These include the construction of the tire, tire material, surface roughness of the road, inflation pressure, and velocity. Discussion of these factors is presented elsewhere [2.8]. The dependence of the rolling resistance on the velocity of the tire is shown in Fig. 2.2. In general for passenger tires, this value ranges from 1.3-1.6% of the normal force at 100 km/h to 1.92.4% of the normal force at 180 km/h [2.10]. Recent work has investigated the dependence of the rolling resistance of a tire as a function of the temperature of the tire carcass [2.9]. Measurements of the level of rolling resistance of a large number of tires have been published [2.10, 2.11] in the form of rolling resistance coefficients (also see Chapter 3).

2.3 Slip, Longitudinal Force, and Lateral Force In free-rolling, straight-line motion, tread elements of a tire undergo small compressive deformation in the plane of tire-road contact as they deform from a free circular surface to the shape of the contact patch, interact with the roadway surface, and then exit the patch. These deformations cause slippage of the tire tangent to, and relative

17

Chapter 2

Figure 2.2 Rolling resistance of radial and cross-ply tires on smooth, level road surfaces under normal load and at a prescribed pressure. (Derived from Adler [2.10]. Used with permission)

to, the surface of the road. They typically are on the order of a few thousandths of an inch, and are referred to collectively as secondary slip. This slippage may be in the longitudinal or lateral direction, or both, and the specific displacement will be a function of the roadway surface as well as the construction of the tire. In contrast to secondary slip, a more appreciable slip occurs in some regions of the tire patch and is referred to as primary slip. Primary slip at the tire-road interface is more a function of steering input to the tire, or an applied wheel torque through braking or accelerating, as opposed to the tire carcass merely traversing the tire patch. Qualitative observations of the primary slip that occurs during a steering input has been reported [2.7]. An area of the contact patch consisting of predominantly secondary slip is called an adhesive area1. An area of the contact patch consisting predominantly of primary slip is called the partial skidding area. These two regions are depicted in Fig. 2.3. The diagrams in Fig. 2.3 illustrate the qualitative change in the patch shape for wheel slip angles of various magnitudes. They also show the changes in the relative size of the non-slip area (stationary portion) of the contact patch and that part of the contact patch undergoing primary slip. The secondary slip occurring in the contact area is often considered negligible in terms of generating tire forces.

Figure 2.3 Slip as a function of slip angle (from Clark [2.7]).

1. Strictly speaking, no significant adhesion (tensile contact forces) occurs over the contact patch surface. The contact forces are tangential over a compressed surface and therefore are frictional in nature. Nevertheless, some tire literature continues to use the term adhesion.

18

Tire Forces

Slip at the tire patch during straight-line braking and acceleration can be assessed conceptually by assessing the extreme conditions. The portion of the patch undergoing primary slip increases from none for a free-rolling tire with zero steering input, to 100% for a tire undergoing a locked-wheel skid. The transition from free rolling (no slip) to locked-wheel skidding (full sliding) is modeled using a continuously changing kinematic wheel slip variable. Figure 2.4 depicts another means to visualize the primary slip of a tire at the patch area due to straight-line braking, under various conditions. The figure depicts a typical passenger car tire undergoing (A) straight-line free-rolling motion; (B) wheel slip angle, α; (C) straight-line braking; and (D) straight-line acceleration.

2.3.1 Longitudinal Slip In a broad sense, the longitudinal force, Fx, causes wheel traction and is responsible for the acceleration and deceleration of a vehicle. The lateral force, Fy, is responsible for steering and lateral control of the vehicle. The longitudinal force is described as a function of a kinematic parameter called the longitudinal slip, s; the lateral force is described in terms of a kinematic parameter called the wheel slip angle, α (sometimes referred to as the tire or wheel sideslip angle). Understanding the longitudinal and lateral forces developed at the tire patch requires an understanding of the longitudinal wheel slip and lateral slip angle.

Figure 2.4 Illustrations of tire patch slip conditions. Typical passenger car tire undergoing (A) straightline free-rolling motion; (B) slip of angle, α; (C) straight-line braking; and (D) straight-line acceleration. (Arrow indicates the direction of wheel velocity.)

Acceleration and deceleration forces on a vehicle are typically developed at each tire patch through the application of torque about the wheel’s hub, or axis of rotation. As the tire reacts in an elastic manner to the application of torque, the tire carcass is expected to deform. It is instructional to look at the deformation of the tire for each of the cases of a braking force and a driving force. Consider the case of the tire with zero lateral wheel slip angle and under the application of a braking force. In this instance, the rotational motion of the tire is accompanied by a rearward elastic displacement of the tire relative to its axle. The longitudinal tire force, Fx, tends to stretch the tread elements just forward of the patch and compress the tread elements just aft of the patch, as shown in Fig. 2.4 C and Fig. 2.5, where Fx = FB. The distribution of the longitudinal shear forces for a free-rolling tire is represented by

19

Chapter 2

curve 1 in Fig. 2.5. Curve 2 represents the additional shear force created by the braking torque. Curve 3 shows the resultant shear force distribution along the length of the contact patch. Note that the resultant of the vertical force distribution for the braking tire is forward of the contact patch center, shown by the distance er in Fig 2.5. (The movement of the resultant vertical force relative to the center of rotation is the effect that is principally responsible for a tire’s rolling resistance.) Figure 2.5 Distributions of longitudinal force and vertical force over the contact patch for braking (from Clark [2.7]).

The net longitudinal force component in the x direction, Fx, is the resultant of a force distributed over the tire patch and is shown in Fig. 2.5 as a braking force, FB. It has been found experimentally to be a nonlinear function of longitudinal wheel slip, s. To understand the characteristics of this force, consider the rolling tire as shown in Fig. 2.6. Figure 2.6 (a) illustrates the velocity components of a rigid wheel rotating about its axle with an angular velocity, ω. Figure 2.6 (b) shows the force and velocity components acting over the contact patch. In Fig. 2.6 (a), R is the rolling radius of the tire, and V is the resultant velocity of the hub, or axle, of the wheel. The slip, s, at the tire patch is called longitudinal slip or simply the wheel slip. Longitudinal wheel slip, s, has been defined in various ways in the literature [2.2]. In this book, it is defined separately for a wheel under the application of a braking torque and for an acceleration torque. The advantage of using two separate definitions for the two different conditions is that the values of the longitudinal slip always range between 0 and +1; that is, 0 ≤ s ≤ 1.

20

Tire Forces

For a braking wheel, the slip, s, is defined as



s=

V px Vx

=

Vx − Rω Rω = 1− Vx Vx

Figure 2.6

(2.1)

where Vx is the longitudinal velocity component of the wheel hub, Vpx is the longitudinal velocity component of the tire at the tire patch, and ω is the angular velocity of the wheel. For a freely rolling wheel under straight-line motion, the hub velocity is Vx = Rω; then by Eq. 2.1 s = 0. That is, no slipping takes place between the wheel and the ground. If the wheel is locked from rotation by braking, ω = 0 and Vx ≠ 0; then by Eq. 2.1, s = 1. This corresponds to a locked-wheel skid. The level of braking torque (controlled by varying the brake pedal force) determines the level of wheel slip between the limits of a free-rolling wheel and a locked wheel. As a consequence of this definition of (longitudinal) wheel slip, all levels of braking can be described by 0 ≤ s ≤ 1.

Velocity components of a rotating and slipping wheel, (a). Force and velocity components at Point P in the tire patch, (b).

For a wheel with positive traction or acceleration such as Fx in Fig. 2.6, wheel slip can be defined as



s=−

V px Rω

=

Rω − Vx V = 1− x Rω Rω

(2.2)

For a freely rolling wheel under straight-line motion, Vx = Rω; then by Eq. 2.2 s = 0. If the forward velocity of the wheel is restrained to zero but the wheel is rotating, or spinning, then Eq. 2.2 gives s = 1. For all other cases, acceleration axle torque (controlled by the driver) causes acceleration wheel slip to be in the range of 0 ≤ s ≤ 1. The reason for defining acceleration slip and braking slip differently is as follows. The limiting case of the traction force applied with a constrained wheel for which Rω ≠ 0 and Vx = 0 will produce an infinite slip from Eq. 2.1. Correspondingly, the case of the limiting condition of a locked-wheel skid, ω = 0 and Vx ≠ 0, will produce an infinite slip from Eq. 2.2. By using two different definitions, longitudinal wheel slip is always bounded between 0 and 1, thereby establishing a single intuitive range for the values of wheel slip, independent of whether the applied torque is acceleration or braking. Before considering the longitudinal and lateral forces generated at the tire, a short section on the characteristics and quantification of the frictional force at the tire is presented. This

21

Chapter 2

section also discusses the notation used throughout this book to represent tire-road friction.

2.3.2 Comments, the Coefficient of Friction, and the Frictional Drag Coefficient Friction is ubiquitous. We continually try to overcome it with methods such as the use of lubricants, yet we cannot walk or drive without it. Engineers and scientists spend much time measuring it and even more time trying to devise effective mathematical models to simulate it. In some ways, friction is very simple (it brings moving objects to a stop), yet in other ways we cannot explain it (pulling a tire segment over a pavement by hand does not provide the same level of drag force as the same tire on a vehicle sliding over the same pavement). Yet, friction must be discussed and modeled. Based on his observations and measurements, Charles Augustin de Coulomb (1736-1806) devised laws governing friction of solids that still are used today. In summary, Coulomb observed that if the normal force over the contact surface between two flat rigid bodies is N, then the force, F, to produce sliding is equal to a constant times the normal (compression) force; that is, F = fN. The constant, f, is referred to as the coefficient of friction. He further observed that the force necessary to cause sliding to begin, FS , is greater than the force, FD, necessary to sustain sliding. This leads to the concepts of the static coefficient of friction, fS , and the dynamic coefficient of friction, f D, where, typically, fS > f D. In addition, Coulomb observed that the friction force, F, is independent of the contact area and that the coefficient of friction differs for sliding of different material-pair combinations. Note that Coulomb’s laws are stated for solid materials. Vehicle tires have significant flexibility. Nevertheless, Coulomb’s laws are applied to tires and used in vehicle dynamic studies and accident reconstruction, treating the coefficient, f, as an experimental quantity. Most engineering books use μ as the symbol to represent the friction coefficient. In this book, the symbol f is used. The use of f is not uncommon in vehicle dynamic applications. To avoid confusion, the symbol μ is reserved to represent a ratio of impulses, later in the book when discussing vehicle collisions, and was used in Chapter 1 to represent the mean of a statistical distribution. Experiments have shown that when the wheels of a forward-moving vehicle are locked by brake application, causing it to skid (longitudinal wheel slip s = 1) over a roadway surface, the retarding force is not a constant. As shown by experimental data in Fig. 2.7, the acceleration (and force) peaks, drops somewhat, continues to vary, and eventually returns to zero when the vehicle comes to a stop. This is not the type of behavior associated with Coulomb’s idealized model. At any instant of time, the ratio of the braking force to the normal force (the vehicle’s weight) can be thought of as an instantaneous value of the coefficient of friction. This means that the friction coefficient developed by sliding tires is not constant with respect to time over a sliding stop. Though this may be true, the situation is much more complicated. As discussed earlier in this chapter, the longitudinal force depends upon wheel slip (which can vary over the contact patch), and the force is not constant, at least not until it reaches a value of s = 1. But other conditions differ from Coulomb’s: Unlike a block of metal or wood, a tire is not a rigid solid. In addition,

22

Tire Forces

friction coefficients are not truly constants and can vary with speed. Therefore, when applied to vehicles, Coulomb’s law is an approximation. On the other hand, such an approximation can be reasonably accurate, particularly when an appropriate value of f is used. In Fig. 2.7, note that an average value of acceleration is shown. The average frictional coefficient corresponding to this average acceleration is f = aavg, where the aavg is expressed in units of g; that is, as a fraction of the acceleration due to gravity. In the figure, this value is f = 0.72. The use of the average value will lead to a reasonably accurate value of stopping distance when used properly in calculations. Figure 2.7 Typical deceleration vs. time of a braking vehicle measured at the center of mass of the vehicle.

To distinguish between a true (instantaneous) Coulomb coefficient of friction and a constant average value, f, appropriate for use in vehicle analysis or accident reconstruction (such as the average value in Fig. 2.7), this book uses the term frictional drag coefficient. This implies a constant value suited to an application. Other, equivalent terms are in common use, such as friction factor and drag factor. In summary, the term coefficient of friction implies an ideal value corresponding to frictional theory, whereas frictional drag coefficient refers to averaged values used in reconstruction applications.

2.3.3 Longitudinal Tire Force Longitudinal tire forces, both acceleration and braking, are found experimentally to be nonlinear functions of the wheel slip, s; that is, Fx = Fx(s). Figure 2.8 shows examples of such measurements [2.12] of a braked wheel and tire for different normal force levels and where the slip, s, is controlled to vary between 0 and 0.85. The slip values are plotted as a percentage; that is, 100 × s. Consider a braked wheel that is locked; that is, s = 1. Such a wheel condition produces a longitudinal force, Fx, such that Fx = fxFz, where fx is the frictional drag coefficient between the tire and the roadway in the longitudinal direction. It is sometimes convenient to display the longitudinal force in a normalized form represented here by Qx(s), where

23

Chapter 2



Qx ( s ) =

Fx ( s ) µ x Fz

(2.3)

This normalized form of the force, Qx(s), has the benefit of producing a force that has all the features and characteristics of an actual longitudinal tire force, but where Qx(s = 1) = 1. This scales the force curve by the constant fxFz and a force distribution can be obtained simply by multiplying an equation representing the normalized curve by the appropriate value of fxFz. Figure 2.8 Measured values of a tire’s longitudinal force for different normal forces (from Salaani [2.12]).

Figure 2.9 Typical modeled normalized longitudinal force, Qx(s).

Figure 2.9 shows a typical modeled distribution of a normalized longitudinal force, Qx(s). Initially, as s increases from 0, the braking force grows rapidly and is approximately linear with slope, Cs. The slope of this linear portion of the curve is often assigned the symbol, Cs, and is referred to as the slip stiffness. Fx(s) continues to increase in a nonlinear fashion

24

Tire Forces

as s increases and reaches a maximum typically in the range 0.10 < s < 0.20. The braking force decreases as the wheel slip, s, increases beyond this range, reaching a value of Q(s) = 1 when s = 1. The shape of this curve, particularly its associated maximum, is one of the reasons for the development of antilock braking systems. In straight-line braking, the antilock brake system maintains the longitudinal force, Fx, near the maximum by controlling the slip. This process maximizes the tire force and decreases the braking distance and time, as compared to a locked-wheel skid. (The antilock braking system also maintains steering control, as seen later.) As before, the normalization process in Eq. 2.3 creates the condition that the actual longitudinal force is equal to fxFzQx(s).

2.3.4 Lateral Tire Force The lateral tire force component, Fy, lies in the plane of the tire patch and is perpendicular to the heading of the wheel in the direction opposing the lateral velocity, Vy, as shown in Fig. 2.6 (b). This lateral force, developed by a tire with sideslip (α ≠ 0), provides directional control and lateral stability of a vehicle. When a lateral velocity exists at any wheel, whether at a steered wheel or a fixed wheel, a lateral tire force develops. During a steering maneuver, the heading angle of the wheel and the resultant velocity, V, differ by an angle, α, which is called the wheel sideslip angle (or simply the slip angle), as shown in Fig. 2.6 (a). The wheel slip angle is defined as



 Vy    Vx 

α = tan −1 

(2.4)

where Vy and Vx are shown in Fig. 2.6 (a). For a freely rolling wheel (s = 0 and Fx = 0), the lateral force, Fy = Fy(α), is a nonlinear function of slip angle α. Examples of experimentally measured values of such lateral forces on a wheel and tire are shown in Fig. 2.10 for different normal forces [2.12]. Forces were measured between slip angles -30° ≤ α ≤ +30°, limits that are usually determined by the testing equipment. The lateral force is also a function of other parameters [2.2], such as velocity, camber angle, and temperature, but these effects are considered second order and not studied here. For zero slip angle (α = 0), the lateral force is zero and corresponds to a wheel velocity aligned with its heading. At the other extreme, when the slip angle α = π/2, the resultant force is normal to the heading of the wheel and is equal to the frictional drag coefficient in the lateral direction times the normal force at the wheel, Fy = fyFz. A normalized version of the lateral force can be defined as follows:

Qy (α ) =

Fy (α ) f y Fz



(2.5)

A representative plot of a normalized lateral force, Qy(α), is shown in Fig. 2.11. This shows that Qy(α) is approximately linear for small values of the wheel slip angle, α. As

25

Chapter 2

α increases, Qy(α) becomes nonlinear, the slope decreases, and the normalized force approaches a value of 1 as α approaches π/2. The initial linear slope of the actual force, Fy(α), is called the slip angle stiffness, Cα, which is also referred to as the cornering stiffness. Fy(α) also approaches the road friction limit, fyFz, at α = π/2. Note that the slip angle stiffness, Cα, depends on the normal force, as seen from the data in Fig. 2.10. The lateral tire force is obtained from the normalized force, such that Fy(α) = fy FzQ(α). Figure 2.10 Measured values of a tire’s lateral force for different normal forces (from Salaani [2.12]).

Figure 2.11 Typical modeled normalized lateral tire force, Qy(α).

26

Tire Forces

According to Coulomb’s theory of rigid body friction, the coefficient of sliding friction is independent of direction and load; i.e., fy = fx = f. However, experimental data have shown that for tires, differences between these values do occur [2.13]. Much of the data on lateral frictional drag are obtained from the field of racing through measurements of lateral vehicle acceleration [2.2]. Hence the lateral and longitudinal coefficients are considered distinct. This particular characteristic of friction at the tire patch is explored next.

2.4 Friction Circle and Friction Ellipse In the previous two sections, the two force components developed by a tire in the plane of the road, the longitudinal force, Fx, and lateral force, Fy, were investigated independently. The situation in which both forces act simultaneously at the tire patch, such as during braking while steering (or steering while braking), is quite common. In this section, it is shown qualitatively how both lateral and longitudinal forces can be plotted on the same diagram and used to define a region of control (lack of skid). The longitudinal force, Fx, is plotted along the vertical axis, and the lateral force, Fy, is plotted along the horizontal axis; the wheel slip, s, and the wheel slip angle, α, become implicit parameters. This orientation of axes is atypical from much of the previous literature, but it has been used elsewhere [2.2]. It is used here because it has the added benefit of having the forces appear on the plot in the direction of their application (as viewed from the perspective of a driver) and thus is associated with looking down at the tire along the z-axis. In Fig. 2.12 (a) and 2.12 (b), the braking force acts towards the bottom of the page and would

Figure 2.12 Depictions of the Friction Circle. Part (a) represents a condition of pure rolling with a lateral force, Fy, only. Part (b) shows a combination of braking and steering with both a lateral force, Fy, and a longitudinal (braking) force, Fx. (Two braking forces are shown, one for light braking, Fx(α,s), and one for a lockedwheel skid, f Fz.) Part (c) shows the condition of steering and light braking superimposed on a vehicle wheel.

27

Chapter 2

have the net effect of slowing the vehicle down; the lateral force acting to the right causes the vehicle to turn right.

2.4.1 Idealized Friction Circle and Idealized Friction Ellipse Suppose, temporarily, that the 1. frictional drag coefficients in the lateral and the longitudinal directions are the same; that is, fx = fy = f, and 2. the lateral force and the braking (or acceleration) force are kept well below their sliding values, fFz. An issue that needs to be addressed is the limit of the resultant tire force, F, where F = Fx (α , s ) 2 + Fy (α , s ) 2

, when steering and traction coexist but when s < 1 and α < π/2 simultaneously. This is the condition when the tire reaches its friction limit without locked-wheel braking. Consider Fig. 2.12. The coordinate axes are the lateral force, Fy(α,s), and longitudinal force, Fx(α,s). The origin represents the condition of free rolling with no traction force or steering force, s = 0 and α = 0. The only force acting on the tire at the tire patch in the plane of the road under these conditions is the rolling resistance, which is small and is neglected. The positive vertical axis represents the traction force during positive acceleration of the vehicle. The largest acceleration force (with no slip angle) that can be developed is fxFz, which is for a spinning tire and wheel with zero translational speed. The negative vertical axis represents the braking force. The largest force magnitude (with no slip angle) that can be developed in this direction is fxFz, for a locked wheel and skidding tire. The positive horizontal axis represents the lateral force developed at the tire patch acting perpendicular to the wheel heading for positive wheel slip angle, α. The maximum force that can be developed in this direction is fyFz for a tire sliding laterally (α = 90°) to its heading. Since it was assumed that the coefficient of friction is independent of direction, fx = fy, the three points (spinning, skidding longitudinally, and skidding sideways) imply the use of a circle as an indicator of a limit force for the resultant force F(α,s) for all combinations of slip angle, α, and wheel slip, s. In the general tire literature, this type of plot has acquired the moniker of the “friction circle” but is referred to here as the idealized friction circle. The term idealized is used because, as is shown later, actual tire force limits can be significantly larger than predicted by fxFz and fyFz. This idealized representation of tire force components under combined traction and steering indicates that for any combination of s and α, as long as the resultant tire force is less than the total available frictional force, fFz, the tire provides directional control of the vehicle. If the resultant force is equal to the total available frictional force, fFz, and hence lies on the friction circle, then the tire skids and directional control of the tire is lost. The condition of the combined force at the tire patch equaling fFz is sometimes referred to as “saturation.” In accordance with intuition and physics, the magnitude of

28

Tire Forces

the resultant force can never exceed fFz for any combination of α and s for the idealized friction circle. Further consideration of the transition of a wheel from tracking (maintaining directional control) to skidding will help develop insight into the characteristics of the combined force. Consider, hypothetically, a free-rolling (s = 0) wheel that is cornering with velocity, V, and with a slip angle, α, as shown in Fig. 2.12 (a). In this case the lateral force developed is acting to the right, along the positive Fy axis. Suppose a light braking force, Fx, is added with wheel slip, s, keeping the direction of V constant and α constant. This combines with the lateral force, and a resultant tire force F(α,s) is developed, as shown in Fig. 2.12 (b). Suppose now that the braking force is steadily increased (without changing V and α). As braking is increased (that is, as s increases) the resultant force reaches the friction circle and a magnitude of fFz, as shown in Fig. 2.12 (b). At this point the tire is sliding, and the direction of the resultant force opposes the velocity vector of the tire and wheel; that is, β = α. The motion of the tire is no longer kinematically constrained to rolling and is now skidding. At this point the tire has reached saturation. Up to this point, it has been assumed that the total available frictional force for a tire in the longitudinal direction equals that in the lateral direction; i.e., fx = fy = f. Unequal lateral and longitudinal frictional drag coefficients, fx ≠ fy, have the net effect of transforming the above concept of the idealized friction circle, as described in the previous paragraph, into an idealized friction ellipse. The idealized friction ellipse is now examined in more detail. An expression for the equation of the friction ellipse can be developed based on the tire forces at the patch, Fx(α,s), Fy(α,s), Fz, and the coefficients of friction, fx and fy. Similar concepts have been developed elsewhere [2.8]. The development here uses the quantities presented above in order to build on the conceptual approach to the friction ellipse. Using the fundamental equation of an ellipse with one axis equal to 2fxFz, and the other axis equal to 2fyFz, the abscissa variable equal to Fy(α,s) and the ordinate variable equal to Fx(α,s) (which is in keeping with the convention developed above) yields

Fy2 (α , s )

f y2 Fz2

+

Fx2 (α , s ) ≤1 f x2 Fz2



(2.6)

The inequality in Eq. 2.6 indicates that as long as the resultant force remains inside the friction ellipse, sliding (skidding) does not occur and directional control is maintained. Equality in Eq. 2.6 corresponds to sliding; for instance, when the wheel slip, s = 1 for any a, or when a = p/2 for any s. Note that for fx = fy, Eq. 2.6 becomes the equation of a circle. Under conditions of sliding, the direction of the resultant force opposes the velocity VP; b = a, Fx = -Fcos a, Fy = -Fsin a (see Fig. 2.12 (b)) and

29

Chapter 2



F 2 cos 2 α F 2 sin 2 α + =1 f x2 Fz2 f y2 Fz2



(2.7)

During sliding, F = fFz; then Eq. 2.7 can be written as



cos 2 α sin 2 α 1 + = 2 2 2 fx fy f



(2.8)

Further simplification yields

f2=

f x2 f y2 f y2 cos 2 α + f x2 sin 2 α



(2.9)



(2.10)

and taking the square root of both sides gives

f =

fx f y f x2 sin 2 α + f y2 cos 2 α

Equation 2.10 provides an equivalent frictional drag coefficient to be determined for any combination of fx, fy, and α. The sliding friction force, fFz, can then be determined for a given value of Fz; Eq. 2.9 reduces to an identity for fx = fy.

2.4.2 Actual Friction Circle and Actual Friction Ellipse The previous discussion is based on an assumption that neither force component Fx(s) nor Fy(α) can exceed its sliding value; that is, Fx(s) ≤ fFz for all s and Fy(α) ≤ fFz for all α. However, experimental measurements of tire forces and experience with racing vehicles show that this is not the case. Figures 2.8, 2.9, 2.10 and 2.11 all show that Fx(s) > fFz and Fy(α) > fFz for ranges of values of s and α (at least for some levels of normal forces). The effect of this on the concept of the friction circle is that, in practice, actual tire forces can reach levels outside of the idealized friction circle, or ellipse, before full sliding or saturation occurs. The true limit curve (when a tire reaches saturation) is the limit of the family of the resultants of the combined steering and braking forces for all combinations of s and α. An example based on experimentally measured tire forces is shown in Fig. 2.13. In Fig. 2.13, the indicated friction ellipse is an approximate fit to the envelope of three measured sets of tire data points and is shown as an exact ellipse. Actually, experimental results would produce a curve which is close to, but not exactly, a true ellipse. The corresponding idealized friction ellipse is shown and is significantly smaller than the true friction ellipse. Figures 2.8 and 2.10 show that at low to mid ranges of slip values tire forces typically peak above the forces generated at forward and lateral sliding conditions alone (s = 1

30

Tire Forces

and α = π/2). These higher tire force levels, particularly braking forces, can be exploited, forming the basis of antilock braking systems (ABS) and as used by electronic stability control (ESC) systems.

Figure 2.13 Idealized friction ellipse, actual friction ellipse, and measured truck tire test data from [2.14].

2.5 Modeling Combined Steering and Braking Tire Forces The discussion up to this point has focused predominantly on the qualitative description of the individual lateral and longitudinal forces developed at the tire road contact patch. In addition to understanding the nature of these forces qualitatively, vehicle engineers and accident reconstructionists must model these forces mathematically. Mathematical models are required for reasons ranging from a simple orderof-magnitude check of a tire force, to the modeling of tire forces for a vehicle dynamics simulation program. In any case, it is important for an accident reconstructionist to understand the kinematics and dynamics of tires, particularly for use in modeling preimpact and postimpact vehicle motion. Various models are used to predict the combined lateral and longitudinal force [2.15, 2.16, 2.17]. A versatile means to model the individual lateral and longitudinal forces is the Bakker-Nyborg-Pacejka (BNP) model [2.18]; it frequently also is referred to as the “Magic Formula.” This model has emerged as the most commonly used approach to model the individual longitudinal and lateral force components generated by a tire. That topic, and the Modified Nicolas-Comstock combined tire force model, are presented in the following sections.

2.5.1 The Bakker-Nyborg-Pacejka Model for Lateral and Longitudinal Tire Forces In the past, and depending on the application, tire forces have been represented by various mathematical equations, including piece-wise linear approximations and exponential functions. For details see Nguyen and Case [2.15]. Work by Bakker, Nyborg and Pacejka [2.18] produced a convenient tire force formula based on a combination of trigonometric and algebraic functions. This formula is referred to in this paper as the BNP model or the BNP equations. The same BNP equations apply equally well to both the longitudinal and lateral forces (as well as contact patch moments). Due to its convenient form and its

31

Chapter 2

ability to fit experimental data reasonably well, the BNP model has supplanted previous methods and has been adopted by various authors. These include Schuring, et al. [2.19] and d’Entremont [2.20]. Because of the versatility of the BNP model, it is used here. A short description of the model follows, with the equations presented in a normalized form. The BNP tire force equation expresses a tire force, P, as a function of an independent variable, u, where 0 ≤ u ≤ 1. The variable, u, can represent longitudinal wheel slip, s, or the slip angle, α, in the form of 2α/π. The variable, Q, the normalized tire force component, is used here, where Q(u) = P(u)∕P(1), and P(1) = P(u)|u=1. The BNP equations in two-part, normalized form, can be written as



−1 P (u ) D sin C tan ( Bφ )  = Q(u ) = P(1) P(1)

(2.11)

where



 E  −1  tan ( BKu ) B

φ = (1 − E ) K u + 

(2.12)

The parameters B, C, D, and E are constants chosen to model specific wheel and tire systems and/or to give forces that correspond to specific experimental data. (The original BNP equations contain additional constants, Sv and Sh, which permit a vertical and horizontal offset of the origin to allow a more accurate match to experimental data; since these are small they are set to zero here.) The constant, K, is given a value of 100 so that 0 ≤ Ku ≤ 100 corresponds to a percentage wheel slip, a common interpretation of the slip used by some authors. For the lateral force, the constant, K, is given a value of 90 so that 0 ≤ Ku ≤ 90 corresponds to degrees of wheel slip. Furthermore, since the forces are normalized to their value at u = 1, the constant, D, is given the value of 1. When modeling a longitudinal force component, u represents longitudinal wheel slip, s. When modeling a lateral force component, the wheel slip angle is such that 0 ≤ α ≤ π∕2 and so u = 2α∕π. The initial slope of Q(u) is BCK ∕P(1) which is the first derivative of Q(u) evaluated at u = 0 with D = 1 (see appendix in [2.17]). Equations 2.11 and 2.12 can be used with different values for the constants B, C, and E to model longitudinal and lateral tire force components. The actual force magnitudes are found simply by multiplying each Q(u) by its appropriate limiting frictional force. That is, Fx(s) = Q(s)fxFz and Fy(α) = Q(α)fyFz. The initial slopes of Fx(s) and Fy(α) are the stiffness coefficients, so

and

32

Cs =

BCK f x Fz P(1)

(2.13)

Tire Forces



Cα =

BCK f y Fz P(1)

(2.14)

respectively. When modeling wheel and tire systems with specific, known stiffness coefficients, Eqs. 2.13 and 2.14 can be used to solve for the corresponding values of B. Figures 2.9 and 2.11 were created using the BNP formula with an appropriate set of parameters determined from experimental data and then normalized as described above. With the appropriate coefficients, B, C, D, E, and K, the BNP equations provide expressions for the longitudinal force component, Fx(s), for no wheel slip angle (α = 0) and for the lateral force component, Fy(α), when there is no wheel slip (s = 0), sometimes referred to as free-rolling cornering. During combined steering and braking, however, each of these components will simultaneously become a function of the longitudinal wheel slip, s, and the wheel slip angle, α. Hence, for generality, the force components Fx(s) and Fy(α) must be described as Fx(α,s) and Fy(α,s), respectively. The magnitude of the resultant tire force, F, tangent to the roadway surface for combined braking and steering is



F (α , s ) = Fx2 (α , s ) + Fy2 (α , s )



(2.15)

where F(α,s) lies in the plane of the road. All models of the combined tire force, regardless of the formulation, must provide realistic results. This consideration, in combination with other performance criteria related to the limiting cases of the combined force, is examined in detail elsewhere [2.17, 2.25]. Although many exist, only one tire model for combined braking and steering is presented here, because of its generality. This is a modified form of the Nicolas-Comstock model [2.21]. A comparison of different tire models used in accident reconstruction simulation software is contained elsewhere in technical papers [2.22] and [2.23]. Two other effective, but more complex, models worth mentioning include the STI model [2.6] and the TMEasy model [2.24]. In addition, a totally different approach referred to as brush models has been published [2.25].

2.5.2 Modified Nicholas-Comstock Combined Tire Force Model While the idealized friction ellipse, Eq. 2.6, is useful for an intuitive and qualitative understanding of the physical conditions at the tire-road interface under combined loading, it provides only an idealized bound on the resultant tire force. More importantly, it does not model tire forces within the idealized friction circle or in the regions where the forces exceed their sliding values. For quantitative development of the combined tire force, F(α,s), a more general theory is needed, not simply an inequality. Various models have been proposed over the years. See for example Nicolas and Comstock [2.21], Bakker, et al. [2.18], Wong [2.8] and Schuring, et al. [2.16].

33

Chapter 2

The model proposed by Nicolas and Comstock is chosen not only for its effectiveness and compact form but also because of its generality. It can be used to provide combined braking and steering forces for any pair of longitudinal and lateral force functions, Fx(s) and Fy(α), even experimental values. It is based on two conditions: the first is that the resultant force, F(α,s) for s = 1, is collinear with and opposite to the resultant velocity, Vp; the second condition is that as α is varied from 0 to π∕2, the resultant force, F, can be described by an ellipse of semi-axes, Fx and Fy. Semi-axis, Fx is the longitudinal friction force defined by an Fx slip curve for α = 0, and semi-axis, Fy, is the lateral friction force predicted by an appropriate equation for s = 0. The following expressions, proposed by Nicolas and Comstock, meet these criteria.

Fx (α , s ) =

Fy (α , s ) =

Fx ( s ) Fy (α ) s s 2 Fy2 (α ) + Fx2 ( s ) tan 2 α



(2.16)



(2.17)

Fx ( s ) Fy (α ) tan α s 2 Fy2 (α ) + Fx2 ( s ) tan 2 α

Note that from these two equations the ratio of Fy(α,s) to Fx(α,s) is always (tanα)/s. Each of Eqs. 2.16 and 2.17 is easily evaluated for a given s and α for any set of appropriate force functions, Fx(s) and Fy(α). However, for either α = 0 or s = 0, the expressions as defined above are undefined. In particular, when α = 0, no lateral force is on the tire; i.e., Fy|α = 0 = 0, by definition. With both α and Fy(α) equal to zero, both the numerator and the denominator of the two equations go to zero. This is inconvenient, as it is expected that Fx(0, s) should reduce to Fx(s). Similarly, with s and Fx(s) equal to zero, both the numerator and the denominator of the two equations go to zero. For combined steering and braking, it is expected that as α → 0, Fx(s,α) → Fx(s); and as s → 0, Fy(s,α) → Fy(α). The Nicolas-Comstock model does not provide this behavior. All reasonably valid tire models should exhibit such behavior. For small values of α and s, the tire is operating in the linear region of the force curves (see Figs. 2.9 and 2.11). Therefore, linear approximations can be used for the longitudinal and lateral forces. Hence, Fx(s) ≈ Css and Fy(α) ≈ Cαα and tanα ≈ α. Introducing these approximations into Eqs. 2.16 and 2.17 yields

Fx (α , s ) = Fx ( s ) Fy (α , s ) = Fy (α )

34

Cα

where 0 < α 0, then this implies that 0 ≤ e ≤ 1 as well. Note that e = 0 does not mean that all of the system’s initial kinetic energy is lost. What about e < 0? According to Eq. 6.22, negative values of -1≤ e ≤ 0 do not violate energy conservation. Examination of Eq. 6.11 indicates that negative values of e imply that the bodies pass through each other. For ordinary collisions, this is impossible. There are some exceptions, but these are rare and are discussed elsewhere [6.1].

6.3.2 Collisions Where Sliding Ends; the Critical Impulse Ratio, μ0 Consider the special case where rough contact surface conditions develop a tangential impulse, Pt, large enough to cause sliding (relative tangential velocity between m1 and m2) to end at, or prior to, separation of the two bodies. This corresponds to the final condition where V2t - V1t = 0. Using this condition with Eqs. 6.15 and 6.16 gives an interesting and very useful result; that is, V2t - V1t = 0 when



µ = µ0 =

r 1+ e

(6.23)

where r is given by Eq. 6.21. This impulse ratio, μ0, is called the critical impulse ratio. Note that μ0 takes on the sign of r which, in turn, is determined by the initial conditions. This means that when μ = μ0, the bodies have the same final velocity components in the t, or tangential, direction. There is another way of looking at this. By definition, Pt =

136

Analysis of Collisions, Impulse-Momentum Theory

μPn, so sliding does not exist at separation when Pt = μ0Pn. It is reasonable to conclude that when |Pt| < |μ0 Pt0|, sliding exists at separation, which implies that sliding exists at separation when |μ| < |μ0|. The absolute value signs are necessary because the sign of μ determines the direction of the tangential impulse, Pt. If now it is assumed that the tangential impulse is generated by a frictional type force with an average frictional drag coefficient, f, and f < |μ0|, then sliding continues through separation; if the average frictional drag coefficient is such that f = |μ0|, then sliding will end at separation; and if f > |μ0|, sliding will still end before separation. Further investigation [6.1] shows that, in general, as μ is increased from 0 to μ0, a maximum kinetic energy loss occurs when μ reaches μ0. To summarize, in a collision of two point masses with a frictional drag coefficient, f, over the contact surface, where f < |μ0|, then μ = f sign(μ). If f ≥ |μ0|, then μ = μ0 and sliding ends at or before separation.

6.3.3 Sideswipe Collisions and Common-Velocity Conditions In the context of vehicle collisions, a value of the impulse ratio μ < μ0 implies a sideswipe collision, because the vehicles continue to slide over the intervehicular contact surface throughout the contact duration. In fact, this is used as a definition of a sideswipe collision in this book; that is, a sideswipe collision is a collision where μ < μ0. When μ = μ0, relative tangential motion (sliding) ends at or before separation. Furthermore, when the coefficient of restitution, e = 0, the bodies (automobiles) do not rebound or separate in the normal direction either. Consequently, at the end of the collision, the bodies have identical velocity components, a condition frequently referred to as a common velocity. So, in this book, the common-velocity conditions are when e = 0 and μ = μ0. For point-mass theory, this means the bodies remain attached after impact. At this point it can be said that the common-velocity conditions are satisfied for most direct, high speed vehicle collisions. This will be discussed later. Two vehicles collide at right angles, as shown in Fig. 6.3. Vehicle 2 initially is at rest; Example 6.2 Vehicle 1 moves in the direction of its heading at a speed of 13.41 m/s (44 ft/s) into Vehicle 2. Following the impact, both move with the same speed in the original direction of Vehicle 1. The solid curve in the graph in Fig. 6.4 is typical of an experimental acceleration record of the mass center of Vehicle 1 from this type of collision. Use this curve and the dashed triangular pulse to estimate the peak (maximum) deceleration of Vehicle 1 and the peak intervehicular force during the impact. Vehicle weights are W1 = 21.81 kN (4900 lb); W2 = 10.04 kN (2255 lb). Assume that the impulse of the tire-roadway frictional forces are negligible. Solution  Because both vehicles have the same final speed, e = 0. Equations 6.13 and 6.14 give V =V = v + 2n 1n 1n

m2 (v2 n − v1n ) = −13.41 + (0.315)13.41 = −9.18 m / s (−30.1 ft / s ) m1 + m2

137

Chapter 6

Figure 6.3 Collision configuration of two vehicles, with Vehicle 2 initially at rest.

Figure 6.4 Normal acceleration component, g, of Vehicle 1, Example 6.2.

The velocity change of Vehicle 1, Δv1 = -4.23 m/s (-13.9 ft/s). The area of the dashed, idealized, triangular peak force is equal to the impulse, Pn. Equation 6.5 gives

Pn = m1 (V1n − v1n ) = 2223(−9.18 + 13.41) = 9403 N -s (2114 lb-s )

138

Analysis of Collisions, Impulse-Momentum Theory

From Fig. 6.4, the impulse duration is 0.15 s. The average force is Pn ∕(τ2 - τ1) = 62689 N (14094 lb); based on a triangular shape, the peak force is 125377 N (28186 lb). The corresponding peak deceleration for Vehicle 1 is 5.75 g. The actual peak of the solid curve is about 15% higher than the peak of the triangle, or 6.61 g. A rule of thumb for vehicle impacts is that the maximum acceleration and force can be considered to be about two to three times the average values. Shapes other than triangles can be used to approximate vehicle crash pulses; for more information see Varat and Husher [6.2]. For a pair of bodies with known properties and with known impact coefficient values (e and μ), the above equations (Eqs. 6.13 through 6.16) permit calculation of the final velocities from known initial velocities. For accident reconstruction purposes, it is often convenient to solve the inverse problem; that is, to determine the initial velocities given final velocities. Despite the fact that Eqs. 13 through 16 are linear, and an inverse mathematical solution is possible, this approach is not practical. The reason is because most vehicle collision applications use the common-velocity conditions. The inverse solution contains terms involving 1∕e which becomes unbounded when e = 0. Consequently, in typical applications, the equations are solved in an iterative fashion to find a solution.

6.4 Controlled Collisions Barrier collisions are used to collect a wide variety of data, including monitoring performance of safety components such as sensors and restraints, measurements of anthropometric accelerations, leakage of flammable liquids, structural crashworthiness of vehicles, and vehicle component designs. Barrier collisions are used to test for conformance to NHTSA regulations known as Federal Motor Vehicle Safety Standards (FMVSS). Certain configurations of crash tests are more common than others. Full frontal impacts into rigid fixed barriers, rear impacts by moving rigid barriers, side impacts with moving rigid and deformable barriers, and frontal offset impacts into rigid and deformable barriers are used the most. Vehicle-to-vehicle collisions also are used for testing, although this type of test is conducted less frequently and finds application in reproducing collisions between specific vehicles in technical work for litigation. The New Car Assessment Program (NCAP) conducted by the NHTSA uses full frontal fixed rigid barrier tests at 35 mph (56.33 km/h) and is said to represent the effect of a vehicle directly colliding at 70 mph (112.65 km/h) with another identical, but stationary, vehicle. It also can be said that the NCAP tests are equivalent to a direct collision with an identical vehicle where both vehicles move toward each other at 35 mph. A point here is that barrier crash tests in some way are viewed as being equivalent to accidental crashes. By and large, such concepts of equivalency are useful in reconstructions in a subjective sense and permit comparisons of actual crashes to hypothetical and/or controlled ones. Before questions of equivalency can be established, the condition(s) of equivalency must be determined. One equivalency criterion may be to obtain the same ΔV from a barrier test as from a vehicle-to-vehicle crash. Another may be the same structural energy

139

Chapter 6

absorption, or loss, from a barrier test as from a vehicle-to-vehicle crash. In most cases, the equations for the collinear point-mass collisions can be used to establish equivalency. Example 6.3 An experimental frontal collision of a vehicle with mass, m1, into a fixed rigid barrier is

intended to reproduce approximately a direct frontal collision into another vehicle with mass, m2 . The initial vehicle-to-vehicle speeds are -v and v, respectively. Determine the equivalent barrier speed vb of m1: A. To have the same ΔV and momentum change for m1 as in the vehicle-to-vehicle collision, and B. To dissipate the same kinetic energy of m1 as the vehicle-to-vehicle collision. Assume perfectly plastic collisions (common-velocity condition) for all collisions. Solution A  For a direct collinear collision, no transverse force or impulse can develop, so μ = 0; Eq. 6.18 gives ΔV1 as

DV1 = m(1 + e)(v2 n − v1n ) / m1 = 2m2 v1 / (m1 + m2 ) From the same equation, ΔV1 for a fixed barrier collision, for m2 → ∞ and v2n = 0, simply is vb. Equating these gives

vb = v1

2m2 m1 + m2

where vb is the equivalent barrier speed. Solution B  In the case of energy dissipation, the kinetic energy loss, TL1, of m1 must first be determined. For e = 0, this is

TL1 =

 1 1 m m1v12n − m1V12n = 2m 1 −  v 2 2 2  m1 

The energy loss to m1 from a barrier collision for e = 0 is ½ m1 vb2 . Equating these gives

vb = 2v

m m1 + m2

The two equivalent barrier speeds in Example 6.3 are different; however, both are useful and practical quantities. If the main interest is injury potential, equivalency based on ΔV makes sense. On the other hand, if structural crashworthiness is important, then ΔV based on energy loss can be more pertinent. The example clearly shows that different

140

Analysis of Collisions, Impulse-Momentum Theory

equivalent barrier speeds can exist for a given collision, and they should be fully defined and explained. An interesting question can be raised in the interpretation of the results of Solution B of Example 6.3. Is the crush energy, Ec, for Vehicle 1 from the barrier test the same as for the collision between Vehicle 1 and Vehicle 2? Without additional information, this question cannot be answered. In the barrier collision, all of the energy loss goes into crushing m1. In the vehicle-to-vehicle crash, the amount of crush depends on the relative stiffness (resistance to deformation) of the two vehicles. The stiffer vehicle will deform less and absorb less energy. In an extreme, where m2 is a rigid movable barrier, m1 will absorb all of the energy loss of the collision, not just its own energy loss, as posed in the problem. This is covered more thoroughly in Chapter 7. Energy equivalent barrier speed (EEBS), equivalent barrier speed (EBS), and barrier equivalent velocity (BEV) are terms commonly used to analogize the severity of an accident reconstruction. These terms typically refer to the forward speed and corresponding kinetic energy with which a vehicle must contact a flat, fixed, rigid barrier at 90° for equivalence to conditions of another collision. For example, the energy may be equal to a specified amount determined from residual crush. It is usually assumed that in the equivalent barrier collision, all energy is absorbed and there is no rebound. The nature of the equivalency should always be spelled out.

6.4.1 Coefficients of Restitution When full frontal rigid barrier crash tests are conducted, it is rare that there is absolutely no rebound of the vehicle. This means that the coefficient of restitution is not actually zero. For most high-speed collisions, rebound is small and often can be ignored, but there always are cases where it should be taken into account. Table 6.1 shows the results of tests carried out for NHTSA from barrier crashes of 26, 1989 and 1990 vehicles. The average and standard deviation are ebavg = 0.112 and sb = 0.028, respectively. Individual collision speeds for these test values were not listed, but typically were 30 or 35 mph (48.3 or 56.3 km/h). Use of these data for reconstructions of collisions requires care. First, application should be restricted to full frontal collisions to maintain similar damage patterns. Second, values of the coefficients of restitution from frontal barrier tests are associated with an individual vehicle, to the specific crush pattern, and, to a small extent, to the physical properties of the barrier. On the other hand, the coefficient of restitution defined and used in the theory of point-mass impact mechanics covered above (and the planar mechanics to be covered later) corresponds to a single collision of two different vehicles. In fact, the coefficient of restitution is an impact parameter, a collision parameter, that characterizes the energy loss of the collision, not the individual vehicles. As will be shown shortly, it is possible to relate individual, vehicle/barrier values, e1 and e2, and determine a single, combined value for a collision of those specific vehicles. In fact, it can be done in more than one way. Other reconstruction techniques, such as SMAC and CRASH [6.4, 6.5], have been extended to use individual-vehicle coefficients of restitution; see McHenry and McHenry [6.6].

141

Chapter 6

Table 6.1

Coefficients of restitution, from rigid barrier tests [6.3]. Year

Make

Model

e

89

Toyota

Cressida

0.115

89

Ford

Bronco

0.036

89

Hyundai

Sonata

0.116

89

Audi

80

0.124

89

Volkswagen

Fox

0.070

89

Peugeot

505

0.118

89

Geo

Metro

0.100

89

Geo

Metro

0.104

89

Nissan/Datsun

Maxima

0.074

89

Nissan/Datsun

Pickup

0.096

90

Lexus

ES250

0.131

90

Hyundai

Excel GLS

0.120

90

Ford

Taurus

0.143

90

Geo

Prizm

0.141

90

Nissan/Datsun

Axxess

0.136

90

Toyota

Celica

0.138

90

Chevrolet

Blazer MPV

0.109

90

Chevrolet

S10 Pickup

0.102

90

Ford

Ranger

0.054

90

Nissan/Datsun

Infinity M30

0.131

90

Lincoln

Town Car

0.166

90

Ford

Mustang

0.106

90

BMW

325 I

0.110

90

Honda

Prelude

0.128

90

Mercedes

190

0.121

90

Buick

Lesabre

0.111

*Prasad [6.3].

6.4.1.1 Stiffness Equivalent Collision Coefficient of Restitution

The two coefficients, e1 and e2, from individual barrier tests of Vehicle 1 and Vehicle 2, respectively, can be related for direct central impacts through their energy losses and their crush stiffnesses. The approach presented here has been used by others; for example, see Prasad [6.3]. Collisions can be viewed as having two phases or stages: approach and rebound. The point of demarcation is the point of maximum compression. The kinetic energy given back (through “restitution”) is e2 times what is stored, and the amount lost is (1 - e2) times what is stored. From Eq. 6.22, for direct central collisions at the time of maximum approach, the energy stored in the combined vehicles at maximum deformation is ET = ½m¯ (v2n - v1n)2. The amount of energy returned as kinetic energy

142

Analysis of Collisions, Impulse-Momentum Theory

during rebound is ½ e2m¯ (v2n - v1n)2, and the amount lost is (1 - e2)ET. The total energy lost during the collision can be written as



TL = (1 − e12 ) E1 + (1 − e22 ) E2 = (1 − e 2 ) ET

(6.24)

where E1 and E2 are the energy losses of each vehicle corresponding to the respective, individual barrier-test coefficients of restitution, e1 and e2 . If it is assumed that the crushing deformation during the approach phase for each vehicle can be modeled as linear elastic springs with stiffness k, then each must have equal and opposite force levels at all times, and k1C1 = k2C2, where Ci represents crush deformation. Squaring this relationship and recognizing that for a linear spring, E = ½kC2, the crush energies in the vehicles are related by



E1k1 = E2 k2 = ET k

(6.25)

where

k =

k1k2 k1 + k2

(6.26)

Using these in Eq. 6.24 and solving for e gives a single stiffness equivalent coefficient or restitution, ek:

ek =

e12 k2 + e22 k1 k1 + k2



(6.27)

6.4.1.2 Mass Equivalent Collision Coefficient of Restitution

Another approach can be used to determine a single, equivalent coefficient of restitution from two barrier-test values. During a direct, central vehicle-to-vehicle collision, at the time of maximum compression (at the end of approach), both vehicles have the same velocity, vc. This velocity can be found by letting V1n = V2n = vc in Eq. 6.9; omitting subscripts n gives



vc =

m1v1 + m2 v2 m1 + m2

(6.28)

A view can be taken that during a direct central vehicle-to-vehicle collision, approach and rebound is equivalent to each vehicle hitting (or being hit by) a rigid barrier moving with closing speed vc. This means that the closing speed for Vehicle 1 is vc - v1 and is v2 vc for Vehicle 2. Equating the sum of the individual energy losses to the collision energy loss using Eq. 6.22 gives

143

Chapter 6

1 1 1 (1 − e12 )m1 (vc − v1 ) 2 + (1 − e22 )m2 (v2 − vc ) 2 = (1 − e 2 )m(v2 − v1 ) 2 2 2 2 (6.29) where

m=

m1m2 m1 + m2

(6.30)

Solving for e from Eq. 6.29 and using Eq. 6.30 gives a single, mass equivalent coefficient of restitution, em:

em =

e12 m2 + e22 m1 m1 + m2



(6.31)

It is clear that, in general, ek and em will give different results. A question immediately arises concerning which gives better results or under what circumstances one is more accurate than the other. There appears to be no comparison made on the basis of experiments. Because stiffness modeling is a common procedure in accident reconstruction based on residual crush (see Chapter 9), it might be inferred that ek might be more realistic. However, it must be kept in mind that the derivation of Eq. 6.27 is based on an assumption that crush deformation behavior during approach is similar to a pair of linear elastic springs. Because of the large amount of plastic deformation, this certainly is not true. On the other hand, Eq. 6.31 is based on an assumption that the energy loss of each vehicle during the vehicle-to-vehicle collision is identical to the individual vehicle-to-barrier collisions. This is not likely to be true, because crush patterns can differ significantly. So, which should be used? Absent specific knowledge in an individual application that favors ek or em, a prudent approach would be to calculate and use both quantities for a reconstruction and use the two results as a measure of uncertainty.

6.5 Planar Impact Mechanics In the earlier equations and applications of this chapter, angular velocities, rotational inertia, and angular (rotational) momentum were ignored. In some applications, such assumptions are acceptable. Fortunately, rotational effects can be treated rigorously without difficulty. In this section, the system equations and solution equations of planar impact mechanics are derived and presented following the approach developed by Brach [6.1]. Sometimes the theory that follows is referred to as rigid-body impact theory. Rigid body theory does not mean that the colliding bodies are not deformable, but rather that rotational inertia is taken into account and each body’s preimpact dimensions are used. Before deriving equations, however, consider some concepts that were learned in the section on point-mass equations:

144

Analysis of Collisions, Impulse-Momentum Theory

• Application of Newton’s second law, in the form of impulse and momentum, provided an algebraic approach to the modeling of impacts; no integration is necessary, • The impact problem was formulated where the final velocities and impulses were calculated from known initial velocities. • Newton’s laws alone produced too few equations to formulate the problem and yield solution equations; equations defining, or characterizing, the normal and tangential impact processes had to be defined. The definitions of the coefficient of restitution and impulse ratio coefficient, respectively, earlier filled the need for additional equations. These concepts carry over into the planar impact problem. The derivation here is a rigorous formulation of the planar problem of the impact between two rigid bodies. However, keep in mind that all applications here are to vehicle collisions that almost always have low values of the coefficient of restitution; that is, they are highly inelastic. Figure 6.5 shows two bodies separated in the form of free body diagrams, representing vehicles with masses m1 and m2, rotational moments of inertia I1 and I2, and with a common contact point, C. Point C is sometimes referred to as the impact center. A local, fixed (x, y) reference coordinate system is attached to the ground. The bodies’ orientations at the time of impact are indicated by heading angles, θ1 and θ2, relative to the (x, y) system. The common point, C, is located relative to the centers of gravity by distances, d1 and d2, and angles, ϕ1 and ϕ2 . A normal and tangential coordinate system, (n, t), is referenced to a common contact or crush surface and is oriented with respect to the (x, y) system by the angle, Γ. Other distances are



d a = d 2 sin(θ 2 + ϕ2 − Γ)

(6.32)



db = d 2 cos(θ 2 + ϕ2 − Γ)

(6.33)



d c = d1 sin(θ1 + ϕ1 − Γ)

(6.34)



d d = d1 cos(θ1 + ϕ1 − Γ)

(6.35)

These distances are the moment arms of the normal and tangential impulse components, Pn and Pt, and are needed shortly. Impulse components and initial and final velocity components can be transformed from the (x, y) to the (n, t) systems using trigonometry. For example, the relationship between impulse components is given by:

145

Chapter 6



Pn = Px cos Γ + Py sin Γ



(6.36)

and

Pt = − Px sin Γ + Py cos Γ



(6.37)

Figure 6.5 Free body diagram of the planar impact of two vehicles showing coordinates and dimensions.

An assumption now is made that the level of force acting over the common contact surface is significantly higher than other forces acting on the vehicle. (In accident reconstructions, these other forces would be aerodynamic force, tire-roadway friction, etc.) Thus, impulses of all forces other than contact forces are neglected. In addition, the contact duration, or more accurately the duration of the contact force impulse, is assumed to be very short. This implies that during contact, accelerations are high, so that velocities change suddenly and displacements (changes in vehicle position and orientation) are negligible. With these assumptions, Newton’s laws in the form of impulse and momentum can be applied directly to the bodies, as shown in Fig. 6.5. The equations of impulse and momentum for m1 and m2 in the x and y directions are

146



m1V1x − m1v1x = Px

(6.38)



m2V2 x − m2 v2 x = − Px

(6.39)

Analysis of Collisions, Impulse-Momentum Theory



m1V1 y − m1v1 y = Py



m2V2 y − m2 v2 y = − Py

(6.40)

(6.41)

According to Newton’s second law, the change in angular momentum is equal to the moments of the impulses on each body. Applying to each mass gives:



m1k12 (Ω1 − ω1 ) = Pn d c − Pd t d

(6.42)



m2 k2 2 (Ω 2 − ω2 ) = Pn d a − Pd t b

(6.43)

The variables, k1 and k2, are the radii of gyration of the bodies about their mass centers. In other words, the corresponding moments of inertia are given by I1 = m1k12 and I2 = m2k22 . Recall that capital or uppercase symbols represent values of variables at the end of the contact duration, and small or lowercase symbols are for the beginning of contact. This is true for the linear velocities, angular velocities, and impulses. (The initial impulses, at the beginning of contact, are zero.) Counting equations and unknowns at this point gives six equations: Eqs. 6.32 through 6.37, and eight unknowns, V1x, V1y, V2x, V2y, Ω1, Ω2, Px, and Py. Two more equations are needed, and so two coefficients are defined. The coefficient of restitution is defined as the ratio of the relative normal velocity at point C at the end of contact to the relative normal velocity at the beginning of contact; that is,



e = −VCrn / vcrn

(6.44)

where the subscripts C indicate the location, r indicate relative velocity, and n indicate the normal velocity components. These relative velocities at point C are

VCrn = V1n + d c Ω1 − V2 n + d a Ω 2

(6.45)

vCrn = v1n + d cω1 − v2 n + d aω2

(6.46)

and

The impulse ratio coefficient, μ, defined earlier as the ratio of the tangential to normal impulse components, is used as well:



Pt = µ Pn

(6.47)

147

Chapter 6

There now are eight equations and eight unknowns. The equations are linear and can be solved; in fact, the solution equations can be written [6.1] explicitly as follows:



V1n = v1n + m(1 + e)vrn q / m1

(6.48)



V1t = v1t + µ m(1 + e)vrn q / m1

(6.49)



V2 n = v2 n − m(1 + e)vrn q / m2

(6.50)



V2t = v2t − µ m(1 + e)vrn q / m2

(6.51)



Ω1 = ω1 + m(1 + e)vrn (d c − µ d d )q / (m1k12 )

(6.52)



Ω 2 = ω2 + m(1 + e)vrn (d a − µ db )q / (m2 k2 2 )

(6.53)

where m is given by Eq. 6.17; vrn is

vr n = (v2 n − d aω2 ) − (v1n + d cω1 )





(6.54)

and q is found from:

 md d md a 2 md c 2 md a db 1 = 1+ + − µ  c 2d + 2 2 q m2 k2 m1k1 m2 k2 2  m1k1



  

(6.55)

Note that the velocities in the above equations, V1n, V2n, V1t, V2t, v1n, v2n, v1t and v2t, all are mass center velocities. With the solution equations available, some important quantities can be determined. The combined kinetic energy, TL , lost by both bodies in the collision can be expressed as

TL =

148

2  md e 2 md f  1 ) + mqvrn 2 (1 + e)  2 + 2 µ r − (1 + e)q (1 + µ 2 + 2 m1k12 m2 k2 2  

(6.56)

Analysis of Collisions, Impulse-Momentum Theory

where

de = dc − µ dd



d f = d a − µ db



(6.57)



(6.58)

and

r=

(v2t − dbω2 ) − (v1t + d d ω1 ) (v2 n − d aω2 ) − (v1n + d cω1 )

(6.59)

An important quantity is the impulse ratio that gives the tangential common-velocity condition—that is, where V1Ct - V2Ct = 0. Note that this condition and the corresponding velocity components are at the contact point C, not at the mass center. This gives the critical impulse ratio, μ0:



µ0 =

rA + (1 + e) B (1 + e)(1 + C ) + rB



(6.60)

where



md c 2 md a 2 A = 1+ + m1k12 m2 k2 2

(6.61)

md c d d md a db + m1k12 m2 k2 2

(6.62)

md d 2 mdb 2 + m1k12 m2 k2 2

(6.63)

B=

C=

Recall that a final relative tangential velocity of zero is one of the common-velocity conditions. Together, the common-velocity conditions are e = 0 and μ = μ0. Impulse components, Px and Py, can be calculated from Equations 6.32 and 6.33 once the final velocity components are determined from the solution equations. An equation that is used often in accident reconstruction is one that gives the velocity change of the mass center of a vehicle during an impact. This velocity change, typically written as ΔV, has often been considered to be a relative indication of the severity of a collision [6.25]. The vector magnitude of the velocity change is given by

149

Chapter 6



DVi = (Vin2 − vin2 ) + (Vit2 − vit2 ), i = 1, 2



(6.64)

Substituting for the final velocity components from Eqs. 6.48 through 6.51 gives the general expression



2m(1 + e)q (1 + µ 2 )TL mi DVi = , i = 1, 2 2  md e2 md f  2 2 + 2 µ r − (1 + e)q 1 + µ + +  2  m k m2 k22  1 1 

(6.65)

Note that ΔVi depends on the total impact energy loss, TL , and also depends explicitly on the coefficient of restitution and the impulse ratio. For a frontal barrier collision, Pt = 0 (and therefore μ = 0) and



mi DVi =

2m(1 + e)TL , i = 1, 2 1− e

(6.66)

Equation 6.66 is one of the fundamental equations of the classical CRASH3 method (Chapter 9), where the coefficient of restitution, e, is assumed to be zero and TL = EC , the energy dissipated by crush.

6.5.1 Overview of Planar Impact Mechanics Model Given certain information, the planar impact mechanics model provides a way of calculating the final velocity components and impulse components of two colliding vehicles. The input information can be grouped into four physical categories: initial velocity components, physical properties of the vehicles, orientation angles (headings), and collision-damage characteristics. In symbols these are: • Initial velocity components: v1x, v1y, ω1, v2x, v2y, ω2 • Vehicle physical properties: m1, I1, m2, I2 • Orientation angles (headings): θ1, θ2 • Collision-damage characteristics: d1, ϕ1, d2, ϕ2, Γ, e, μ The last group of parameters (d1, ϕ1, d2, ϕ2, Γ) relates the damage and intervehicular contact surface of the combined vehicles to the collision, and (e, μ) to the level of energy loss. That e and μ characterize the energy loss is seen by the fact that when e = 1 and μ = 0, the collision is perfectly elastic and frictionless, and when e = 0 and μ = μ0, the energy loss is a maximum [6.1].

150

Analysis of Collisions, Impulse-Momentum Theory

Vehicle 1, a pickup truck with a weight of W1 = 4720 lb (21.0 kN) is westbound at the Example 6.4 speed of v1 = 30 mph (48.3 km/h), as shown in Fig. 6.6. It collides at a right angle with a northbound sedan, W2 = 2450 lb (10.9 kN) with the speed of v2 = 30 mph (48.3 km/h). The moments of inertia about the centers of gravity for the vehicles are I1 = 3655.6 ft-lb-s2 (4956.3 kg-m2) and I2 = 1317.5 ft-lb-s2 (1786.3 kg-m2). The vehicles are shown in Fig. 6.6 with the amount of penetration indicated. Neither vehicle has an initial yaw velocity. Use both point-mass theory and planar impact mechanics to calculate the final velocity components of both vehicles using common-velocity conditions. Compare the solutions, including the kinetic energy loss, TL . Figure 6.6 Collision configuration, Example 6.4.

Solution  Taking the normal axis positive to the right and the tangential axis positive upward, the initial velocity components are v1n = -44.00 ft/s (-13.51 m/s), v1t = 0.00, v2n = 0.00, and v2t = 44.00 ft/s (13.41 m/s). The common-velocity conditions are e = 0 and μ = μ0. Point-mass equations: For e = 0, Eq. 6.23 for the critical impulse ratio gives μ0 = r∕(1 + e) = (v2t - v1t)∕(v2n - v1n) = 44∕44 = 1. Using Eqs. 13 through 16, the final velocity components are: V1n = -28.97 ft/s (8.83 m/s), V1t = 15.04 ft/s (4.58 m/s), V2n = -28.97 ft/s (8.83 m/s), and V2t = 15.04 ft/s (4.58 m/s). According to point-mass impact theory, both vehicles have identical postimpact mass center velocities and no rotational velocities. From Eq. 6.20, the kinetic energy loss is TL = 97,047 ft-lb (131.6 kJ), which is 45% of the original system kinetic energy. Planar impact mechanics: The planar impact mechanics solution equations can be solved using spreadsheet technology, and the input must be determined. The common-velocity conditions are e = 0 and μ = 100% μ0. Both the x-y and n-t axes coincide, and the crush surface angle Γ = 0. Vehicle 1 is oriented with a heading to the left (negative n direction), and so from Fig. 6.5, θ1 = 0°; whereas Vehicle 2 is headed in the positive t direction, so

151

Chapter 6

θ2 = 90°. The angles ϕ1 and ϕ2 give the angles of the position of the distance vectors with lengths, d1 and d2, to the common point on the crush surface. From Fig. 6.6, d1 = 7.30 ft (2.23 m) and d2 = 2.23 ft (0.68 m). Furthermore, the vectors from the centers of gravity to point C make angles ϕ1 = 0° and ϕ2 = -90° (see Fig. 6.5). These data form the input to the solution of the planar impact mechanics equations. Figure 6.7 shows the results for this input. The final velocity components are V1n = -28.96 ft/s (-8.83 m/s), V1t = 7.83 ft/s (2.39 m/s), V2n = -28.96 ft/s (-8.83 m/s), and V2t = 28.91 ft/s (8.81 m/s). Because angular momentum is taken into account in this solution, final angular velocities are calculated. These are Ω1 = -131.44 °/s and Ω2 = -111.41 °/s; the signs indicate a clockwise final yaw velocity for both vehicles. Finally, the initial kinetic energy of the two vehicles combined is 215,719 ft-lb (292 kJ) and the final kinetic energy is 141,928 ft-lb (192 kJ), so the energy loss is TL = 34.2%.

Figure 6.7 Results of a planar impact analysis.

152

Analysis of Collisions, Impulse-Momentum Theory

It is apparent that there are similarities and differences between the point-mass and rigid body solutions. The final velocity components in the east-west direction are identical for both solutions. Due to the inclusion of rotational inertia, the vehicles have final angular velocities from the planar solution, which are not included in the point-mass solution. The freedom to rotate and the fact that the contact area is not at the centers of gravity not only causes nonzero final angular velocities but also causes the final tangential velocity components to differ from the point-mass solution. Because of rotational effects, less momentum in the tangential direction is transferred to Vehicle 1 from the impact, and so V1t = 7.83 ft/s (2.39 m/s) instead of V1t = 15.04 ft/s (4.59 m/s) from the pointmass solution. Because momentum in the t direction is conserved, less momentum is consequently lost by Vehicle 2, and so V2t = 28.91 ft/s (8.81 m/s) instead of the point-mass V2t = 15.04 ft/s (4.58 m/s). It is important to note that the kinetic energy loss from the point mass solution (45%) is greater than the kinetic energy loss from the planar impact mechanics solution (34%). This is because the final rotational kinetic energy of the bodies is neglected in the point mass solution.

6.5.2 Application Issues: Coefficients, Dimensions, and Angles 6.5.2.1 Coefficient of Restitution and Impulse Ratio

It was seen that for point-mass collisions, (Newton’s or the kinematic) coefficient of restitution, e, was bounded such that 0 ≤ e ≤ 1. Unfortunately, this is not the case for planar, rigid-body collisions [6.7, 6.8]; based on energy conservation, the upper bound sometimes can exceed unity and sometimes can be limited by a number less than 1. Fortunately for applications to vehicle collisions, e never approaches values greater than about 0.4 in practice, so a non-unity upper bound on e should not cause a problem in applications to accident reconstruction. If the planar impact solution equations are ever used for applications other than vehicle collisions, the user should ensure that the total energy loss never becomes negative for any combination of e and μ ≤ |μ0|. This will guarantee a realistic solution [6.9]. With respect to the lower bound of zero, the value of e should never be negative in vehicle crash applications. A negative value of e implies that the vehicles pass through each other. Determination of an appropriate value for the coefficient of restitution for a specific collision is a problem faced frequently in accident reconstruction. Some insights are gained from examination of experimental vehicle collisions, as is done in Section 6.6 below. A study [6.10] contains useful information and data from controlled frontal and side barrier collisions as well as vehicle-to-vehicle collisions. Ishikawa [6.11] has developed an impact model similar to the planar impact model presented above. Ishikawa [6.12, 6.13] also presents values of the coefficient of restitution normal to the crush surface from experiments conducted in Japan for two categories of collisions. Thirty-two values are computed from experimental side collisions and thirteen from frontal collisions. The values from the frontal collisions ranged from 0.00 to 0.15 and had an average of 0.07. The values from the side collisions ranged from -0.32 to 0.51 and had an average

153

Chapter 6

of 0.10. According to Ishikawa, “ a negative normal restitution coefficient implies that the vehicles continue to penetrate each other following the end of the collision at the impact center” (a “virtual” impact with the vehicles passing through each other). Such a collision for real vehicles is clearly impossible, so the utility of Ishikawa’s experimental values of e is open to question. As for the point-mass impact problem, a bounding, or limiting, value of the impulse ratio also exists for the planar impact mechanics problem. When the final relative tangential velocity component of the two vehicles is zero (sliding over the contact surface ends), the impulse ratio, μ, takes on a special, or critical, value, μ0. This value is given by Equation 6.60. For the planar impact mechanics solution for a two-vehicle collision, common-velocity conditions become e = 0 and μ = μ0. For a sideswipe collision, μ < μ0 (independent of the value of e). Values of μ > μ0 should not be used for vehicle collisions because under some conditions these values can cause an unrealistic energy loss for the collision. Recall that μ is not a friction coefficient but represents the retardation impulse that controls sliding along the tangential plane representing the crush surface designated by the angle, Γ, in Fig. 6.5. For a given collision, the critical impulse ratio, μ0, depends on the vehicle masses, their moments of inertia, the impact configuration, the coefficient of restitution, and the initial conditions. It can vary drastically from collision to collision. Consequently, an arbitrary value of an intervehicular frictional drag coefficient, f, must never be used for the proper analysis and reconstruction of a collision satisfying the common-velocity conditions. Although impact analysis models and reconstruction software may recommend typical values of f, μ0 should always be used as one of the common-velocity conditions. This is true even when a value of the coefficient of restitution other than e = 0 is appropriate. A value of μ = f < μ0 should be used only when sliding of the vehicles over the contact surface continues throughout the contact duration and exists at separation. A convenient way of handling the selection of a value of μ is to express it as a percentage of μ0 that ranges between 0% and 100%. This not only keeps μ ≤ μ0, but also ensures that it has the proper sign. The equations of planar impact mechanics are more general and more rigorous than the point-mass equations. Both sets of solution equations are fairly simple to apply. Despite the clear advantages of the planar impact mechanics solution, the point-mass equations are encountered frequently in accident reconstruction publications and in accident reconstruction applications. Which should be used? In some cases (low speed, inline collisions, frontal barrier collisions, etc.) the results of the two approaches are the same. But for most collisions configurations, the point-mass solution is inadequate. This is particularly true for any collision where occupant motion is important because the point-mass equations ignore rotational velocities. It is always prudent to use planar impact mechanics for vehicle impact analysis and reconstructions [6.14]. Finally, what about three-dimensional impact models. Unfortunately, only a few general models of three dimensional impact models exist [6.15] and none have been adapted and verified experimentally for applications to vehicle collisions.

154

Analysis of Collisions, Impulse-Momentum Theory

6.5.2.2 Distances, Angles, and Point C

A basic assumption of planar impact mechanics is that the position and configuration of the bodies do not change during contact and that all of the body dimensions remain constant. The deformation of vehicles from high-speed collisions is never completely elastic and rarely small. In fact, most vehicle structures are designed to have a controlled crush for the purposes of occupant protection. Figures 6.8 and 6.9 show some typical crush surfaces. A residual crush surface of a vehicle is reached in a relatively short time, but still represents a change from the undeformed shape as a function of time and space. That is, it is a three-dimensional, dynamic event. To apply planar impact mechanics, it is necessary to make assumptions. One is that the intervehicular crush surface can be represented by a vertical plane, and that a common point C exists that represents the point of application of the intervehicular impulse, P. Point C, often referred to as the impact center, is easy to define mathematically; it is the point of application of the resultant vector impulse, P = Pn n + Pt t (bold characters represent vectors). It represents the point of application of the intervehicular force averaged over space and time. Unfortunately, the location of point C never is exactly known, and must be estimated. One approach is to lay out a plan view of the residual crush and use the centroid of the crushed area. Another is to use a point on the residual crush surface, because some springback of the body occurs. Another is to use the maximum crush surface [6.13]. Whatever method is used, judgment of the analysis is necessary. Consequently, the selection of point C, d1, d2, ϕ1, ϕ2, and Γ all require measurements and some judgment. Another aspect of this topic is that vehicle dimensions, the moments of inertia I1 and I2, and their counterparts, the radii of gyrations k1 and k2, are assumed to remain constant during the impact. In reality, they change with time as the vehicles deform. There has never been a study of the uncertainty associated with the changes in these inertial parameters. Methods exist [6.1] to treat such changes in the context of planar impact mechanics but have rarely been applied.

Figure 6.8 Example of residual crush of a vehicle struck from behind in an offset frontto-rear collision.

6.5.3 Work of Impulses and Energy Loss (Crush Energy)

A very useful feature of planar impact theory has to do with the

155

Chapter 6

Figure 6.9 Example of residual crush to a vehicle struck on the passenger’s side in an intersection collision.

work of impulses and energy loss. Thompson (Lord Kelvin) and Tait [6.16] determined that, in general, the work, WP, done by an impulse, P, acting at a point is

WP =



1 P (v + V ) 2

(6.67)

where v is the initial velocity at the point and V is the final velocity at the point, both in the direction of P. In words, this states that the work of an impulse, P, is equal to the product of the impulse with the average velocity along the line of action of the impulse. When applying this to the contact impulse, P, whose components are Pn and Pt, the respective relative velocity components at point C must be used; that is,

WP =

1  1 Pn ( v1Cn − v2Cn ) + (V1Cn − V2Cn ) + Pt ( v1Ct − v2Ct ) + (V1Ct − V2Ct ) 2 2

or

WP =

156

1  P ( v + dcω1 ) − ( v2 n − d aω2 ) + (V1n + dcΩ1 ) − (V2 n − d aΩ2 ) 2 n 1n

(6.68a)

Analysis of Collisions, Impulse-Momentum Theory

1 + Pt ( v1t − d dω1 ) − ( v2t + dbω2 ) + (V1t − d d Ω1 ) − (V2t + dbΩ2 ) 2

(6.68b)

Because Pn and Pt are the only impulse components acting on the colliding bodies, their combined work, WP, must equal the energy loss, TL , of the collision. Actually, the work of the impulse is negative and energy loss is positive, so WP = - TL . Various methods such as CRASH3 [6.18] have been formulated to estimate, by measurements and calculation, the impact energy loss of a vehicle associated with residual crush normal to a damaged surface. Then the CRASH3 method uses a correction factor to determine the energy loss associated with tangential effects. Energy and energy loss are scalar quantities and not vectors and cannot be formally split into normal and tangential components. As an approximation, however, the first term of Eq. 6.68a can be associated with the normal impulse and the second term with the tangential impulse. In this way, energy loss can be apportioned to “normal effects” and to “tangential effects” and used to compare with independent energy calculations. Using this approach, the work of Pn can be compared directly to the crush energy calculated by the CRASH3 algorithm and the work of Pt to the tangential correction factor. Such an approach is covered in the next chapter. This concept has been analyzed more thoroughly [6.17]. Using the same conditions as in Example 6.4, calculate the kinetic energy loss by Example 6.5 subtracting the final from the initial energy of the vehicles. Use these values to show that Eq. 6.68b gives the same results. Solution  The energy loss can be calculated as

TL =

1 1 1 1 m1 ( v12n + v12t ) + m2 ( v22n+ v22t) − m1 (V12n + V12t ) − m2 (V22n + V22t ) 2 2 2 2 TL = 215719 − 141928 = 73791 ft -lb (100.1 kJ )

The impulse components, moment arms, and velocities can be obtained from the calculation sheet. In particular, v1n = -44.00 ft/s (-13.41 m/s), v1t = 0.00, ω1 = 0.00, v2n = 0.00 ft/s , v2t = 44.00 ft/s (13.41 m/s), ω2 = 0.00, V1n = -28.97 ft/s (-8.83 m/s), V1t = 7.83 ft/s, Ω1 = -131.41 °/s, V2n = -28.97 ft/s (-8.83 m/s), V2t = 28.92 ft/s (8.81 m/s), Ω2 = -111.53 °/s, da = 0.00 ft, db = 2.23 ft (0.68 m), dc = 0.00 ft, dd = 7.30 ft (2.23 m), Pn = 1105.6 lb-s (4918.0 N-s) and Pt = 1148.5 lb-s (5108.8 N-s). Substituting these values into Eq. 6.68b gives

−WP = TL = 48524 + 25267 = 73791 ft -lb (100.1 kJ ) The first work term in Eq. 6.68b is 48524 ft-lb (65.8 kJ) and can be considered to be the energy of both vehicles lost due to crush normal to the common crush surface.

157

Chapter 6

The second term, 25267 ft-lb (34.3 kJ), is the energy loss associated with the tangential resistance generated over the crush surface, leading to a common relative tangential velocity. Note that the common-velocity conditions lead to a normal component of velocity of VCn = -28.97 ft/s (8.83 m/s) and a tangential component velocity of VCt = 24.57 ft/s (7.49 m/s) for both vehicles.

6.6 RICSAC Collisions In the 1970’s, a series of staged, two-vehicle collisions was conducted for the National Highway Traffic Safety Administration and was reported [6.19] for the purpose of collecting data for collision analysis. The name of the project was Research Input for Computer Simulation of Automobile Collisions and is more familiarly known by the acronym, RICSAC. This acronym is used here. Data from eleven of the collisions were analyzed by Brach [6.20, 6.21, 6.22], McHenry and McHenry [6.23], and Woolley [6.24]. In the 1987 paper Brach presented values of experimentally measured velocities different from the original RICSAC publication, in that velocity components were corrected to include the effects of accelerometer locations remote from the vehicle centers of gravity. This issue also was addressed in the paper by McHenry and McHenry. Another problem with the data stems from the use of a vehicle-based coordinate system. Such a coordinate system is necessary in the data reduction stage because accelerometers are fixed to the vehicles, and the accelerations and velocities must be transformed to a fixed, earth-based system. A listing of the experimentally measured initial and final velocities from the RICSAC collisions is included in Tables 6.2 (a) and (b). The tables include information concerning the kinetic energy and energy loss values. Various aspects of the RICSAC data will be used and discussed in many examples to follow. Values of coefficients of restitution and impulse ratio coefficients were established from each collision by using a method of least squares to fit experimental data to the planar impact solution equations. This was done by finding the coefficient values that gave the best fit of the overall data to the equations of planar impact mechanics (Eqs. 6.48 to 6.53). Part of the fitting procedure was that constraints were used on the coefficient of restitution such that e2 ≤ 1. A constraint was placed on the impulse ratio such that 0 ≤ |μ| ≤ |μ0|. Tables 6.2 (a) and (b) show that the values of e had some consistency within collision categories. The most consistent was for the 60° front-to-side collisions where all three coefficients were zero. Two of the coefficient values of the three 90° front-to-side collisions were considerably higher, at 0.400 and 0.419, while the third was 0.079. It is quite possible that an impact to a “structurally hard” wheel area led to the two higher values. With the exception of one value at 0.217, the coefficients for the front-to-rear and front-to-front all ranged from 0.000 to 0.100. The impulse ratio coefficients, μ, were very consistent. Although their actual values ranged from small negative values (collisions 3, 4, and 5) to near a value of 1 (collisions 1, 6, and 7), they all are equal to their critical value, μ0. This indicates that for all RICSAC collisions, relative tangential sliding ended at or before separation.

158

Analysis of Collisions, Impulse-Momentum Theory

Table 6.2 (a)

RICSAC staged collision analysis [6.21]. Collision Number

1

6

7

8

9

10

Collision Geometry

60° front to side

60° front to side

60° front to side

90 ° front to side

90 ° front to side

90 ° front to side

Initial System Energy (A)

100900

106300

178580

132820

107410

260340

Collision Energy Loss (A)

64877

58949

98756

71189

41352

100750

CE, Crush Energy (B)

48392

45365

71995

34805

26732

32190

Corrected CE (B)

92013

123053

137651

55833

35920

45629

Ratio, Corrected/CE (B)

1.9

2.7

1.9

1.6

1.3

1.4

Restitution Coefficient, e (C)

0.000

0.000

0.000

0.079

0.400

0.419

Impulse Ratio, μ (C)

0.966

0.824

0.772

0.413

0.486

0.590

μ ∕μ0 (C)

1.0

1.0

1.0

1.0

1.0

1.0

Impact Energy Loss, % (C)

51.9

48.3

48.8

36.1

28.8

31.0

Impact Energy Loss (C)

52367

51343

87147

47948

30934

80705

Normal Energy Loss (C)

19575

21047

37323

33072

16863

40092

Tangential Energy Loss (C)

32793

30296

49824

14876

14071

40613

Ratio, Impact to Normal (C)

2.7

2.4

2.3

1.4

1.8

2.0

Notes: All energy values expressed in ft-lb. A. Values are from RICSAC Tests. B. Values are calculated from CRASH3 algorithm with actual PDOF and reported crush stiffness coefficients. C. Values are based on data from tests and fit to the planar impact model by the method of least squares.

Y 2

1

X COLLISIONS 1, 6, 7

1

Y

2

X COLLISIONS 8, 9, 10

This example consists of using the equations of planar impact mechanics to reproduce Example 6.6 the results of an experimental crash of the side of a vehicle against a narrow, rigid barrier and a comparison of results. Germane and Gee [6.26] report on a staged crash test of a 1995 Hyundai Elantra traveling at 42.8 mph (68.9 km/h) into the end face of a rigid, L-shaped, concrete barrier. Figure 6.10 is a video frame of the vehicle at the beginning of contact with the barrier. The vehicle had a heading of 10°, counterclockwise from the vertical and was translating horizontally to the left, striking the barrier at the driver’s side rear wheel. Figure 6.10 shows the chosen (x, y) coordinate system selected for this application. Because the end face of the barrier is flat, rigid, and perpendicular to the x axis, the (n, t) coordinate system is placed coincident with the (x, y) coordinates; that is,

159

Chapter 6

Table 6.2 (b)

RICSAC staged collision analysis [6.21]. Collision Number

11

12

3

4

5

Collision Geometry

10° front to front

10° front to front

10° front to front

10° front to rear

10° front to rear

Initial System Energy (A)

109700

253300

74292

249170

2424400

Collision Energy Loss (A)

103231

229765

25482

128819

102793

CE, Crush Energy (B)

78127

122700

17634

118855

117020

Corrected CE (B)

78908

124818

17646

120349

117035

Ratio, Corrected/CE (B)

1.0

1.0

1.0

1.0

1.0

Restitution Coefficient, e (C)

0.000

0.100

0.217

0.045

0.053

Impulse Ratio, μ (C)

0.038

0.031

-0.065

-0.050

-0.090

μ ∕μ 0 (C)

1.0

1.0

1.0

1.0

1.0

Impact Energy Loss, % (C)

90.9

91.9

34.2

36.3

32.0

Impact Energy Loss (C)

99717

232801

25408

90449

77581

Normal Energy Loss (C)

100046

233561

25779

91196

78793

Tangential Energy Loss (C)

-329

-760

-371

-747

-1212

Ratio, Impact to Normal (C)

1.0

1.0

1.0

1.0

1.0

Notes: All energy values expressed in ft-lb. Negative tangential energy losses are physically unrealistic, result from approximations and are small (near zero). A. Values are from RICSAC Tests. B. Values are calculated from CRASH3 algorithm with actual PDOF and crush stiffness coefficients. C. Values are based on data from tests and fit to the planar impact model by the method of least squares.

Y

2

1

X COLLISIONS 11, 12

2

Y 1

X COLLISIONS 3, 4, 5

Γ = 0. Figure 6.11 shows the input dimensions and data that represent the experimental conditions and correspond to Fig. 6.5. Solution  Analysis of the experimental data indicated a coefficient of restitution of e = 0.14. Figure 6.11 shows the final velocities, energy loss, ΔV, and other results from the planar impact mechanics solution using the experimental value of e. In the analysis, the Hyundai is designated as Vehicle 1 and the barrier is represented by Vehicle 2 through the use of an extremely large mass and moment of inertia. Table 6.3 contains a comparison of the experimental measurements and the results of the calculations.

160

Analysis of Collisions, Impulse-Momentum Theory

Figure 6.10 Crash test of a 1995 Hyundai into a massive fixed barrier.

Figure 6.11 Planar impact mechanics solution of the barrier impact of Fig. 6.10.

161

Chapter 6

Table 6.3

Comparison of calculations and experimental measurements for Example 6.5. Experimental Data [6.25]

Differences

US

SI

29.9 ft/s

30.3 ft/s

9.2 m/s

20.4 mph

20.7 mph

33.3 km/h

+1.5 %

Final angular velocity

491 deg/s

472 deg/s

8.23 rad/s

-3.9 %

ΔV

22.6 mph

24.3 mph

39.1 km/h

33.2 ft/s

35.7 ft/s

10.9 m/s

+7.5 %

74,435 ft-lb

81,956 ft-lb

111.2 kJ

+10.1 %

42.6 %

46.9 %

46.9 %

Final mass center velocity

Kinetic energy loss

162

Planar Impact Mechanics

100(calc - exp)/exp

CHAPTER

7 Reconstruction Applications, ImpulseMomentum Theory

7.1 Introduction Although Chapter 6 includes examples, it covers considerable theory. This chapter develops additional applications and examples. Point-mass theory applications are covered first, followed by applications of planar impact mechanics where rotations of the impacting vehicles are taken into account. The point-mass equations are solved using spreadsheet technology. Point-mass solutions are useful when vehicle rotations are inconsequential or when used for a preliminary analysis. Planar impact solution equations, which include the effect of rotations and rotational inertia, are clearly the most important set of equations for the analysis and reconstruction of vehicle collisions. The solution of the impact equations is an algebraic problem and is easily carried out using computer technology. The coverage here includes such use with some of the advanced features of spreadsheets. Low-speed, front-to-rear impacts are covered and require additional terms in the impulse-momentum equations. The planar impact equations presented in the last chapter are in the form directly applicable to the analysis of vehicle collisions. That is—given the vehicle properties, the collision configuration, and initial conditions—the final velocities, ΔV values, intervehicular impulse, and energy loss can be found. This is in contrast to the reconstruction of vehicle collisions where, more often than not, the final postimpact velocity components are known and the initial, preimpact velocities are sought. This means that the impact equations have to be evaluated and used in an iterative fashion. Although not ideal, this approach is not overly difficult and even has some advantages; examination of the multiple solutions often provides an understanding of the sensitivity as variables are changed when searching for a solution. Moreover, spreadsheet technology presents tools

163

Chapter 7

that can be used to directly provide input quantities for desirable, or target, values of selected output. Examples illustrating the use of these tools are provided in this chapter.

7.2 Point-Mass Collision Applications In some reconstructions, point-mass impact theory can provide useful information. One case is when the postimpact rotational velocities of vehicles are relatively small. Another is when carrying out preliminary calculations prior to a full planar impact analysis. Point-mass impact theory can sometimes be used for carrying out comparative, “what if” analyses of the feasibility of different collision scenarios. Figure 7.1 shows two vehicles with preimpact velocities, v1 and v2, and postimpact velocities, V1 and V2 . These velocities have angles θ1, θ2, ϕ1, and ϕ2, respectively. It occurs frequently that the initial directions of travel, θ1 and θ2, of two vehicles are known. Following the collision, it is assumed that each vehicle travels to rest over a known, approximately straight-line path. The distances traveled by the centers of gravity during the skid to rest are x1, y1, x2, and y2, and the path angles are ϕ1 and ϕ2 . In addition, postimpact motion is over a relatively flat surface with uniform drag coefficients, f1 and f2 . These coefficients can be actual or equivalent values. Under these conditions, the postimpact speeds, V1 and V2, can be computed using the stopping distance formula, Eq. 3.6 (b). If, during contact, the effect of forces other than the intervehicular contact force can be neglected (forces such as the friction between the wheels and the ground), conservation of momentum can be applied. This usually means that the collision should be at a relatively high speed and involve considerable damage. If this is the case, then conservation in the x direction is

m1v1 cos θ1 + m2 v2 cos θ 2 = m1V1 cos ϕ1 + m2V2 cos ϕ2

(7.1)

Likewise, in the y direction:

m1v1 sin θ1 + m2 v2 sin θ 2 = m1V1 sin ϕ1 + m2V2 sin ϕ2

(7.2)

Figure 7.1 also shows the angle Γ used to define the orientation of a common intervehicular crush plane or surface. This also defines the normal and tangential axes, (n, t), as used in Chapter 6. An important point to recognize about the point-mass approach is that after the postimpact speeds are calculated using the skid-to-rest equation for each vehicle, only two unknowns remain. Because the initial directions of motion (θ1 and θ2) are assumed known, only the initial speeds of Vehicle 1 and Vehicle 2 are unknown. This means that only two equations are needed: Eqs. 7.1 and 7.2. It also means that Eqs. 6.11 and 6.12, and the coefficients, e and μ, are not needed. Checking their values, after a solution has been found, can be used to advantage. If the values of e and μ corresponding to the reconstructed velocities are unrealistic, this can be a warning that the reconstructed velocities may also be unrealistic. A computerized solution can be set to automatically

164

Reconstruction Applications, Impulse-Momentum Theory

Figure 7.1 Coordinates and variables for point-mass collision analysis.

calculate values of e and μ. For example, a negative value of e is unrealistic. A value of |μ| greater than |μ0| or a sign of μ different from μ0 is unrealistic. Figure 7.2 shows an accident scene where two vehicles collide in an intersection where Example 7.1 traffic drives on the left side of the road. The postimpact distances (shown with dashed lines) are measured at the scene to have coordinates x1, y1 = 18.3, 11.4 m (60.0, 37.4 ft) and x2, y2 = 21.7, 11.2 m (71.2, 36.7 ft). It is assumed that the vehicles collided as they entered the intersection and while traveling in their respective lanes and directions of travel, with no evasive maneuvers; therefore, (θ1, θ2) = (0°, 90°). The vehicles’ weights are W1 = 16.63 kN (3738.3 lb) and W2 = 15.68 kN (3524.4 lb). The road surface is dry asphalt. The coefficient of friction was not measured at the site, and so f1 = f2 = 0.7 are used as representative values. The crush plane (common crash surface) is selected to be parallel to the left side of Vehicle 2, so Γ = 0o. Determine the preimpact speeds of the vehicles. Solution  The solution of this problem is carried out using a spreadsheet implementation of the point-mass impact equations. The results are shown in Fig. 7.3 and show initial speeds of Vehicle 1 and Vehicle 2 to be 108 km/h (67 mph) and 65 km/h (40 mph), respectively. The energy loss, in percent, is TL = 49%. This means that 49% of the combined preimpact kinetic energy of the vehicles was dissipated in the collision, whereas the remainder was dissipated during the slide to rest of each vehicle. It is important to note that for Γ = 0, the values of e and μ are e = 0.06 and μ = -0.59, where the critical value is -0.57. That these coefficients are close to the common-velocity conditions imparts a sense of confidence that the reconstruction is realistic. If either e is not near 0 (or near the expected value) or μ does not have a value near μ0 (except for a sideswipe collision), then the solution should be examined to find the cause of unrealistic values.

165

Chapter 7

Figure 7.2 Intersection collision (left-hand motorway) showing impact and rest positions.

Example 7.2 Consider an example of a collision in an intersection with a 30° relative angle, as shown

in Fig. 7.4. Vehicle 1 is a sedan weighing 2800 lb (12.46 kN) and Vehicle 2 is a pickup truck with a weight of 3800 lb (16.90 kN). An x-y coordinate system is chosen, with y to the north and x to the east. Postimpact center-of-gravity travel distances are measured as x1, y1 = -38.3, 2.1 ft (-11.7, 0.6 m) for Vehicle 1 and x2, y2 = -42.30, -10.8 ft (-12.9, -3.3 m) for Vehicle 2. The original vehicle headings are assumed to be parallel to the respective roads and are θ1 = 90° and θ2 = 210°. Vehicle 2 skids to rest entirely on a pavement with frictional drag coefficient f = 0.7. Vehicle 1 skids into thick, heavy bushes that are likely to have significantly slowed it by an unknown amount. An equivalent value of f = 0.7 is used for both vehicles to initiate a reconstruction. The crush surface angle, Γ, is aligned with the side of Vehicle 1, so Γ = 90°. Estimate the preimpact speeds of both vehicles based on point-mass collisions. Solution  Figure 7.5 shows the results of using the calculation sheet with f1 = f2 = 0.7. This shows that the initial speeds were 31 mph (49 km/h) for Vehicle 1 and 58 mph (93 km/h) for Vehicle 2. The calculation further shows the coefficient of restitution, e = -0.02, the impulse ratio, μ = 1.03, and the critical impulse ratio, μ0 = 1.21. Although near zero, a negative coefficient of restitution is physically impossible, so some examination of the

166

Reconstruction Applications, Impulse-Momentum Theory

Figure 7.3 Results of point-mass collision analysis, Example 7.1.

reconstruction is warranted. Remember that the postimpact frictional drag coefficient of Vehicle 1 involved the bushes and therefore has large uncertainty. An approach that sometimes can provide useful results in a situation such as this is to use a “what-if” analysis tool of the spreadsheet. This allows the user to specify a goal of e = 0 and find the value of f1 that gives this value. In this case, the results in Fig.7.6 show a corresponding value of f1 = 0.75. This can be interpreted that retardation by the bushes caused an effective average drag of f1 = 0.75 for Vehicle 1. The impulse ratio value, 1.01, is not equal to its critical value, 1.19, but is not significantly different, particularly considering this is a point-mass solution. Additional iterations could be done to meet both criteria.

167

Chapter 7

Figure 7.4 Intersection collision showing impact positions and orientations on the roadway and rest positions.

Although the point-mass solution is not as accurate as the solution of the planar impact problem, it often is adequate when the circumstances of the accident satisfy the assumptions. A point-mass solution can be used to provide an initial or starting point for a planar impact mechanics problem.

7.3 Rigid Body, Planar Impact Mechanics Applications; Vehicle Collisions with Rotation The equations of planar impact mechanics were derived and solved in Chapter 6. The solution equations have a form where the vehicle dimensions, collision damage, initial velocity components, and coefficients are considered as known quantities and the final velocities, impulses, energy loss, ΔV’s, and other information are calculated. A spreadsheet that permits numerical solutions of the planar impact equations is relatively simple to set up. For accident reconstructions, a postimpact trajectory analysis often is done first and provides the final velocities of the impact. In such cases, it is desirable to carry out an impact solution where the final velocities are known and the initial velocities are unknown. The system equations are linear and algebraic and theoretically can be solved in an inverse fashion; that is, it is possible to arrange the equations algebraically to determine the initial velocities for a given set of final velocities. Unfortunately, the solution is invalid when the coefficient of restitution is zero or near zero, one of the two common-velocity conditions used frequently for accident reconstruction. So other approaches are necessary. An effective way to reconstruct a collision is to use a forward approach where the equations are solved for the desired final velocities in an iterative

168

Reconstruction Applications, Impulse-Momentum Theory

Figure 7.5 Results of point-mass collision analysis.

fashion by varying the initial velocities. Later it will be seen that excellent reconstruction results can often be reached following an inverse search technique where the final velocities are matched using a minimum least-squares approach. Although variations exist, a common procedure of using the planar impact equations in a forward approach to reconstruct a two-vehicle collision is summarized here (and discussed shortly in more detail): 1. Determine the vehicle dimensions and physical properties. 2. Examine and/or measure the crash damage of both vehicles to determine a common, average planar crush surface and the distances and angles from the

169

Chapter 7

Figure 7.6 Results of point-mass collision analysis using spreadsheet Goal Seek feature.

centers of gravity to the common point (the impact center, C) on or near the crush surface. 3. Determine the impact configuration (positions and orientations of the vehicles during the contact duration). 4. Determine which, if any, velocities may be known (particularly initial angular velocities that are frequently zero) and which are unknown; and determine

170

Reconstruction Applications, Impulse-Momentum Theory

what other quantities, such as crush energy and initial heading directions, may be known. 5. Choose the appropriate impact coefficients; for example, the use of the commonvelocity conditions, e = 0 and μ = μc. 6. Use the collision information to solve the impact equations. 7. Change the unknown initial conditions until they produce the knowns of the reconstruction. Variations of the above procedure can also be followed. As mentioned earlier, a good starting solution sometimes can be found using a point-mass solution. After establishing a reconstruction, it always is a good practice to examine the results to be sure they match all important conditions and are physically realistic. It also is good practice to estimate uncertainty and sensitivity.

7.4 Collision Reconstruction Using a Solution of the Planar Impact Equations Evaluation of the solution equations, Eq. 6.48 through Eq 6.53, along with other results following from the solution (ΔV, TL , PDOF, etc.), can be used to reconstruct a planar impact of two vehicles. The usual procedure is to solve for the final velocity components given the physical information for the vehicles, the impact configuration variables, the coefficients, and the initial velocity components. All of the required input information is contained in the shaded sections (such as in Fig. 6.11). Input information consists of the masses, m1, m2; moments of inertia, I1, I2; distances d1, d2; angles ϕ1, ϕ2; angles θ1, θ2; the common surface angle, Γ; coefficients e and μ; and the six initial velocity conditions for Vehicles 1 and 2. All other quantities, such as energy loss and PDOF, are calculated from the input. All input quantities with dimensions are expressed in units of pounds, feet, seconds, and degrees, except where otherwise indicated. SI units can also be used. It is a good idea to construct and place a diagram on the solution sheet to illustrate the main features of the planar impact model. The coordinates and vehicle orientations are arranged so that a head-on collision (nose to nose) has orientation angles θ1 = θ2 = 0, and all angles and angular velocities are positive in a counterclockwise sense. As a first example of the solution of the planar impact equations, consider a frontal Example 7.3 offset impact of a 4340 lb (19.31 kN) pickup truck with a flat rigid barrier at a speed of 30 mph (44 ft/s, 13.41 m/s) as shown in Fig. 7.7. The objective here is to determine the final velocity components, ΔV, impulse, and energy loss from such a collision. The physical information for the vehicle and collision is listed in Table 7.1. This information corresponds to the general collision depicted in Fig. 6.5. In this example the impact is with a rigid barrier and not another vehicle. To represent a rigid barrier, Vehicle 2 in the impact equations can be given “infinite” mass, m2, and moment of inertia, I2 . In practice, any set of large values that results in negligible final velocities of m2 will do the job. Values of 1.0 × 106 will be used for m2 and I2 . Initial velocity components must be established according to the (x, y) coordinate system in Fig. 7.7. This gives v1x = -44 ft/s, v1y = ω1 = v2x = v2y = ω2 = 0. Because the barrier is rigid, the (common) crush surface has

171

Chapter 7

an orientation such that Γ = 0° and where the (x, y) and (n, t) coordinate systems are aligned. It is typical for rigid barrier collisions that there will be some small rebound of the vehicle from the barrier. So a value of e = 0.1 is used. The critical value of μ = μ0 is used. (Because v1y = 0, this is a normal collision and only a small tangential impulse can be developed.) Figure 7.7 Offset frontal impact with a fixed rigid barrier.

Solution  The results are shown in the form of a spreadsheet, Fig. 7.8. The final angular velocity of the pickup is 100.9° and is positive (counterclockwise). The final, normal contact velocity is Vcn = - e vcn = 4.40 ft/s (0.45 m/s). Because e ≠ 0, ΔV = 46.1 ft/s (14.1 m/s); keep in mind that this is ΔV of the mass center (not at the point of contact). If the ΔV of some other point in the vehicle is needed, it must be calculated separately. The energy loss is 93.0% of the original kinetic energy. A large percentage of the residual kinetic energy, ½m1 v12 + ½I1 Ω2, is associated with the final rotational velocity, where Ω = 100.9°/s. The normal impulse is equal to 6136.9 lb-s (27.3 kN-s) and the tangential impulse is 1034.2 lb-s (4.6 kN-s). Because the initial tangential contact velocity (vct) is zero and the final value (Vct) also is zero, no work is done by the tangential impulse and all of the energy loss is associated with damage (normal residual crush).

7.5 Reconstructions Using a Spreadsheet Solution of the Planar Impact Equations The above examples are not strictly reconstructions because the final velocities are calculated from initial conditions. A reconstruction works in reverse to find initial velocities from final ones. An example of a reconstruction is now presented. Example 7.4 An example collision reconstruction is presented that is based on RICSAC staged

collision #9. All vehicle dimensions and characteristics have been published [7.1] and all initial and final velocities were measured. This collision reconstruction takes the perspective that the final impact velocities are known and the initial impact velocities are to be reconstructed. RICSAC 9 was a controlled 90° front-to-side crash between a 1975 Honda Civic (Vehicle 1) and a four-door 1974 Ford Torino (Vehicle 2). A photograph of the impact configuration is shown in Fig. 7.9. A diagram of the impact

172

Reconstruction Applications, Impulse-Momentum Theory

Table 7.1

Input information, Example 7.3 Mass, m1 Moment of Inertia, I1 Orientation, θ1 Distance, d1 Angle, ϕ1 Crush Surface Angle, Γ

135 lb-s2/ft 3265 ft-lb-s2 0˚ 4.66 ft 21° 0

1968 kg 4425 kg-m2 0˚ 1.42 m 21 ° 0°

Figure 7.8 Results of a planar impact mechanics analysis of a frontal impact.

173

Chapter 7

configuration in Fig. 7.10 shows the distances from the centers of gravity and the impact center. Figures 7.11 and 7.12 show photographs of the vehicles with residual damage. Examination of the impact configuration and frontal crush indicates that the area of primary impact between the vehicles was between the front center of the Honda and the right-front wheel of the Ford. The impact center shown in Fig. 7.10 was chosen to reflect this. Pertinent vehicle dimensions and physical properties, taken from published test conditions, are listed in Table 7.2. The distances d1 and d2 and the angles θ1, θ2, ϕ1, and ϕ2 in Table 7.2 reflect the dimensions of the vehicles and the coordinates and dimension defined and illustrated in Fig. 6.5 [7.2]. For this collision it is known that v1y = ω1 = v2x = ω2 = 0. Measured values of the final velocity components and the ΔV’s also are listed in Table 7.2. The reconstruction for this example is based on an assumption that commonvelocity conditions are appropriate for this collision, so values of e = 0 and μ = 100% μ0 are used. This may or may not be a valid assumption and is reexamined later in another example. Figure 7.9 Positions and orientations of vehicles at initiation of contact, RICSAC 9.

Solution  A gallant, but not too successful, effort was made to find by iteration the initial velocity components v1x and v2y that produces the final velocity components approximately equal to the measured values in Table 7.2. Results in the form of a spreadsheet are given in Fig. 7.13. The velocity, V1x, should be near -5.1 ft/s (-1.6 m/s) but is -7.2 ft/s (-2.2 m/s); V1y should be near 17.6 ft/s (5.4 m/s) but is 10.4 ft/s (3.2 m/s), and so on. The initial velocities reached by trial and error are v1x = -19.9 ft/s (-6.1 m/s) and v2y = 27.9 ft/s (8.5 m/s). These compare to the measured values from the RICSAC tests of v1x = - v2y = - 31.1 ft/s (9.5 m/s). This is an unacceptable match. In addition, the ΔV values for Vehicle 1 and Vehicle 2 should have been 28.6 and 12.6 ft/s (8.7 and 3.8 m/s) but were found to be 16.3 and 7.5 ft/s (5.0 and 2.3 m/s), respectively. After the collision, Vehicle 1 had a negative angular velocity, Ω1 = -174.9°/s and Vehicle 2 had a positive

174

Reconstruction Applications, Impulse-Momentum Theory

Figure 7.10 Diagram of positions and orientations of vehicles, RICSAC 9.

Figure 7.11 Photograph of Vehicle 1 showing damage, RICSAC 9.

angular velocity, Ω2 = 33.9°/s. These compare to measured values of Ω1 = -180°/s and Ω2 = 45°/s. Note that the velocity components, Vcn, at the common point and normal to the crush surface for Vehicle 1 and Vehicle 2 are equal. This is a consequence of the use of the common-velocity conditions. The coefficient of restitution, e, is equal to zero, so the vehicles do not rebound at the contact point. The velocity components, Vct, tangent to the crush surface are equal because relative tangential sliding ended before separation. Note that the normal and tangential velocities at the contact point (point of action of the impulse) are not zero because they are the velocities of the vehicle in the n and t directions relative to a fixed reference.

175

Chapter 7

Figure 7.12 Photograph of Vehicle 2 showing damage, RICSAC 9.

Test conditions, RICSAC 9.

Table 7.2 Veh 1 (Honda)

Veh 2 (Ford)

Veh 1 (Honda)

Veh 2 (Ford)

m1, m2

70.1

152.2

lb-s2 ∕ft

1029

2221

kg

I1, I2

976

3953

ft-lb-s2

1323

5359

kg-m2

d1, d2

4.8

5.6

ft

1.5

1.7

m

θ1, θ2

0

90

˚

0

90

˚

ϕ1, ϕ2

6

-29.7

˚

6

-29.7

˚

vx

-31.09

0

ft/s

-9.48

0

m/s

vy

0

31.09

ft/s

0

9.48

m/s

ω

0

0

˚/s

0

0

˚/s

Vx

-5.1

-7.3

ft/s

-1.6

-2.2

m/s

Vy

17.6

20.2

ft/s

5.4

6.2

m/s

Ω

-180

45

˚/s

-180

45

˚/s

ΔV

28.6

12.6

ft/s

8.72

3.84

m/s

As discussed earlier, the work done by the normal impulse, Pn, can be used to estimate the residual crush energy and the work of the tangential impulse component, Pt, to estimate the energy loss due to relative tangential motion over the vehicle-to-vehicle crush surface. For example, the results in Table 7.3, from Brach and Brach [7.4], show that if a vehicle’s ΔV is calculated from (normal) crush energy alone, it can be significantly low because the energy loss due to tangential effects is ignored. On the other hand, if the work of the tangential impulse is included, the results are more accurate. The CRASH3 algorithm offers a correction, but it is arbitrary and requires a process that depends on visual estimation of the principal direction of force (PDOF). The planar impact mechanics solution determines the energy loss and ΔV’s properly based on the choice of the coefficients, e and μ.

176

Reconstruction Applications, Impulse-Momentum Theory

Figure 7.13 Results of a planar impact mechanics analysis of RICSAC 9, with commonvelocity conditions.

Note that the impulse components (Pn and Pt) are found from the planar impact mechanics solution equations. The line of action of the total impulse is what is referred to as the PDOF. The line of impulse for RICSAC 9 is shown in Fig. 7.10. In contrast to the CRASH3 method, the PDOF is not estimated but is determined by the damage surface (through the angle Γ) and the collision conditions. Finally, the results of the reconstruction of RICSAC 9 in Fig. 7.13 show initial velocities considerably different from the measured values: (V1x, V2y) = (-19.9, 27.9), calculated, compared to (V1x, V2y) = (-31.1, 31.1), measured. Many reasons exist for the differences. One is that this collision borders on being a low-speed collision; the closing velocity was approximately 21 mph (34 km/h), and tire-roadway friction impulses may have been

177

Chapter 7

Table 7.3

ΔV values for RICSAC 9 (ft/s). Vehicle

impact.xls

CRASH3

No tangential effects

1

16.2

16.6

2

7.4

7.6

Including tangential effects

1

28.8

18.5*

2

13.3

Measured test values

1

31.4

2

13.1

8.5*

Source: [7.2] * Corrected according to the CRASH3 algorithm.

significant. But, as will be seen shortly, common-velocity conditions, particularly e = 0, may not be a good condition for this impact. Example 7.5 In this example, the previous analysis of RICSAC 9 is continued. An approach utilizing

features of some spreadsheet tools is used to obtain a much better match between the planar impact analysis and the experimental results. A question arises as to whether a different value of e can provide a better match of the computed and measured ΔV values. Again, the initial conditions are v1y = ω1 = v2x = ω2 = 0.

Solution  In an analysis of the RICSAC collisions using optimal search techniques, Brach [7.3] fit the planar impact equations to the measured data from eleven collisions. A result of this analysis was to recover the values of the impact coefficients, e and μ, that best fit the experimental data. Results for all collisions are summarized in Table 6.3. One of the main conclusions was that all of the collisions were best fit by the commonvelocity condition that μ = μ0. This was not true, however, for the other common-velocity condition; that is, that e = 0. In fact, a value of e = 0.4 was found to be appropriate for RICSAC 9. A calculation can be set up in the form of a spreadsheet to calculate the sum of squares of differences, Q, where 2



Q = ∑ (Vix − Vixm ) 2 + (Viy − Viym ) 2  i =1



(7.3)

In the computation of Q, Vix and Viy are the final velocity components of the planar impact mechanics solution equations, and Vixm and Viym are the corresponding experimentally measured values. Initial velocity components are given their experimental values. The quantity, Q, is minimized by using a spreadsheet tool (not covered here) where, for this example, the minimization procedure determines the coefficient of restitution value, e, that minimizes Q and where the coefficient is constrained such that 0 ≤ e2 ≤ 1. (This approach is similar but not identical to that of Brach [7.3] and the results in Table 6.3.)

178

Reconstruction Applications, Impulse-Momentum Theory

Figure 7.14 Results of a planar impact analysis of RICSAC 9, least-square fit to the solution equations.

Figure 7.14 shows the output of a solution for a minimum Q. The coefficient of restitution is e = 0.355. This gives values of ΔV1 = 19.1 mph (30.7 km/h) and ΔV2 = 8.8 mph (14.2 km/h) compared to measured values of ΔV1 = 19.5 mph (31.4 km/h) and ΔV2 = 8.6 mph (14.2 km/h). This example points out that although the coefficient of restitution typically is zero for high-speed collisions, exceptions can occur. Such exceptions can occur when the collision surface includes “hard” structural parts. In RICSAC 9, the front of the Honda struck the right-front wheel assembly of the Ford. In addition, as noted before, the initial speeds of this collision are relatively low. Another example follows using the solution of the planar impact equations, in which, in addition to the usual information gathered at an accident site, one of the vehicles had an

179

Chapter 7

Event Data Recorder (EDR). Such information can be used to improve the accuracy of an accident speed reconstruction. Example 7.6 Suppose a collision occurs similar to that of RICSAC 7, but in this case collision

information from one vehicle is available from an EDR record. The EDR record illustrated here is for Vehicle 1. It is fictitious but includes actual data from RICSAC 7, a 60° front-to-side collision. Some of the collision conditions are illustrated in Fig. 7.15. The EDR information, shown in Figure 7.16, consists of time traces of the acceleration and corresponding ΔV curves taken from the longitudinal axis of Vehicle 1 [7.1].

The heading axis of Vehicle 1 is oriented in the negative x direction, as shown in Fig. 7.15. Except for a change in angular position during the contact duration (neglected here), the acceleration and ΔV correspond to the x-components of the impact motion of Vehicle 1; that is, ΔV1x = 21.96 ft/s (6.7 m/s). For this example, it is assumed that the postimpact motion of the vehicles was not observed or recorded, but residual crush measurements of the vehicles were made and analyzed. Assume that a crush-energy analysis (see Chapter 8) indicates that the total crush energy dissipated in the collision (combined crush energy of both vehicles) was Ec = 37260 ft-lb (50606 J). The objective is to determine the initial speeds of both vehicles using these values of ΔV1x and Ec. Other assumptions and/or conditions are that the initial vehicle headings are known, the initial angular velocities are zero, both common-velocity conditions apply, and the contact surface is chosen tangent to the passenger’s side of Vehicle 2. Values of other physical and dimensional properties, such as mass values and moments of inertia, are shown in Fig. 7.17. Solution  A spreadsheet is used in this example along with the spreadsheet’s “whatif” tool. A sum of squares, Q, is set up to minimize the difference between the value of ΔV1x computed by the planar impact equations, and the target value of 21.96 ft/s (15.0 mph, 6.7 m/s) with a constraint applied such that the crush energy Ec, is held at a value of 37260 ft-lb (50606 J). The initial speeds, V1 and V2, of the vehicles are the unknowns. The spreadsheet’s Solver tool is directed to determine unknowns by minimizing Q and simultaneously satisfying the energy constraint. Initial guesses of 20 ft/s are used for the values of V1 and V2 . The results are shown in Fig. 7.17. The spreadsheet minimized Q, furnishing a final value of ΔV1x = 22.2 ft/s (6.8 m/s), practically identical to the target value. The resulting initial speeds are v1 = 42.66 ft/s (29.1 mph, 13.0 m/s) and v2 = 42.66 ft/s (29.1 mph, 13.0 m/s). The measured initial speeds from RICSAC 7 were v1 = 42.68 ft/s (13.01 m/s) and v2 = 42.68 ft/s (13.01 m/s). Because initial speeds have been found that satisfy the stated conditions (and which are known to accurately reflect the measured values from RICSAC 7), it could be concluded that the reconstruction is complete. In cases where the “true” values are not known, it is prudent to examine the solution and investigate the uncertainty associated with the solution. This is done in the next example. Example 7.7 In this example, the uncertainty and sensitivity of the results of the reconstruction in the

previous example are examined by making changes to the input variables and observing

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Reconstruction Applications, Impulse-Momentum Theory

Figure 7.15 Impact orientation of vehicles showing the principal direction of force (PDOF).

the changes in the reconstructed initial velocity values. In accident reconstructions, input information rarely is known exactly. Sometimes variations in input do not result in large uncertainty, but in cases with high sensitivity, small changes can render a reconstruction useless. A residual crush energy constraint was used in the previous example such that Ec = 37260 ft-lb (50606 J). Suppose that Ec is known only to within ±10%; that is, 33534 ≤ Ec ≤40986 ft-lb (45465 ≤ Ec ≤ 55569 J). As a second part of this example, reconstruct the initial speeds for a ±5% variation in the value of ΔV1x from the EDR. Solution  Repeating the previous solution for changed values of Ec produces some surprises. Table 7.4 shows the resulting initial speeds from the previous example and those found for the higher and lower values of Ec. For a ±10% change in Ec, the corresponding changes in v1 are -6.4% and + 6.1%, and in v2 are +19.0% and -18.1%, respectively. Now suppose that there could be a ±5% variation in the value of ΔV1x from the EDR; that is, Figure 7.16 Event data record of Vehicle 1 showing the measured acceleration in the x direction (dashed curve), the integral of the acceleration, ΔV, ft/s (solid curve), initial engine speed (rpm), and brake and transmission status at the beginning of record.

181

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Figure 7.17 Results of a planar impact mechanics analysis of a vehicle collision, Example 7.6.

20.86 ≤ ΔV1x ≤ 23.06 ft/s. Table 7.4 shows that from repeating the reconstruction for a ±5% change in ΔV1x, there is a corresponding change of about ±1.2% in V1 and of about ±23.5% in V2 . To find out what happens when there are simultaneous changes in ΔV1x and Ec, a more complete uncertainty analysis (such as using Monte Carlo techniques) is needed. Based on the above individual variations for Ec and ΔV1x it can be concluded that the initial speeds can be reconstructed with uncertainty of V1 in the range of ±23.5%. The above examples show that some features of modern spreadsheets are useful to carry out collision reconstructions. The use of spreadsheet “what if” tools raises questions, however. These include:

182

Reconstruction Applications, Impulse-Momentum Theory

• How many unknown quantities can be found and how much information must be provided to find them? • Is minimization of a sum of squares always the best procedure? • Are there any cases where a direct inverse solution of the planar impact equations can be used? • How is it known if an answer is unique? • If there is a unique answer, how is it identified? • What is the sensitivity to input quantities and what is the uncertainty of the solution? Although answering these questions is beyond the scope of this book, the user of these spreadsheet tools should consider such questions. Figure 7.18 is a scale diagram that shows the intersection where Vehicle 1 and Vehicle 2 Example 7.8 collided in a “T-bone” configuration, as shown. Vehicle 1, a sedan, is westbound crossing the intersection when the front of Vehicle 2, a sport utility vehicle, strikes Vehicle 1 at the driver side door. As a result of the impact, Vehicle 2 rotates 90° counterclockwise while traveling about 38 ft (11.6 m) almost directly northbound. Vehicle 1 rotates approximately 109° after impact and initially moves in a northwesterly direction and rolls backward without driver input to its rest position (often Values of V1 and V2 for ±5% changes in ΔV. Table 7.4 termed a “run-out”). The runout of Vehicle 1, shown in three Crush V1 V2 positions in light outline, is along Energy, ft-lb ft/s ft/s a circular path to rest. The distance 33534 39.93 50.77 Vehicle 1 moves from impact to 37260 42.66 42.67 rest is about 134 ft (40.8 m). The 40986 45.27 34.93 postimpact tire marks of Vehicle 2 are shown in the drawing. The ΔV1x, ft/s V1 V2 tire marks of Vehicle 1, used to 20.86 43.20 32.59 determine its postimpact path, are 21.96 42.66 42.67 not shown. The roadway surface 23.06 42.13 52.72 was asphalt and was dry at the * nominal values in italics time of the accident. A frictional drag coefficient value of f = 0.75 is used for the analysis. This is treated as a speed reconstruction. However, an important objective is to investigate the issue of whether Vehicle 1 did or did not stop before entering the intersection. The following solution presents two reconstructions of the preimpact speeds of the vehicles based on the physical evidence, one using planar impact mechanics and another using point-mass mechanics. The results are then compared.

183

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Figure 7.18 Scale diagram for Example 7.9.

Solution  The planar impact mechanics solution starts by modeling the postimpact motion of Vehicle 2 using vehicle dynamics simulation (see Example 13.5, Chapter 13). The vehicle dynamics simulation gives speed components of Vehicle 2 at separation as Vx = -4.20 ft/s (1.3 m/s), Vy = 42.5 ft/s (13.0 m/s), and Ω2 = 76°/s. These values are then treated as the postimpact speed of Vehicle 2 in a planar impact analysis. The impact analysis used the spreadsheet’s “what if” feature to match these speeds in a least-squares fashion while simultaneously determining the coefficient of restitution, e, and the impulse ratio, μ. Table 7.5 lists the input and output information from planar impact mechanics. In summary, the initial speed components of both vehicles that provide (V2x, V2y, Ω2) = (-4.38, 42.50, 76.00) are (v1x, v1y, ω1) = (-32.15, 0, 0) and (v2x, v2y, ω2) = (0, 75.43, 0), all in units of ft/s and °/s. The initial speed components of both vehicles that provide (V2x, V2y, Ω2) = (-1.34, 13.0, 76.0) are (v1x, v1y, ω1) = (-9.8, 0, 0) and (v2x, v2y, ω2) = (0, 23.0, 0) in units of m/s and °/s. The corresponding values of the impact coefficients are e = 0.055 and μ = μ0 = -0.130.

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Reconstruction Applications, Impulse-Momentum Theory

Input information and results for planar impact analysis (e = 0.055, μ = μ0 = -0.130, Γ = 90°). Veh 1

Veh 2

Veh 1

Veh 2

m

122.99

146.95

lb-s2/ft

1805

2157

kg

I

2045.71

3362.04

ft-lb-s

2773

4557

kg-m2

θ

0

90

°

0

90

°

d

3.29

7.09

ft

1.00

2.16

m

ϕ

118.71

0

°

118.71

0

°

vx

-32.15

0

ft/s

-9.80

0

m/s

vy

0

75.43

ft/s

0

22.99

m/s

2

ω

0

0

°/s

0

0

°/s

Vx

-27.03

-4.38

ft/s

8.24

1.34

m/s

Vy

39.35

42.50

ft/s

11.99

12.95

m/s

Ω

265.06

76.00

°/s

256.05

76.00

°/s

Table 7.5

The collision is now analyzed using point-mass impact mechanics using the approach presented in Section 7.2. The straight-line distance of postimpact motion of each vehicle and the associated frictional drag is necessary for input. The postimpact motion of Vehicle 1 is along a curved path and so the assumption of a straight-line path is not truly met. Moreover, the scene photographs show that the left rear wheel of Vehicle 1 was damaged during impact and the front wheels were straight. Thus, it is necessary to determine and use an equivalent frictional drag coefficient. The straight-line path issue is handled by effectively straightening out the curved postimpact path of the vehicle and aligning it along the direction of the initial postimpact motion, in this case about 104°. Estimating a frictional drag for Vehicle 1 over its postimpact motion is a difficult task due to the elevation change during the motion, the different surfaces over which the vehicle travels, and particularly the damage to the left rear wheel assembly. The equivalent frictional drag coefficient is determined from the above planar impact mechanics solution and found to be f = 0.26. The postimpact speed of Vehicle 1 from that solution is 47.74 ft/s (14.6 m/s); over 134 ft (40.8 m) of travel, the effective frictional drag is f = 0.26 g. Table 7.6 lists the input information used in the point-mass analysis and Table 7.6

Input information and results for point mass impact analysis (Γ = 90°). Veh 1

Veh 2

W

3957

4728

θ

180

90

(x, y)

Veh 1

Veh 2

lb

17601.6

21031.2

N

°

180

90

°

(-30.2, 130.9)

(-4.4, 37.9)

ft

(-9.2, 39.9)

(-1.3, 11.6)

m

f

0.26

0.75

g

0.26

0.75

g

vx

-16.60

0

ft/s

-5.06

0

m/s

vy

0

81.59

ft/s

0

24.87

m/s

Vx

-10.74

-4.90

ft/s

-3.27

-1.49

m/s

Vy

46.54

42.64

ft/s

14.19

13.00

m/s

185

Chapter 7

the preimpact and postimpact speeds. It is seen that the best and closest match to the final velocity components of Vehicle 2 (obtained from the postimpact vehicle dynamics simulation) is (V2x, V2y) = (-4.90, 42.64) ft/s, (-1.5, 13.0) m/s. Comparison of the results of the two solutions shows some similarities. For example, the preimpact velocities of Vehicle 2 are similar, 51.4 mph (82.7 km/h) for planar impact mechanics and 55.6 mph (89.5 km/h) for the point-mass solution. The coefficient of restitution is calculated as e = 0.048 for the point-mass solution and is determined via optimization in the planar impact mechanics solution to be e = 0.055. However, one important, and consequential, difference between the two solutions is the preimpact speed of Vehicle 1. For the point-mass solution, the speed is 11.3 mph (18.2 km/h), whereas for the planar impact solution, the speed is 21.9 mph (35.2 km/h); these differ almost by a factor of two. The reason for this difference is relatively easy to determine intuitively in that the planar impact solution accounts for the (nontrivial in this situation) postimpact rotation of Vehicle 2. This postimpact rotational velocity of Vehicle 2 requires a higher preimpact velocity of Vehicle 1. The point-mass solution ignores this aspect of the collision. This difference in the preimpact speed of Vehicle 1 has an important consequence with respect to the issue of Vehicle 1 stopping, or not stopping, before entering the intersection. Based on the assumption that Vehicle 1 came to a stop 20 feet east of its location at the time of impact, the acceleration level can be calculated for the preimpact velocities of the two methods. For point-mass impact solution, the acceleration level is a = 0.21 g, and for the planar impact solution, the acceleration level is a = 0.80 g. The former acceleration level is within the capabilities of the vehicle, whereas the latter is not. The planar impact mechanics solution provides more accurate results because it accounts for the postimpact rotation of the vehicles. Therefore, based on the more accurate reconstruction (using planar impact mechanics), Vehicle 1 did not stop prior to entering the intersection. This example shows that use of the point-mass impact model in reconstructing some collisions can lead to incorrect results and incorrect conclusions. Except for some special cases (such as the following topic, low-speed collisions) the use of the point-mass impact model for reconstructions is superfluous.

7.6 Low-Speed In-Line (Central) Collisions While most planar collisions can be analyzed using the planar impact equations, a different approach sometimes is necessary for collisions at low speed. The terms lowspeed and high-speed collisions are to some extent arbitrary. Specifically, the word speed here refers to the closing speed. For planar impacts, this is the initial relative normal component of the speed of approach, vrn, given by Eq. 6.54. Generally, a value less than 20 mph (32 km/h) is considered low and greater than 20 mph (32 km/h) is considered high. At least two special characteristics of low-speed collisions require attention. These characteristics are 1) the need to take into account values of the restitution coefficient significantly greater than zero and 2) the potential significance of tire forces (and their

186

Reconstruction Applications, Impulse-Momentum Theory

impulses) due to brake slip during the duration of contact. The collisions considered here typically occur when the vehicles are in line; that is, they have the same heading and have little or no lateral offset. Such collisions can be front-to-front and front-torear; however, at times front-to-side collisions can fit this category as well. The vehicles in these collisions have little or no rotation during contact. Often there is little or no visible or residual damage to the vehicles. Although low-speed collisions may appear to be relatively minor, they involve a considerably large number of claims of significant injury, often referred to as whiplash, or WAD (Whiplash Associated Disorders). The usual categories of information and data to be considered in analyzing such collisions are the vehicles, humans, and environment. Typical vehicle parameters are design and condition of the bumper systems; physical characteristics of the vehicles such as weight; seat and headrest designs and conditions; and restraint systems. Examples of human parameters include age, gender, physical characteristics such as size and weight, neckhead-spinal physiology (length, strength, etc.), stature, muscularity, physical condition, awareness, anticipation and response, seating position and posture of head and torso, arm actions (reaching, supported by steering wheel, etc.), injury tolerance, and pain tolerance. Environmental parameters include static and transient bumper heights (vehicle loading, braking, etc.), vehicle lateral offset, brake application, brake application effort, pavement conditions, vehicle angular alignment at contact, target vehicle stopped or moving, and closing speed. Discussion of types of injury and the relationship between vehicle motion and level of injury is beyond the scope of this book. Given the proper physical information, the methods covered here will allow estimation of the vehicles’ velocity changes and, to some extent, the corresponding peak accelerations. Figure 7.19 represents two vehicles during contact in a front-to-rear collision of Vehicle 2 into Vehicle 1. The forces FB1 and FB2 are the forces due to braking on Vehicles 1 and 2, respectively, during the contact duration, Δτ. The impulses of these forces correspond to P1 and P2 . The collision generates an intervehicular impulse, P, of the force, Fc, over the contact duration, Δτ. In addition to P, each vehicle can receive an externally applied impulse, P2 due to the impulses of the forces from braking with wheel slip or locked wheels. In the following, these external impulses are assumed to be known quantities. The laws of impulse and momentum can be applied. For Vehicle 1,

m1 (V1 − v1 ) = P − P1

(7.4) Figure 7.19 Free body diagram for a front-to-rear collision of Vehicle 2 into Vehicle 1.

187

Chapter 7

For Vehicle 2,

m2 (V2 − v2 ) = − P − P2

(7.5)

From the definition of the coefficient of restitution,

V2 − V1 = −e(v2 − v1 )

(7.6)

Equations 7.4, 7.5 and 7.6 form a set of 3 equations with 3 unknowns, V1, V2 and P. The solution for the final velocities is

V1 = v1 + m2 (1 + e)(v2 − v1 ) / mT − PT / mT

(7.7)



V2 = v2 − m1 (1 + e)(v2 − v1 ) / mT − PT / mT

(7.8)

P = m(1 + e)(v2 − v1 ) + (m2 P1 − m1 P2 ) / mT

(7.9)

and where



m1m2 m1 + m2

(7.10)



mT = m1 + m2

(7.11)

PT = P1 + P2

(7.12)

m=

and

Expressions for velocity changes ΔV1 and ΔV2 can be obtained from Eqs.7.7 and 7.8, respectively. Equations 7.7 and 7.8, which are extensions of Eqs. 6.13 and 6.14, include effects of the impulses from externally applied forces. Applications of Eqs. 7.7, 7.8, and 7.9 are straightforward. However, care must be taken to ensure that results are meaningful and realistic. For example, suppose that Vehicle 2 collides, as shown, into the rear of Vehicle 1. Suppose, further, that the brakes of Vehicle 2 are not applied, and that the brakes of Vehicle 1 are locked. In such a case P2 = 0, because there is no impulse applied to Vehicle 2 other than P. An impulse from the skidding wheels of Vehicle 1 over the duration of contact does exist. It would be expected that it would have a magnitude equal to the product of the frictional force and time; that is, P1 = f1W1Δτ. However, if the initial speed of Vehicle 2 is low or the frictional drag coefficient for Vehicle 1 is high, or both, and the collision doesn’t cause Vehicle 1 to begin to slide, then V1 = 0 and P1 = m2 (1 + e) (v2 - v1) < f1W1Δτ. Consequently, it is necessary to exercise care in the application of these equations.

188

Reconstruction Applications, Impulse-Momentum Theory

Suppose that Vehicle 2 has a weight of 2400 lb (10.7 kN), is moving initially at a speed of Example 7.9 v1 = 5 mph (2.2 m/s), and strikes a stationary Vehicle 1. The brakes of the struck vehicle are applied lightly, providing a frictional drag coefficient of f = 0.2. The weight of Vehicle 1 is 3350 lb (14.9 kN). It is known that the contact duration for this collision is Δτ = 0.125 s and it is assumed that the coefficient of restitution ranges between e = 0.3 and e = 0.5. Determine the ΔV of each vehicle and estimate the peak acceleration of the vehicles. Solution  Figure 7.20 contains a spreadsheet implementation of Eqs. 7.4 through 7.12. For the conditions given, ΔV1 is in the range of 5.22 ft/s to 6.07 ft/s (2.3 m/s to 2.7 m/s) and ΔV2 is in the range of -4.31 ft/s to -4.93 ft/s (-1.9 m/s to -2.2 m/s). If the peak acceleration is between two to three times the average acceleration, then a1peak = 2.6 to 4.5 g and a2peak = -1.07 to -1.23 g. When braking is present, the above equations can be used, but they must be applied with caution. There are several reasons for using caution. 1. First, the coefficient of restitution, e, is defined for a collision of the specific vehicles [7.6]. Barrier collision restitution coefficients differ from vehicle-tovehicle collision coefficients. Conversion of barrier coefficients to vehicle-tovehicle coefficients was discussed in Chapter 6 through the stiffness or mass equivalent coefficients. 2. Secondly, the coefficient of restitution for a vehicle-to-vehicle collision without braking can change significantly when braking is present because the collision conditions change [7.6]. 3. The contact duration of a collision can change when braking exists [7.6]. 4. The frictional impulse used in a reconstruction is not arbitrary. That is, it is possible to choose a frictional drag coefficient and impulse duration that are physically impossible to attain for conditions of a given collision. Consider items 2 and 3 above. For the same closing velocity as a collision without braking, as brakes of the struck vehicle are applied more strongly, the frictional drag increases. This causes the vehicles to stay in contact longer. Because the frictional drag removes energy from the system, the final velocities of both vehicles approach each other, and so the coefficient of restitution decreases. Eventually, if the frictional drag becomes high enough, the struck vehicle begins to act more as a barrier (not a rigid barrier, however, because the “barrier,” i.e., the struck vehicle, has an absorbing bumper), the contact duration then begins to drop, and the restitution begins to rise. In other words, e and ΔV depend on f in a relatively complex manner. The reason for this complexity is that the equations for low-speed collisions deal with external frictional impulses, not directly with forces, and the external impulses can be over-specified. Additional discussion of the characteristics of low-speed collisions can be found in the works of Heinrichs and Goulet [7.7] and Happer, et al. [7.8].

7.7 Airbags, Event Data Recorders (EDR), and ΔV Data retrieved from an EDR of one or more vehicles following an accident can be very useful to a reconstructionist. All light vehicles now are equipped with supplemental

189

Chapter 7

Figure 7.20 Solution of low-speed front-to-rear impact equations.

restraint systems, commonly known as airbags. The process by which airbag deployment is initiated during a crash is based on information from various sensors in the vehicle, including a signal from one or more vehicle-mounted accelerometers. The information from these sensors is analyzed by an algorithm (computational scheme) to determine if the criteria set by the manufacturer of the vehicle is met such that an airbag should be deployed [7.9, 7.10]. When the airbag is deployed (a deployment event), many vehicles now analyze and preserve various sensor signal histories, including the acceleration data, and record the vehicle’s change in velocity, ΔV, and in some cases, precrash information. The ΔV and other crash data often are stored and made available for download after

190

Reconstruction Applications, Impulse-Momentum Theory

an event. Depending on the manufacturer, make, and model of vehicle, such data are accessed from what is known generically as an event data recorder (EDR). Modules are self-contained electronic devices on vehicles and generally perform a specific function including storage. These modules carry various names, frequently vehicle manufacturer specific, including: Powertrain Control Module (PCM), Airbag Control Module (ACM), Sensing and Diagnostic Module (SDM), Engine Control Module (ECM), and Roll-Over Sensor (ROS), among others. The time history and plot of the ΔV can be used to great advantage as a criterion in the reconstruction of a collision (see Example 7.6). Data from EDRs are typically categorized into crash data and precrash data [7.11]. Following airbag deployment, crash data are collected with millisecond precision during the contact phase of the crash, whereas precrash data are preserved over a duration of seconds prior to the deployment, typically with a precision of seconds rather than milliseconds. In some vehicles, such data are stored without airbag actuation (usually referred to as a nondeployment event).

7.7.1 Crash Data If a vehicle has a single accelerometer as part of the frontal airbag supplemental restraint system, it measures the acceleration along the heading (front to rear) axis of a vehicle to sense a frontal collision. If a vehicle is equipped with one or more side airbags, acceleration perpendicular to the heading axis (side to side) is also measured. The sideto-side acceleration may or may not be stored and made accessible. Figure 6.5 shows the heading axes of colliding vehicles at angles θ1 and θ2 from the reference x axis. Heading angles and the corresponding Principal Directions of Force (PDOF) angles also are illustrated in Fig. 7.15. The ΔV value provided by the EDR from the front-to-rear axis corresponds to a component of the resultant ΔV such that

DV frEDR = DV cos( PDOF )



(7.13)

The ΔV component provided by the EDR from the side-to-side axis is

DVssEDR = DV sin( PDOF )

(7.14)

The ΔV in Eqs. 7.13 and 7.14 is the resultant velocity change of the mass center of a vehicle computed using planar impact mechanics (see Eq. 6.5), and PDOF is the angle between the resultant impact impulse and the vehicle heading axis, as shown in Fig. 7.15. Note that the accelerometer, typically part of the sensing module itself, may not be located at the mass center of the vehicle. The application of Eqs. 7.13 and 7.14 may require transformation of the acceleration data involving the angular velocity of the vehicle and the location of the module relative to the vehicle center of gravity. Except for some special cases of collisions, the ΔV computed using point mass mechanics (see section 6.3) will overestimate a vehicle’s ΔV (see Example 6.4). This is because the final rotational kinetic energy is ignored in point mass mechanics and thus the collision

191

Chapter 7

energy loss, TL , is overestimated; from Eq. 6.65 it can be seen that, under these conditions, the ΔV will also be overestimated. For impacts with a large vehicle-to-vehicle collision offset, this overestimation can be significant. Experiments [7.11, 7.12, 7.13] have determined that the ΔV value reported from vehicle EDRs is typically biased low relative to the actual crash ΔV. One factor causing this error is that the signal processing that computes the crash ΔV (necessarily) begins a short time interval after the crash actually begins, and therefore a small portion of the signal is not included in the integration scheme.

7.7.2 Precrash Data Output from some light-vehicle EDRs contains information of the vehicle and driver for a time interval prior to the crash; that is, prior to the issuing of the command for airbag actuation (sometimes referred to as Algorithm Enable, or AE). This time interval ranges typically from 5 s to 20 s. A sample is shown in Table 7.7. This information is useful and informative in its downloaded form but also can be analyzed to provide additional information. For example, the Vehicle Speed data as a function of time Table 7.7

Sample of a Segment of Precrash Data Output from an EDR.

can be numerically integrated to give the position of the vehicle as a function of time as it approaches the crash location. The Vehicle Speed data can be used to estimate acceleration (or deceleration) by dividing speed change by the corresponding time interval. This augments the Brake Switch information which indicates brake pedal application but not the level of braking. On the other hand, the Accelerator Pedal data does give a continuous indication of the level applied to the accelerator as a percentage of full throttle, and the Engine Throttle data gives a continuous indication of the throttle position. Some cautions are necessary. The precrash Vehicle Speed may not be accurate if the vehicle’s wheels are in a condition of high slip. Inaccuracies can occur in all of the EDR’s signals. More information is available on this subject; see [7.14, 7.15].

192

CHAPTER

8 Collisions of Articulated Vehicles, ImpulseMomentum Theory

8.1 Introduction In Chapter 6, Newton’s second law in the form of impulse and momentum was applied to the impact between two rigid bodies. Examples in that chapter, and in Chapter 7, illustrate the application of impact mechanics to the analysis and reconstruction of collisions between light vehicles; i.e., cars, pickup trucks, vans, etc. Unfortunately, not all vehicle collisions involve only light vehicles. Collisions involving vehicles such as heavy vehicles, straight trucks, and articulated vehicles are also encountered. The planar impact mechanics model presented in Chapter 6 cannot handle the multiple degrees of freedom associated with articulated vehicles (a vehicle comprised of two or more distinct interconnected bodies coupled by a pin joint). In addition, collisions involving one or two heavy (and/or articulated) vehicles may not satisfy the basic assumptions of planar impact mechanics. In particular, the assumption that impulses of external forces (such as tire-road friction) are negligible compared to the impulse developed over the intervehicular contact surface may not be valid. The large masses, long dimensions, presence of the pinned joint, or all of these factors, require special considerations and more flexible model capabilities than provided in the model presented thus far. This chapter considers impacts between articulated vehicles. Specifically, the general impact equations involving a pair of pinned rigid bodies are derived using the principle of impulse-momentum. These equations form a set of linear algebraic equations that requires a numerical solution. An example is presented that demonstrates the need to include the capability of modeling the impulses of forces external to the intervehicular contact surface. Results of the articulated vehicle model, correlated with data from a controlled experimental collision, follow the presentation of the equations. Another example is then presented that illustrates the application of a velocity constraint to one

193

Chapter 8

of the bodies. The chapter concludes with two examples in which the model is applied to the reconstruction of actual collisions. A plethora of information is available about the methods that permit the reconstruction of a wide range of collisions involving the impact of single (nonarticulated) vehicles such as cars and light trucks. These methods include those based on Newton’s laws (planar impact mechanics) and those based on the estimation of the energy associated with residual crush. Accidents involving light vehicles comprise the majority of automotive collisions, so these methods provide great utility to the reconstructionist. Methods based on planar impact mechanics also can be applied to the analysis of collisions involving larger, heavier vehicles, such as straight trucks, as long as the underlying assumptions are reasonably satisfied. However, analysis of collisions that involve articulated vehicles, such as an over-the-road truck tractor and an attached semitrailer or a light vehicle pulling a semitrailer, might not meet the underlying assumptions. Therefore, a more sophisticated impact model is required to analyze such collisions. An analysis of the planar motion of a single body requires the use of three coordinates: the two rectilinear coordinates of the mass center and the body’s rotational coordinate. In engineering mechanics, such a body is said to possess three degrees of freedom. Two bodies, each with three degrees of freedom, comprise a single system that has a total of six degrees of freedom. However, if the two bodies are interconnected by a pinned joint, the pin constrains the combined motion. Such a constraint reduces the freedom of the motion in two directions, and so the total degrees of freedom of the combined, articulated system of two rigid bodies is reduced from six coordinates to four. Hence, four coordinates are necessary and sufficient to fully describe the position and orientation of a single articulated vehicle. In addition, an impact involving one or more articulated vehicles may not satisfy the same assumptions as for two vehicles, each comprised of a single mass. In particular, the effect of the frictional forces and their impulses between some or all of the vehicle tires and the roadway may not be negligible. The effect of additional degrees of freedom and the significance of external impulses must be addressed explicitly to perform an accurate analysis of an impact involving one or more articulated vehicles. Before considering the general problem of articulated vehicle impact, it is noted that the planar impact mechanics model presented in Chapter 6 may still be used to reconstruct some collisions involving one or more articulated vehicles. Indeed, for an accident where a tractor semitrailer is involved in an impact with a nonarticulated vehicle and in which the velocities of the centers of mass of the tractor and the semitrailer remain essentially collinear between the onset of impact and at separation (such as the nature of the collision shown in Fig. 8.1-a), nonarticulated vehicle impact mechanics typically can be used to accurately model the impact. Judgement on the part of the accident reconstructionist is required to assess the appropriate applicability of nonarticulated vehicle impact mechanics for accidents involving one or two articulated vehicles. A model is presented in this chapter that can be applied to the impact of a pair of vehicles of which none, one, or both vehicles may be articulated. (The mass and inertia of the

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non-contacting bodies can be set to zero when appropriate.) The development is a generalization of the planar impact mechanics model presented in Chapter 6. However, the model developed here is more general than the previous development in that it takes into account the constrained degrees of freedom of pin-connected bodies. The model was originally presented by Brach [8.1] and is a general formulation of the problem because the treatment incorporates a moment impulse at the intervehicular contact surface, as presented in Brach [8.2, 8.3]. The more general treatment also introduces additional modeling flexibility by placing an externally applied impulse and/or velocity constraints on one of the rigid bodies of each of the articulated vehicle pairs. A moment impulse [8.2] is included at the impact center of the articulated vehicle impact model for generality. A value of the moment impulse equal to zero, in combination with no pinconnected bodies and no application of external impulses, reduces this model to that presented in Chapter 6. Accident reconstruction literature concerning the topic of impact between two vehicles in which at least one vehicle is articulated is scant, particularly in comparison to the literature related to the modeling of an impact between two nonarticulated vehicles. To date, the CRASH3 ΔV mechanics model has not been extended to include articulated vehicles. Certain implementations of the software program SMAC (Simulation Model of Automobile Collisions) include the capability of modeling the impact of articulated vehicles. This capability is highlighted in an article by Leonard, et al. [8.4] which considers the capability of the HVE computer program [8.5], with comparison of the results of the program to experimental data. Another version of the SMAC algorithm, referred to as m-smac [8.6], includes the capability to analyze collisions that involve articulated vehicles. An impact model that includes articulated vehicles, which uses an approach similar to that presented here, was presented by Steffan and Moser [8.7]. In that publication, the authors consider some of the issues that will be presented later, including the topic of the necessity and use of external impulses. None of those analysis models include the capability to impose velocity constraints, external impulses, or a moment impulse. This chapter initially presents the assumptions that underlie the general application of the principle of impulse and momentum to a planar collision between rigid bodies. Prior to the development of the impact equations, an example is presented that demonstrates the need to include the capability of modeling the impulses of forces external to the contact surface when considering impact involving a heavy vehicle(s). An example, which validates the model using experimental data, follows the presentation of the equations. Also presented is an example that shows the use of the application of a constraint to one of the vehicles. The chapter ends with two examples that apply the model to the reconstruction of actual accidents.

8.2 Assumptions for Application of Planar Impact Mechanics to Articulated Vehicles The application of planar impact mechanics to a vehicle collision requires that various assumptions about the nature of the collision be satisfied. These assumptions are

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1. A single impact occurs between only two of the articulated rigid bodies (if multiple impacts occur, each must be analyzed separately). 2. The time duration of intervehicular contact is short (typically on the order of 100 to 200 ms); this short time duration implies that a. changes in both linear and angular positions of all vehicles during the contact duration are small, and b. impulses acting on the vehicles due to external forces (typically tirepavement frictional forces) are small in comparison to the impulse due to the intervehicular force. 3. The location on each vehicle of the resultant intervehicular impulse is known or can be reasonably estimated. 4. Changes in the physical geometry of the masses (such as due to the crush deformation) either are small and can be neglected or are known and can be taken into account. 5. The effects of any out of (horizontal) plane dynamics are small. Relatively high closing speed (~ 20 mph [32 km/h] or higher) of collisions between light vehicles (cars, sport utility vehicles, pickup trucks, etc.) typically meet all of these assumptions, and planar impact mechanics can be applied directly. However, collisions involving one or two articulated vehicles may not meet all of the above assumptions. Any combination of large masses, long dimensions, or pinned joints, which occur with either of the two masses that are involved with an articulated vehicle, may require special methods as in assumption 2-b. As an illustration of this concept, consider an accident in which a tractor semitrailer with no cargo is traveling through an intersection. A large pickup truck, traveling at a high speed, collides with the semitrailer at, and perpendicular to, its rearmost axle. As a result of the impact, the lateral forces of the tires of the semitrailer overcome pavement friction and develop significant lateral slip such that the semitrailer undergoes a rotational velocity change. While the semitrailer undergoes this change in rotational motion, the tractor and its wheels continue to roll in their preimpact direction, without any change in rotational (yaw) velocity or a significant change in lateral velocity at the wheels. Application of the planar impact mechanics model to a collision such as this will fail to accurately predict the velocity changes of both of the masses. The effect of large tire-pavement frictional forces and impulses, such as those that maintain the heading of the tractor, in this example, can be taken into account using a lateral zero-velocity-change constraint and other special methods introduced in the following sections. Before covering collisions with articulated vehicles and lateral constraints, a better understanding of the magnitude of external impulses associated with a heavy vehicle is illustrated in the following example. The example compares the magnitude of the tire force impulses and the intervehicular contact impulse for an impact between two light vehicles to an impact between a light vehicle and a heavy vehicle. The example

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illustrates that, under certain circumstances, assumption 2-b above is not met—even for nonarticulated collisions. The impulse of the tire forces generated by a heavy vehicle can be non-negligible in comparison to the impulse associated with the intervehicular force. Therefore, the assumption of negligible external impulses may not always be valid. Consider the two collision geometries shown in Figs. 8.1 (a) and 8.1 (b). Each shows a Example 8.1 pair of vehicles at the onset of contact for an inline, head-on collision. The first figure shows a tractor semitrailer and a sedan, and the second figure shows two sedans. These two collisions are compared under the conditions of no rebound (zero restitution), no preimpact and postimpact rotational velocities, and collision duration of 0.2 s. Table 8.1 lists the relevant vehicular physical data, initial and final velocities, and intervehicular impulses using planar impact mechanics, although point mass mechanics gives the same results.

Figure 8.1 (a) Inline head-on collision between a tractor semitrailer and a sedan.

Figure 8.1 (b) Inline head-on collision between two sedans.

Data for vehicles of Example 8.1. Vehicle 1

Table 8.1 Vehicle 2

Mass (Weight)

1800 kg (3969 lb)

27,000 kg (59,530 lb)

Initial Speed

15.0 m/s (33.6 mph)

-15.0 m/s (-33.6 mph)

Final Speed

-13.1 m/s (-29.4 mph)

-13.1 m/s (-29.4 mph)

Intervehicular Impulse

-50.7 kN-s (-11,400 lb-s)

50.7 kN-s (11,400 lb-s)

Mass (Weight)

1800 kg (3969 lb)

2400 kg (5292 lb)

Initial Speed

15.0 m/s (33.6 mph)

-15.0 m/s (-33.6 mph)

Final Speed

-2.14 m/s (-4.8 mph)

-2.14 m/s (-4.8 mph)

Intervehicular Impulse

-30.9 kN-s (-6950 lb-s)

30.9 kN-s (6950 lb-s)

Vehicle 3

Vehicle 4

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Solution  The final speeds and the intervehicular impulses generated for these two collisions were obtained based on the premise that the assumptions for planar impact mechanics (nonarticulated vehicle impact mechanics) were satisfied. In particular, the analyses neglected the impulses generated by any external forces such as tire-road friction. Suppose that each of Vehicles 2 and 4 were in a locked-wheel skid throughout the entire duration of contact, with tire-to-roadway frictional drag coefficients of f = 0.6. The frictional forces generated by skidding tires of Vehicles 2 and 4 are F2 = 158.9 kN (35,718 lb) and F4 = 14.1 kN (3175 lb), respectively. Assuming that these forces are constant during the 0.2 s of contact, their impulses are 31.8 kN-s (7144 lb-s) and 2.8 kN-s (635 lb-s), respectively. Note that the impulse due to the frictional force for Vehicle 2 in collision 1-a, 31.8 kN-s (7144 lb-s), is about 63% of the intervehicular contact impulse, 50.6 kN-s (11,367.5 lb-s). Further note that the impulse due to the frictional force for Vehicle 4 in collision 1-b, 2.8 kN-s (635 lb-s), is only about 9% of the impulse due to the intervehicular force. The equations derived earlier in Section 7.6, Low-Speed Collisions, can be used to take into account the frictional impulses. The above example illustrates that impulses from forces external to the contact surface may not be negligible when analyzing a collision that involves a heavy vehicle. This forms a rationale for why an external impulse may need to be applied. Other circumstances can arise where the impulse due to a force other than the intervehicular force may be significant and restrict the motion of the vehicle; thus a velocity constraint may be required. Consider a 90° front-to-side collision between two vehicles, with one of the vehicles pulling a semitrailer. (This collision geometry is also considered in a later example—see Fig. 8.4.) Suppose, further, that the wheels of the semitrailer are freely rolling prior to impact and remain in that condition from the onset of contact to separation. This implies that the lateral forces generated by the wheels on the semitrailer do not exceed the lateral frictional limit, and the semitrailer axle does not develop a lateral velocity change due to the impact. If the initial longitudinal velocity of the tow vehicle and the semitrailer are small, this situation acts, in effect, as though the semitrailer is constrained in the lateral direction by tire friction. These illustrations and Example 8.1 demonstrate that a need exists for the articulated vehicle impact model to handle velocity constraints as well as external impulses.

8.3 Articulated Vehicle Impact Equations In this section a generalized four-body impact model is presented. Figure 8.2 shows the free body diagrams of the four masses that comprise two articulated vehicles, with the two vehicles labeled Vehicle A and Vehicle B. Vehicle A consists of Bodies 1 and 3 (shown shaded in the figure) and Vehicle B consists of Bodies 2 and 4. In addition to the various impulses required for the analysis, the figure also shows the variables and the coordinate systems associated with the model development. The x-y coordinate system is fixed to the ground, and the n-t coordinate system at the intervehicular contact surface is at an angle, Γ, relative to the x-y system. Dynamic contact is between Body 1 and Body 2 and creates an impulse, P, with components Px and Py, and a moment impulse, M. Bodies 1 and 3 are interconnected by a frictionless pin that transmits an impulse, R,

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with components R x and Ry. Bodies 2 and 4 are interconnected by a frictionless pin that transmits an impulse, Q, with components Qx and Qy. Allowance is made for an external impulse, C3, applied to Body 3 and an external impulse, C4 , applied to Body 4. Impulses are shown in the figure using their x and y components. Impulses C3 and C4 are arbitrary in location, direction, and magnitude (they can be zero) and are developed by placing final velocity constraints at respective locations on Bodies 3 and 4. The model requires that the contact take place between Bodies 1 and 2 only. However, this requirement does not imply that either Body 1 or 2 necessarily represents a tow vehicle. In this way, the model can accommodate the impact between the two tractors, the two semitrailers, or between the tractor of one tractor-semitrailer combination and the semitrailer of the other combination. Figure 8.2 Free body diagrams for the four masses in a collision between two articulated vehicles.

The variables, P, Q, R, C3, and C4 , in Fig. 2 represent the impulses of the forces that act at those locations, not the forces themselves. Similarly, M represents the moment impulse that can act at the contact surface. The inclusion of the moment impulse at the contact surface permits the modeling of, for example, structural engagement over the intervehicular contact surface that can transmit angular momentum from one vehicle to the other. While experience indicates that a non-zero contact moment is not necessarily a common occurrence during vehicular collisions, the inclusion of the moment impulse in the model provides for greater generality. As will be shown later, a moment coefficient, e’, is needed in the formulation of the problem to relate the final angular velocities of the vehicles [8.2, 8.3]. The impulse, P, represents the resultant intervehicular impulse that acts at a point, C, on the contact surface, called the impact center. This point represents the spatial and temporal average of the location of the application of the impulse, P, and moment impulse, M, during contact between the two bodies. Determination of the location of the impulse center requires judgement by the user in applications of this model, based on the location and nature of the damaged surfaces of the vehicles.

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The orientations of the bodies relative to the ground at the time of impact are indicated by the heading angles, θ1, θ2, θ3, and θ4 , all defined relative to the x-y coordinate axes. All impulses are located relative to the mass center of each body by a distance, d, with orientation angle, ϕ. For example, the impulse, P, and the impact center are located relative to the centers of mass of Bodies 1 and 3 by distances d1 and d3, respectively, acting at angles ϕ1 and ϕ3, respectively. The location of the hitch pin, where reaction impulse, R, acts, is defined on Body 1 by d1R and ϕ1R, and on Body 3 by d3R and ϕ3R . The hitch pin location is defined on Body 2 with d2Q and ϕ2Q, and on Body 4 with d4Q and ϕ4Q. The external impulses C3 and C4 act on Bodies 3 and 4 at distances d3C and d4C from the center of mass, at angles ϕ3C and ϕ4C , respectively. A normal and tangential coordinate system, (n, t), is referenced to the crush surface and is oriented with respect to the x-y coordinate system by the angle Γ. Bodies 1 through 4 have mass m1, m2, m3, and m4 , and centroidal rotational yaw inertias I1, I2, I3, and I4 , respectively. Before proceeding with the development of the equations that govern the impact between two articulated vehicles, some additional discussion of the impulses shown on the free body diagrams is necessary. The impulses C3 and C4 are included in the free body diagrams of Bodies 3 and 4, respectively, and are intended to be externally applied impulses, generally remote from the intervehicular contact impulse, P. Impulses C3 and C4 are found by imposing constraints at those points. The impulses Q and R are the impulses generated at the hitch pins that maintain a common translational velocity at each pin. Therefore, the components of these impulses, R x and Ry for Bodies 1 and 3 and Qx and Qy for Bodies 2 and 4, are shown in the figure acting equal and opposite on each pair of bodies. The components of the intervehicular impulse, Px and Py, are shown on Bodies 1 and 3. The lines of action of these impulse components are the same on each body, but the direction of the action of each is opposite, consistent with Newton’s third law. Similarly, the moment impulse, M, acting along the contact surface is also shown acting on both Bodies 1 and 2 with opposing sense. The development that follows is a direct application of the principle of impulse and momentum [8.8] to each body of the system of pairs of (pinned) rigid bodies. This principle states that the change in (linear and angular) momentum of a rigid body is equal to the sum of the external (linear and angular) impulses acting on that body. The application of the principle in x and y component form initially produces twelve scalar equations, three for each of the four masses. As will be seen, the development of the equations involves more than twelve unknowns, and additional equations are needed to solve the problem. Consistent with the notation used in the formulation of the planar impact mechanics solution in Chapter 6, the variables associated with the preimpact (initial) velocities are shown in lower case, such as v3x, and the variables associated with the postimpact (final) velocities are shown in upper case, such as V2y. The impulse and momentum equations are as follows. For Body 1:

200

m1 (V1x − v1x ) = Px + Rx

(8.1)

Collisions of Articulated Vehicles, Impulse-Momentum Theory



m1 (V1 y − v1 y ) = Py + Ry



(8.2)

I1 (Ω1 − ω1 ) = M + d1 sin(θ1 + ϕ1 ) Px − d1 cos(θ1 + ϕ1 ) Py

−d1R sin(θ1 + ϕ1R ) Rx + d1R cos(θ1 + ϕ1R ) Ry



(8.3)

For Body 2:





m2 (V2 x − v2 x ) = − Px − Qx m2 (V2 y − v2 y ) = − Py − Qy

(8.4)



(8.5)

I 2 (Ω 2 − ω2 ) = − M + d 2 sin(θ 2 + ϕ2 ) Px −d 2 cos(θ 2 + ϕ2 ) Py −d 2Q sin(θ 2 + ϕ2Q )Qx

+ d 2Q cos(θ 2 + ϕ2Q )Qy



(8.6)

For Body 3:

m3 (V3 x − v3 x ) = − Rx + C3 cos α 3 m3 (V3 y − v3 y ) = − Ry + C3 sin α 3

(8.7)



(8.8)

I 3 (Ω3 − ω3 ) = −d3 R sin(θ3 + ϕ3 R ) Rx + d3 R cos(θ3 + ϕ3 R ) Ry + d3C sin(θ3 + ϕ3C )C3 cos α 3 − d3C cos(θ3 + ϕ3C )C3 sin α 3

(8.9)

For Body 4:



m4 (V4 x − v4 x ) = −Qx + C4 cosα 4

m4 (V4 y − v4 y ) = Qy + C4 sin α 4



(8.10)



(8.11)

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I 4 (Ω 4 − ω4 ) = −d 4Q sin(θ 4 + ϕ4Q )Qx + d 4Q cos(θ 4 + ϕ4Q )Qy − d 4C sin(θ 4 + ϕ4C )C4 cos α 4

+ d 4C cos(θ 4 + ϕ4C )C4 sin α 4



(8.12)

Counting equations and unknowns gives 12 equations, 8.1 through 8.12, and 21 unknowns: V1x, V1y, V2x, V2y, V3x, V3y, V4x, V4y, Ω1, Ω2, Ω3, Ω4, M, Px, Py, R x, Ry, Qx, Qy, C3, and C4. For the general problem that involves all of the unknowns, nine more equations are needed to make the problem tractable. Note that while the general problem involves the 21 unknowns listed above, most problems will likely involve only a subset of these. For example, if one of the vehicles is not articulated, three of the unknown velocities and the associated hitch impulses do not appear in the formulation. The remaining nine equations required in the general formulation are introduced from four sources: 1) consideration of the normal and tangential contact processes over the intervehicular contact surface; that is, the introduction of coefficients e and μ 2) the existence of a moment impulse over the intervehicular contact surface and the introduction of a moment impulse coefficient, e’ , 3) the constraints imposed on the system at the hitch 4) the velocity constraints, if any, imposed on Bodies 3 and 4 Impulses R x, Ry, Qx, Qy, C3, and C4 are related to the velocity constraints, and are determined through the imposition of constraint equations. A thirteenth equation is obtained from the impulse ratio coefficient, μ, defined previously as the ratio of the tangential to normal impulse components. This is

Pt = µ Pn

(8.13a)

or, using the x-y components of the impulses,



Py cos Γ − Px sin Γ = µ ( Py sin Γ + Px cos Γ)

(8.13b)

Another equation is obtained from the definition of the coefficient of restitution in the normal direction, n (as defined by Γ), at the impact center, C. Restitution is defined as the ratio of the relative normal velocity at the impact center (shown as point C in Fig. 2) at the end of contact to the relative normal velocity at the impact center at the beginning of contact:

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e=−

VCn vCn

(8.14)

where

VCn = −[V1x + d1 sin(θ1 + ϕ1 )Ω1 ]cos Γ + [V1 y − d1 cos(θ1 + ϕ1 )Ω1 ]sin Γ −[V2 x − d 2 sin(θ 2 + ϕ 2 )Ω 2 ]cos Γ − [V2 y + d 2 cos(θ 2 + ϕ2 )Ω 2 ]sin Γ

(8.15)

vCn = [v1x + d1 sin(θ1 + ϕ1 )ω1 ]cos Γ + [v1 y − d1 cos(θ1 + ϕ1 )ω1 ]sin Γ − [v2 x − d 2 sin(θ 2 + ϕ2 )ω2 ]cos Γ − [v2 y + d 2 cos(θ 2 + ϕ2 )ω2 ]sin Γ

(8.16)

The moment restitution at the surface [8.2, 8.3] gives the following equation:

e ' M = −(1 + e ')(Ω2 − Ω1 ) I

(8.17a)

where

I=

I1I 2 I1 + I 2



(8.17b)

Note that for e’ = 0, the vehicles will have the same postimpact angular velocity; that is, Ω1 = Ω2. This condition corresponds to a perfectly inelastic angular impact, and is independent of the coefficient of restitution, e, defined in Eq. 14. For e’ = -1, the moment impulse, M, is zero and corresponds to a rigid body collision in which the moment impulse is neglected. Two constraints imposed on the system require that the hitch point for each of the two pairs of bodies have the same linear velocity components. It is assumed that no impulsive moments act at the hitch location. The velocity constraint imposed on one pair of masses yields two equations, one for each velocity component along the x-y coordinate system. This gives a total of four more equations, two for each vehicle. The hitch constraint is imposed on the postimpact velocities. (Note that to be physically realistic, compatible initial velocities must be used to satisfy the pin constraints.) For Vehicle A at the hitch point, R, in the x-direction:



V1x − d1R sin(θ1 + ϕ1R )Ω1 = V3 x + d3R sin(θ3 + ϕ3R )Ω3

(8.18)

For Vehicle A at the hitch point, R, in the y-direction:

V1 y + d1R cos(θ1 + ϕ1R )Ω1 = V3 y − d3R cos(θ3 + ϕ3R )Ω3

(8.19)

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For Vehicle B at the hitch point, Q, in the x-direction:



V2 x + d 2Q sin(θ 2 + ϕ2Q )Ω2 = V4 x − d 4Q sin(θ 4 + ϕ4Q )Ω4

(8.20)

For Vehicle B at the hitch point, Q, in the y-direction:



V2 y − d 2Q cos(θ 2 + ϕ2Q )Ω2 = V4 y + d 4Q cos(θ 4 + ϕ4Q )Ω4

(8.21)

As described previously, circumstances in a collision involving an articulated vehicle may require that the final velocity of one of the bodies be constrained to a certain direction due to interactions with other environmental objects (such as a curb, friction, etc.). The development of the impact model, therefore, includes additional impulses, C3 and C4 , to facilitate the inclusion of constraints on the velocities of Bodies 3 and 4. Of course, these constraints may not exist or may not be used in all problems and, therefore, C3 and C4 may each or both be zero. Note that one restriction of the development of the model is that the location of a velocity constraint and the impact center cannot be the same. For example, consider an impact between the tractor of one tractor-semitrailer combination into the semitrailer of another tractor-semitrailer combination. In this case, Body 1 of Vehicle A is the tractor involved in the impact, with Body 3 assigned to the semitrailer. Body 2 (the other body involved in the impact) will be assigned as the semitrailer of Vehicle B, with Body 4 as the tractor of Vehicle B. Because the impact is between Body 1 and Body 2, a velocity constraint can be applied only to the semitrailer of Vehicle A (Body 3) or the tractor of Vehicle B (Body 4). While the model can be applied to two single vehicles (m3 = m4 = 0 and I3 = I4 = 0), the model development dictates that velocity constraints cannot be applied to the two single vehicles. However, with m3 = m4 = 0 and I3 = I4 = 0, the hitch impulses, R and Q, are permitted to be specified, depending on the conditions of the crash. A later example demonstrates this capability. Figure 8.3 shows the geometric configuration of a velocity constraint. Figure 8.3(a) shows the velocity constraint, and Fig. 8.3(b) shows the corresponding impulse. The variables in the figure are given a general subscript, i, which can take on the value of 3 or 4 depending on the body to which the constraint applies. The angle αi is defined in the x-y coordinate system, as shown. The velocity constraint, from Fig. 8.3(a), takes the form



204

Vix cos α i + Viy sin α i = 0, i = 3, 4



(8.22)

Collisions of Articulated Vehicles, Impulse-Momentum Theory

Figure 8.3 Geometry of the velocity constraints.

The direction of the constraint impulse at the location where the velocity constraint is imposed acts perpendicular to the direction defined by αi, as shown in Fig. 8.3(b). Expanding this expression for each of the Bodies 3 and 4 with the appropriate kinematics gives

[V3 x + d3C sin(θ3 + ϕ3C )Ω3 ]cos α 3 +

[V3 y − d3C cos(θ3 + ϕ3C )Ω3 ]sin α 3 = 0

(8.23)



[V4 x − d 4C sin(θ 4 + ϕ4C )Ω4 ]cos α 4 + [V4 y + d 4C cos(θ 4 + ϕ4C )Ω4 ]sin α 4 = 0

(8.24)

Figure 8.3(b) shows the direction of the components of the impulse that acts to restrict the velocity to the direction specified by αi. In addition to the original 12 equations from impulse momentum, an inventory of the equations now gives the additional nine equations, 8.13, 8.14, 8.17, 8.18, 8.19, 8.20, 8.21, 8.23, and 8.24, bringing the total number of equations to 21. For the full complement of masses and conditions, the number of equations equals the number of unknowns, and the problem can be solved. Additional complexities exist due to the various configurations of vehicles and combination of conditions that may be required in the reconstruction of a given collision. For example, if there are no velocity constraints (C3 = C4 = 0), then the total number of equations and unknowns is reduced by two and becomes 19. Equations 8.1 through 8.24 (with the exception of Eqs. 8.15, 8.16, and 8.22) provide a system of linear algebraic equations with the 21 unknown final velocity components

205

Chapter 8

and impulse components listed above. If the initial velocities, impact coefficients, vehicle physical properties, body configurations, and constraint velocities are known, the equations can be used to solve for the unknown final velocity components and impulses for a given set of initial velocities. If only one final velocity constraint exists, the number of equations and unknowns is reduced by one; with no constraints, the number reduces to 19. Under any circumstances, a numerical solution is necessary. As presented in Chapter 6, a critical value of the impulse ratio, μ = μ0, corresponds to the condition that the relative tangential velocity between the bodies in contact ceases prior to, or at, separation. The equations are algebraic and linear, and the solution can be implemented in a computer program. Such an implementation permits the analysis of collisions of various geometries involving combinations of articulated and nonarticulated vehicles. Several examples are now presented to demonstrate the utility of the model.

8.4 Validation of the Articulated Vehicle Impact Equations Using Experimental Data The equations presented above are examined and compared to experimental data. Data [8.9] used for model validation by Steffan and Moser [8.7] is also used here. In the staged collision, an Alfa Romeo 164 sedan pulling a Hobby 495 travel trailer is impacted on the right front door by the front of an Opel Ascona C four-door sedan. Figure 8.4 shows the relative locations and orientations of the vehicles at the onset of contact, while the rest positions are shown in gray. Table 8.2 lists some of the physical parameters of the test

Figure 8.4 Diagram of the relative positions and orientations of the test vehicles at the onset of impact.

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Collisions of Articulated Vehicles, Impulse-Momentum Theory

vehicles and also some of the test values. Figure 8.5 shows the two vehicles at the onset of contact and depicts several of the model parameters that are needed for the analysis. The postimpact speeds from the test were available for the Alfa Romeo and the Ascona only. No data were reported for the travel trailer. Therefore, the comparison between the test speeds and the speeds predicted by the analysis using the articulated vehicle impact model will be done for Bodies 1 and 2 only. Comparison will be made between the preimpact and postimpact kinetic energies of the system. The graphs in Figs. 8.6a and 8.6b show the magnitude of the velocity components of the center of mass, vx, vy, and the rotational velocity of the vehicles determined from the test data as a function of time for the Alfa Romeo and the Ascona, respectively. The data indicate that the impulse duration was approximately 0.15 s. The change of the velocity of the center of mass of the Alfa Romeo, ΔV, is about 22.5 km/h (14.0 mph), with individual values of the velocity changes of ΔVx = 6.5 km/h (4.0 mph) and ΔVy = 21.5 km/h (13.4 mph). The change of the velocity of the center of mass of the Opel Ascona, ΔV, is about 34.3 km/h (21.3 mph), with individual values of the velocity changes of ΔVx = 7.5 km/h (4.7 mph) and ΔVy = 33.5 km/h (20.8 mph). Vehicle input physical parameters for articulated vehicle impact model validation. Vehicle A—Body 1

Vehicle A—Body 3

Vehicle B—Body 2

Weight

13.7 kN (3087.5 lb)

9.8 kN (2196.6 lb)

10.4 kN (2337.7 lb)

Yaw Moment

2152.0 kg-m2

2727.0 kg-m2

1510.0 kg-m2

of Inertia

(1587.2 ft-lb-s )

(2011.3 ft-lb-s )

(1113.7 ft-lb-s2)

Wheelbase

2.7 m (8.7 ft)

3.5 m (11.5 ft) [Hitch to axle]

2.6 m (8.4 ft)

Initial Speed

22.0 km/h (13.7 mph)

22.0 km/h (13.7 mph)

45.0 km/h (28.0 mph)

2

2

Table 8.2

Figure 8.5 Diagram that shows the vehicles at the onset of contact, the location of the impact center, and several of the model parameters needed for the analysis.

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Sufficient information from the experimental data is present to determine the coefficient of restitution for the collision. Using the y-direction as shown in Fig. 8.5 as the positive normal direction, the value of e can be determined from the following equation:

−e = Figure 8.6a Data showing the velocity of the mass center of the Alfa Romeo and its angular velocity as a function of time, with contact initiated at time = 0 seconds.

Figure 8.6b Data showing the velocity of the mass center of the Ascona and its angular velocity as a function of time, with contact initiated at time = 0 seconds. (Note that due to instrumentation convention, the velocity in the x-direction is shown as negative in the data, but is positive with respect to the coordinate system shown in Fig. 8.5.)

208

VCrn V1n + dcΩ1 −V2n + daΩ2 = vCrn v1n + dcω1 − v2n + daω2



(8.25)

Collisions of Articulated Vehicles, Impulse-Momentum Theory

For values of V1n = 21.5 km/h (13.4 mph), V2n = 11.5 km/h (7.1 mph), v1n = 0.0 km/h, v2n = 45.0 km/h (28.0 mph), Ω1 = 0.55 rad/s, Ω2 = -2.2 rad/s, ω1 = 0.0 rad/s, and ω2 = 0.0 rad/s, the coefficient of restitution for the collision is determined to be approximately e = 0.2. Therefore, e = 0.2 was used in the analysis that was performed to compare the model with the test data. The impulse ratio was given the value μ = μ0. For this collision the moment impulse coefficient is set to e’ = -1, which makes the moment at the contact surface M = 0. Table 8.3 shows the comparison of the experimental data with the results of the analyses. Figure 8.7 shows the spreadsheet results of the analysis for e = 0.2 and μ = μ0. No velocity constraint was used in this analysis. The data in Table 8.3 show good agreement between the experimental results and the analysis. No significant refinement of the analysis was performed to obtain the numbers that were used for comparison. A closer match between the analysis and the experimental data might be obtained with small changes in the location chosen for the impact center. This example presents the analysis of the speed changes associated with an impact Example 8.2 between a tractor semitrailer and a full-size pickup truck where a velocity constraint is used. Consider the impact geometry depicted in Fig. 8.8. The figure shows a pickup truck and a tractor semitrailer at the onset of contact between the front of the pickup truck and the right front wheel of the tractor. Table 8.4 lists the relevant vehicular physical data, initial and final velocities, and the impulse required to impose the velocity constraint determined in the analysis. The impact is modeled for e = 0.2 and μ = μ0. Solution  Note that the tractor semitrailer is located immediately adjacent to a utility pole, as shown in Fig. 8.8. It is assumed that the utility pole does not fail during the impact and therefore acts to restrict the postimpact velocity of the contact point between the semitrailer and the pole to have a zero final velocity in the y-direction. Therefore, a Comparison of the test results with analytical results. Alfa Romeo 164

Test Results

Analysis with e = 0.2 and μ = μ0

ΔVx of CG

-6.5 km/h (-4.0 mph)

-4.4 km/h (-2.7 mph)

ΔVy of CG

21.5 km/h (13.4 mph)

22.8 km/h (14.2 mph)

| ΔV | of CG

22.5 km/h (14.0 mph)

23.2 km/h (14.4 mph)

PDOF

---

74.5°

ΔVx of CG

7.5 km/h (4.7 mph)

9.7 km/h (6.0 mph)

ΔVy of CG

-33.5 km/h (-20.8 mph)

-32.7 km/h (-20.3 mph)

| ΔV | of CG

34.3 km/h (21.3 mph)

34.1 km/h (21.2 mph)

System Preimpact Energy

133.3 kJ (98.3×10 ft-lb)

133.3 kJ (98.3×103 ft-lb)

System Postimpact Energy

71.5 kJ (52.7×103 ft-lb)

75.7 kJ (55.8×103 ft-lb)

System Energy Loss

46.4%

43.2%

PDOF

---

-16.5°

Table 8.3

Opel Ascona C

3

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Figure 8.7 Spreadsheet results for model validation with e = 0.2 and μ = μ0.

velocity constraint will be imposed at this point of the semitrailer as part of the impact analysis. Note that the location of the utility pole, and therefore the point of application of the constraint, was selected here as collinear with the y-axis and the center of mass of the semitrailer. This is for simplicity and to facilitate interpretation of the results. Note (as seen in Fig. 8.9) that the final velocity of the center of mass of the semitrailer in the y-direction, V3y, is zero. This is consistent with the velocity constraint imposed on the semitrailer located at a point collwinear along the y-axis with the center of mass of the semitrailer. For this geometry, both the point of application of the constraint and the center of mass of the semitrailer will have the same postimpact velocity in the y-direction. The output of the program also lists the magnitude of the impulse applied

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Figure 8.8 Relative location and orientation of the vehicles at the onset of contact for Example 8.2.

Table 8.4

Vehicle input and output parameters for Example 8.2. Vehicle A—Body 1 Tractor

Vehicle A—Body 3 Semitrailer

Vehicle B—Body 2 Pickup Truck

Mass

6800.8 kg (466 lb-s2/ft)

11,324.9 kg (776 lb-s2/ft) 2262.1 kg (155 lb-s2/ft)

Yaw Inertia

6779.1 N-m-s2

20,337.3 N-m-s2

4880.9 N-m-s2

(5000 ft-lb-s )

(15,000 ft-lb-s )

(3600 ft-lb-s2)

Initial Velocity (cg)

0.0 km/h

0.0 km/h

80.5 km/h (50 mph)

Final Velocity (cg)

4.6 m/s (10.3 mph)

0.14 m/s (0.31 mph)

11.2 m/s (25.1 mph)

Magnitude of the impulse to meet the velocity constraint

N/A

5874.0 N-s (1320.5 lb-s)

N/A

2

2

by the utility pole to the side of the semitrailer to impose the velocity constraint. In this case the impulse is determined to be 5861.9 N-s (1317.8 lb-s). This example considers a collision between Vehicle A, a 4180 lb (18.6 kN) pickup truck Example 8.3 and Vehicle B, a loaded straight truck weighing 27,000 lb (120.2 kN). Neither vehicle in this example is articulated. This example is presented to illustrate the flexibility of use of these model equations; specifically, the use of an external impulse. The pickup truck is traveling southbound on a divided rural highway. The highway has two lanes in each direction and forms an intersection with a two-lane county road, see Fig. 8.10. The traffic on the county road in both directions is required to stop prior to crossing or entering the highway. As Vehicle B crossed the southbound lanes from west to east

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Figure 8.9 Spreadsheet results from the analysis of Example 8.2.

to enter the northbound lanes of the highway, the front of Vehicle A struck Vehicle B behind the driver side door at the fuel tank. A plan view of the geometry of the relative vehicle position and orientation at the onset of contact is depicted in Fig. 8.11. This figure also shows several of the parameters required for input to the analysis, including the orientation of the x-y and n-t coordinate systems. The objective of the analysis is to determine the preimpact speed of the pickup truck based on the physical evidence. Table 8.5 contains the relevant input physical parameters for the vehicles. The physical evidence at the accident scene indicates that the straight truck, Vehicle B, did not move laterally as a result of the impact. This suggests that the impulses generated by the lateral frictional forces at the tires of Vehicle B during the impact did not exceed the frictional limit. Consequently, the impulse of these forces cannot be neglected in

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Figure 8.10 Diagram of an intersection collision showing the position of the vehicles at impact and at rest for Example 8.3.

the determination of the change in speed of the vehicles. The fact that the heading angle of Vehicle B does not change, or changes very little, as a result of the exchange of momentum implies that any speed change will be essentially along its original direction of travel, with little or no change in speed perpendicular to this direction. This limitation of the velocity change of Vehicle B also affects the velocity changes of Vehicle A. Under circumstances where the weights of the vehicles are of the same order of magnitude and, as a consequence, the external impulses (tire-road frictional impulses) are typically small compared to the intervehicular impulse, the expectation is that the postimpact motion of both vehicles will be in a southeasterly direction. However, the physical evidence indicates that Vehicle B continued on its northeast heading after Vehicle input physical parameters used for Example 8.3 Vehicle A—Pickup Truck

Table 8.5

Vehicle B—Straight Truck

Weight

4180 lb (18.6 kN)

27,000 lb (120.2 kN)

Yaw Moment of Inertia

2643.1 ft-lb-s2 (3583.6 kg-m2)

99,833ft-lb-s2 (135,355.4 kg-m2)

Wheelbase

10.9 ft (3.3 m)

22.6 ft (6.9 m) [1st axle to 3rd axle]

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Figure 8.11 Plan view diagram for Example 8.3 showing the relative positions and orientations of the vehicles at the onset of contact.

impact. The center of mass of Vehicle A also moved in a northeasterly direction after impact while the vehicle rotated counterclockwise. Solution  It is known that Vehicle B started its travel across the southbound lanes of the highway from a stopped position west of the highway, as shown in Fig. 8.10. It traveled approximately 50 ft (15.2 m) from its stopped position to its location at the time of the onset of contact. Assuming an acceleration of 0.075 g over that distance, Vehicle B had a speed of approximately 10.0 mph (4.5 m/s) at the onset of contact. The center of mass of Vehicle A moved northeast about 18.0 ft (5.5 m) from its location at the onset of contact to its location at rest and rotated clockwise slightly more than 90° to its rest orientation. Employing the common-velocity conditions and the knowledge that the heading of Vehicle B changed little or not at all as a result of the impact, the postimpact velocity of Vehicle A at the contact surface will invariably be the velocity of Vehicle B. The velocity change of Vehicle B along its heading is expected to be small, independent of the preimpact velocity of Vehicle A, due to the fact that the preimpact velocities of the vehicles are nearly perpendicular. Therefore, the postimpact velocity of Vehicle A is essentially independent of its preimpact velocity. This renders the (typical) use of postimpact velocity of Vehicle A, as determined by analysis of the postimpact trajectory of the vehicle, inappropriate as a criterion on which to base the acceptability of the analysis. Therefore, an alternative

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criterion is needed to assess the acceptability of the reconstruction. As will be shown, a criterion that utilizes the fact that Vehicle B experiences little or no velocity change perpendicular to its heading during the impact will be used. Physically, this is due to the fact that the lateral intervehicular impulse acting on Vehicle B from the collision is insufficient to overcome the impulse of tire-to-ground friction force. The condition of zero final lateral velocity of Vehicle B could be handled with a velocity constraint applied to the vehicle. However, as was stated previously, the impact model was developed such that the velocity constraint cannot be imposed on the bodies in direct contact (see Fig. 8.2). Therefore, another means must be found to constrain the velocity of Vehicle B to remain along its preimpact heading. Ordinarily, the hitch pin impulses R and Q are unknowns. Alternatively, for the case when a vehicle does not have a semitrailer, the solution strategy of the algebraic equations can be chosen to treat the hitch pin impulse of the vehicle as being known; i.e., as inputs to the problem. As known quantities, the point of application and the direction of the applied impulse become arbitrary. Therefore, the impulse components, Qx and Qy, instead of being used as hitch pin impulses, will be used in this analysis to represent the resultant lateral impulse of the tire-roadway frictional forces of all of the tires of Vehicle B during the impact. Two conditions must be imposed: 1. The externally applied impulse, Q, must be such that it acts perpendicular to the preimpact velocity of Vehicle B (Body 2); that is, Qx/Qy = -vx/vy. 2. The final angular velocity of Body 2 is such that Ω2 ≈ 0 (this can be done by fictitiously making the yaw inertia very large; that is, I2 → ∞). The point of application of Q is chosen to be the center of gravity of Vehicle B (Body 2). The magnitude of the impulse can be computed in a two-step process. First, the weight of the vehicle, 27,000 lb (120.2 kN), is multiplied by the frictional drag, f = 0.5, giving a frictional force of 13,500 lb (60.1 kN). Second, the impulse is obtained by multiplying the frictional force by the time duration of the momentum exchange. In this case the duration is assumed to be 0.15 s. This gives a frictional impulse of 2025 lb-s (9.0 kN-s). This impulse is applied to the center of mass of the vehicle by setting d2Q = 0.0. The line of action of the impulse is perpendicular to the heading of the vehicle prior to impact. Figure 8.12 is the spreadsheet showing the input data and output data of the analysis for the conditions outlined. The components of the impulse Q, Qx and Qy, are entered in the appropriate cells. (The positive direction of the impulse components is shown in Fig. 8.2.) The coefficient of restitution, e, is selected equal to zero as the impact was into the left, side fuel tank of the straight truck with considerable damage and little restitution. Similarly, the relative tangential velocity of the vehicles at the crush surface at separation is assumed to be zero. Therefore, the impulse ratio, μ/μ0, is set to 1, or μ is 100% of μ0. The impulsive moment, M, at the contact surface is assumed to be zero, which is accomplished by setting e’ = -1. Inspection of the vehicles showed no evidence that a secondary impact, a sideslap, between the right side of the pickup and the left side of the straight truck occurred. If a secondary impact had taken place, its severity would need

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Chapter 8

Figure 8.12 Spreadsheet results for Example 8.3.

to be assessed and, if required, the impact would need to be modeled separately from the initial impact to determine the changes in the velocities of the vehicles associated with this impact. The final results were determined by successively changing the preimpact speed of Vehicle A, while keeping the preimpact velocity of Vehicle B constant at 10 mph (4.5 m/s), until the postimpact direction of motion of Vehicle B remains essentially unchanged. In addition, the direction of the postimpact velocity of the center of mass of Vehicle A should be similar to the direction of the velocity of Vehicle B. For a preimpact speed of

216

Collisions of Articulated Vehicles, Impulse-Momentum Theory

Vehicle B of 10 mph (4.5 m/s), this condition is met with a preimpact speed of Vehicle A of about 13.0 mph (5.9 m/s). For this speed, the direction of the velocity of Vehicle B changes by about 2°, from a preimpact value of 25.4° to postimpact value of 23.4°, relative to the x-axis. This is a relatively small directional change that satisfies the requirement dictated by the physical evidence. The postimpact direction of the final velocity of the center of mass of Vehicle A is 25.3°. Thus, the direction of the postimpact velocities of the centers of mass of the two vehicles are nearly the same, consistent with the physical evidence. The speed of Vehicle B changes from 10 mph (4.5 m/s) to 9.1 mph (4.1 m/s), a relatively small change, as would be expected because of large difference in the masses of the two vehicles and the geometry of the collision. Earlier attempts to reconstruct this collision without the consideration of external impulses acting on Vehicle B proved extremely difficult, if not impossible. Without the inclusion of the external impulse, the direction of the postimpact velocity of Vehicle B invariably deviated significantly from the direction of its preimpact velocity other than for very small speeds of Vehicle A. The analysis presented above gives the speed of Vehicle A that meets the velocity threshold criterion that was established for an acceptable speed reconstruction. However, speeds of Vehicle A less than the speed at the threshold can also produce acceptable results. In this way, the analysis presented above yields a maximum speed of Vehicle A for the conditions. Therefore, the reconstruction gives an upper bound of the speed of Vehicle A rather than an estimate of the specific speed, or range of speed, at which the vehicle was traveling. Several areas of uncertainty could be evaluated in the reconstruction of this accident. Most notably, a range of acceleration of Vehicle B during entry into the intersection, rather than a single value, could be selected in the determination of the preimpact speed of the straight truck. This range of preimpact velocity, in turn, will lead to a range of the preimpact speed of the pickup truck. The magnitude of the frictional impulse from the tires of the straight truck is a function of the frictional drag and the duration of impulse. The frictional drag coefficient, f = 0.5, could have been given an appropriate range to reflect uncertainty. Similarly, a range of the duration of the intervehicular impulse could have been chosen. The time duration of the impulse selected in the calculations above was 0.15 s. Such an uncertainty analysis is not presented in this example. The above example illustrates the application of the articulated vehicle impact model to a collision in which neither vehicle was articulated. The example was structured to show that creative solution strategies of the system equations can be used to solve difficult, but commonly encountered, accident reconstruction problems. The next example reconstructs the preimpact speed of a sport utility vehicle (SUV) that collided with a recreational vehicle (RV) towing a trailer. This example makes use of the capability of adding an external impulse to one of the tow vehicles. While the RV is towing a trailer at the start of the collision, the two bodies separate during the collision and are found unattached after the accident. The articulated vehicle impact model can handle this situation through the use of externally applied impulses, as will be shown.

217

Chapter 8

Example 8.4 Consider a collision between the front of Vehicle B, a 17,592.6 N (3955 lb) SUV (single

vehicle, Body 2) into the right side of Vehicle A (an articulated vehicle), with a 73,840 N (16,600 lb) RV tow vehicle (Body 1). Figure 8.13 shows the location and orientation of the vehicles at the time of impact and at rest. Vehicle B carried four passengers and Vehicle A towed a custom semitrailer (Body 3) which weighed 4,448.2 N (1000 lb) empty. The semitrailer carried an 11,120.5 N (2500 lb) car, which is shown on the semitrailer in the preimpact position only; the car was still attached securely to the semitrailer after the accident. Vehicle A was backing out of a north-facing driveway to the north and east onto a two-lane road. The driver of Vehicle A was intending to eventually travel westbound on the road.

Figure 8.13 Accident diagram showing the positions of the vehicles at impact and at rest for Example 8.4

The physical evidence indicates that at the time of the collision, Vehicle A was oriented at about a 75° angle from the edge of the roadway, with the semitrailer at approximately a 124° angle relative to the longitudinal axis of the RV. The RV was positioned such that it blocked about 75% of the two-lane roadway (including the south shoulder), with the semitrailer blocking the remaining 25% (including the north shoulder). The SUV approached the RV from the west, impacting it in front of the right rear axle, as shown in Fig. 8.13. Witness accounts indicate that no evasive maneuvers or brake lights of the SUV were observed prior to impact. The SUV engaged the rear axle of Body 1 during impact, displacing the axle laterally, as shown in its rest position in Fig. 8.13. During the collision the hitch on the Body 3 (the semitrailer) released from the ball on Body 1, subsequently engaging and breaking the safety chain. The semitrailer was released from the RV (with the car remaining attached to the semitrailer). Relatively small secondary contact occurred between the left rear corner of Body 2 and the right side of

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Collisions of Articulated Vehicles, Impulse-Momentum Theory

the semitrailer, Body 3. The objective of this analysis is to determine the preimpact speed of the SUV, based on the physical evidence. Table 8.6 contains relevant input data for the vehicles. Figure 8.14 shows the vehicles in their relative locations and orientations at the onset of contact, with several of the physical dimensions required for the analysis shown on the figure. The approach taken in the analysis of this crash is first to select postimpact trajectory conditions of one or more of the vehicles to use as postimpact conditions for the impact analysis. The postimpact motion of Vehicle A (rather than the postimpact motion of Vehicle B) is used for this purpose, as the secondary impact between Body 2 and Body 3 makes the postimpact motion of Vehicle B more complex to analyze.

Table 8.6

Vehicle input physical parameters for Example 8.4. Vehicle A—Body 1 RV

Vehicle A—Body 3 Semitrailer w/ Car

Vehicle B—Body 2 SUV

Weight

73.8 kN (16,600 lb)

15.6 kN (3500 lb)

19.4 kN (4355 lb)

Yaw Moment

41,942.2 kg-m

3066.9 kg-m

3702.3 kg-m2

of Inertia

(30,935 ft-lb-s2)

(2262.0 ft-lb-s2)

(2730.7 ft-lb-s2)

Wheelbase

5.7 m (18.66 ft)

4.5 m (14.92 ft)

2.3 m (7.67 ft)

2

2

[hitch to midpoint between axles]

Figure 8.14 Diagram showing the Vehicle and Body designations and several reconstruction parameters.

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Chapter 8

After impact, Body 1 rotated clockwise approximately 90°, and the mass center moved eastward about 5.5 m (18.0 ft). Analysis of this motion to determine the velocity of the mass center of Body 1 alone (without an attached semitrailer) after impact was performed using vehicle dynamic simulation. The evidence indicates that there was little or no postimpact roll-out of the vehicle after the rotational motion ended. A frictional drag coefficient of f = 0.5 was used in the postimpact trajectory analysis, which yields a postimpact speed of Body 1 of about 6.0 m/s (13.5 mph). It is known that the driver managed to maintain brake application throughout the entire event, which is consistent with no rollout of the vehicle after impact. With the postimpact conditions of Body 1 established, the reconstruction can now proceed to the analysis of the articulated vehicle impact between Vehicles A and B. The analysis is performed for three different cases for comparison purposes. The cases consider impact between the Vehicles A and B under the following conditions: A. with no semitrailer attached to Body 1 (the RV) B. with the semitrailer attached to Body 1 throughout the entire contact duration C. with no semitrailer attached to Body 1, but with an external impulse applied at the hitch location of Body 1 chosen to represent the impulse applied to Body 1 prior to the separation of Body 2 from Body 1 Solution  Analysis of these three cases serves as a means to compare the results of a collision under the various conditions. In particular, Cases A and B serve as bounds to the actual conditions. It is anticipated that the preimpact velocity of Vehicle B determined in Case A will yield a lower preimpact velocity than Case B due to the presence of the semitrailer. Furthermore, it is anticipated that the introduction of appropriate external impulses at the hitch location in Case C will yield a preimpact speed of Vehicle B between the speeds of Cases A and B. Moreover, the magnitude and direction of the impulses at the hitch for Vehicle A in Case B will assist in the determination of the appropriate magnitude and direction of the external impulses to impose on the system for Case C. In all cases, Body 1, the RV, is assumed to be stationary prior to impact, which is consistent with witness and participant accounts. The reconstruction criterion for the impact analysis is a reasonable match of the postimpact speed of the Body 1 predicted by the impact analysis to the postimpact speed determined by the vehicle dynamic simulation. It is assumed that the common-velocity conditions apply at the impact center of the intervehicular contact surface; i.e., the relative tangential motion between the two vehicles stops prior to separation (μ = μ0), and that the coefficient of restitution, e, is zero. The impulsive moment, M, at the contact surface is assumed to be zero (e’ = -1). Sufficient information is available at this point to run Cases A and B. Case C, however, requires that the magnitude and direction of the externally applied impulse be determined. It is known that the ball released from the hitch and the safety chain between the semitrailer and the hitch on the RV picked up the load and fractured

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Collisions of Articulated Vehicles, Impulse-Momentum Theory

during the collision. The chain was a size ½, closed-link crane chain. This size chain is listed as having an approximate braking strain [sic] of 80,000 N (18,000 pounds) [8.10]. Assuming that this is the actual failure force, or load, the impulse at the hitch can be estimated. Assuming that the impulse duration is 0.15 s and that the load increases from zero to 80,000 N (18,000 lb) over that time, the impulse associated with the load is 6000 N-s (1350 lb-s). Using this information, an analysis of Case C can be carried out. The direction of the impulse used in the analysis of Case C is the same as the direction of the impulse that is calculated in Case B where the trailer remains attached to the hitch through the entirety of the impulse. The results from the analysis of each of the three cases are summarized in Table 8.7. The Appendix, Section 8.5, contains the input and output data sheets from the spreadsheet runs for the three cases. Examination of the data shows that the impulse calculated for the handbook failure load value of the chain is less than the impulse that is calculated at the hitch location if the hitch does not fail, Case B. The results shown in Table 8.7 indicate that, as expected, Case A produced the lowest preimpact speed of Vehicle B, whereas Case B gave the highest preimpact speed of Vehicle B. Also as expected, the speed for Case C was between the two other cases. It is interesting to note that the difference between the preimpact speed of Vehicle B reconstructed for Cases A and B is almost 4.5 m/s (10.0 mph). The preimpact speed predicted by Case C is a little less than half of the preimpact speed for the two other cases. Several areas of uncertainty in the analysis would ordinarily be addressed to complete the reconstruction. These uncertainties lead to a range of preimpact speeds of Vehicle B as opposed to a singular value, as was done in the example. One of the areas of uncertainty in the approach taken here is use of the duration of the collision in determining the externally applied impulse at the hitch of Vehicle A. The duration used in the calculations was 0.15 s, but an appropriate range representing uncertainty should be selected for the analysis. A range of frictional drag could also have been used in the reconstruction of

Table 8.7

Summary of the results for Example 8.4. Case A No Semitrailer

Case B Semitrailer remains attached to the RV

Case C Semitrailer replaced by external impulse

Initial Velocity—SUV

24.5 m/s (54.9 mph)

28.7 m/s (64.2 mph)

25.1 m/s (56.2 mph)

Final Velocity—SUV

8.9 m/s (19.9 mph)

6.5 m/s (14.5 mph)

9.0 m/s (20.1 mph)

Final Velocity—RV

4.1 m/s (9.2 mph)

4.1 m/s (9.2 mph )

4.1 m/s (9.2 mph)

at -1.0°

at -6.2°

at -3.3°

Final Angular Velocity—RV

-107.2°/s

-53.9 °/s

-111.1 °/s

Magnitude of the impulse at the hitch of Vehicle A

0.0 kN-s (0.0 lb-s)

13.5 kN-s (3043.8 lb-s) [output of the analysis]

6.0 kN-s (1350 lb-s) [input to the analysis]

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the postimpact motion of Body 1. Other areas of uncertainty may also be identified and all serve to create a range of preimpact speed for Vehicle B. It should be noted that the last two examples illustrate only two applications of the articulated vehicle impact model. Different combinations of the number of bodies and different combinations of choices of known and unknown impulses and final velocity constraints create flexibility in the model that can be exploited to cover a wide variety of different applications. The model presented here reduces to the single body planar impact mechanics model of Chapter 6 when the semitrailers are eliminated, no external impulses are applied, and no moment is present at the contact surface.

8.5 Appendix: Data Sheets for Example 8.4 Figure 8A.1 Spreadsheet results for Case A of Example 8.4.

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Collisions of Articulated Vehicles, Impulse-Momentum Theory

Figure 8A.2 Spreadsheet results for Case B of Example 8.4.

223

Chapter 8

Figure 8A.3 Spreadsheet results for Case C of Example 8.4.

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CHAPTER

9 Crush Energy and DV

9.1 Introduction An early study [9.1] of data from frontal, fixed rigid barrier tests (at speeds above about 20 mph, 32 km/h) noted a nearly linear relationship between test speed and the amount of residual crush. An example of the linear data trend is shown in Fig. 9.1 [9.2]. At speeds higher than about 30 mph (48 km/h) there is typically little restitution, and the energy lost in a barrier collision is close to the entire kinetic energy of approach. Because kinetic energy is proportional to the square of speed, the trend noted in the early study implies that the square root of the kinetic energy is approximately linearly proportional to the residual crush. The crush energy, EC , per unit width, w, of a crushed vehicle can then be expressed as a linear function of crush, C [9.3]:



2 EC = d 0 + d1C w

(9.1)

where the residual crush, C, is measured normal (perpendicular) to, and from, the nominal undeformed vehicle surface. The constants d0 and d1 are called the crush stiffness coefficients and are determined experimentally from staged barrier impact tests. They differ from vehicle to vehicle and between the front, side, and rear of each vehicle. They can differ for different regions along a vehicle, although this is usually not taken into account, primarily because most existing methods do not accommodate such variations and available test data generally do not accommodate determination of the coefficients by regions. Such variations in stiffness are considered elsewhere [9.4].

225

Chapter 9

Figure 9.1 Variation of residual crush with impact speed, full frontal rigid barrier tests, 1971-72 full-sized GM vehicles (used with permission from Cheng, et al. [9.2]).

Example 9.1 A 1972 Buick LeSabre has a curb weight of 4500 lb (20.0 kN) and a width of 79 in.

(2.0 m). A direct frontal collision into a flat, rigid concrete abutment produces a fairly uniform side-to-side residual crush of 42 in. (1.07 m). The rebound is negligible. A. Use the straight line equation from Fig. 9.1 to estimate the impact speed. B. Use the equation from Fig. 9.1, and the answer to part A, to determine the coefficients d0 and d1 in Eq. 9.1.

Solution A  For a uniform residual crush of 42 in., the equation from Fig. 9.1 gives

V = 6.85 + 0.88 × 42 = 43.8 mph = 64.2 ft / s = 70.5 km / h Solution B  From the equation in Fig. 9.1, C = 0 occurs at a speed of V = 6.85 mph (11.0 km/h) when the kinetic energy is 7059.9 ft-lb (9.57 kJ). Using Eq. 9.1 with C = 0 gives

d0 =

226

2(7059.9) = 46.31 lb1/2 = 97.7 N 1/2 6.58

Crush Energy and DV

From A, at 43.8 mph (70.5 km/h) the kinetic energy loss is 288,595 ft-lb (391.3 kJ). Solving Eq. 9.1 for d1 gives

d1 =

 1  2(288595)  1  2 EC − d0  = − 31.58     3.5  6.58 C w   

= 71.39 lb1/2 / ft = 4.59 N 1/2 / cm Vehicles do not always collide head-on into flat rigid barriers, so the above relationship between energy (and speed) and residual crush is not generally applicable for accident reconstruction unless it can be adapted to nonuniform crush profiles and vehicle-tovehicle collisions. This relationship was exploited and a method was developed for the National Highway Traffic Safety Administration [9.5] called CRASH3 (Calspan Reconstruction of Accident Speeds on the Highway, version 3). CRASH3 was developed for collisions of light vehicles. The measurement process of residual crush is intended to follow a specific measurement protocol [9.6] for measuring residual crush. The protocol typically uses a series of six crush measurements spaced equally over a deformed surface of a damaged vehicle at a uniform height from the ground at a level corresponding to where vehicles are designed to resist and develop controlled crush forces. This is usually at front and rear bumper heights. In the theoretical development, residual crush is usually assumed to be uniform from top to bottom of the vehicle surface. This is rarely the case in practice, particularly for damage to the side of the vehicle. For side damage, measurements are made at the vehicle’s structure near floor sill height, but if the damage is significantly greater above the sill, incorporation of the crush above the sill is often averaged with the crush at the sill level to improve correlation with crash tests [9.7]. Residual crush measurements are made over the full lateral extent (beginning to end) of damage. This includes direct contact damage and induced damage. Induced damage is residual deformation that occurs, not due to direct vehicle-to-vehicle contact, but because of contact forces at adjacent areas. If the profile of the initial undeformed body shape is curved (such as looking at a bumper from above), it is important to measure each crush value from the corresponding curved, original, undeformed surface of the vehicle. CRASH3 specifically is not applicable to underride situations where contact is not at or near bumper height. Some limited work has been done in evaluating energy loss in underride/override collisions [9.8, 9.9, 9.10]. CRASH3 is not typically applicable to collisions with narrow objects, although it has been successfully applied if certain corrections are made [9.7, 9.11, 9.12, 9.13, 9.14]. CRASH3 was originally formulated to be used with four, or even two, measurements for reasonably uniform damage profiles. The development presented here uses six measurements at equal lateral intervals. Figure 9.2 illustrates a series of six crush measurements made along damage of extent L of the front and the side of a vehicle where C1 through C6 are the measured distances from the undeformed, as manufactured, surface of the vehicle to the point of the crushed surface.

227

Chapter 9

Figure 9.2 Example side and front (or rear) six-point crush measurements protocol.

Prasad [9.3, 9.15, 9.16] shows that if the crush profile is approximated by linear segments of crush between the six equally spaced points, and Eq. 9.1 is at least approximately true, then the crush energy can be computed from:

EC = d 02 K1 + d 0 d1 K 2 + d12 K 3

(9.2)

K1 = L / 2

(9.3)

where d1 =

and

   1  2 EC 1  2( 288595) − 31.58 − d0  =   3.5 C  w 6.58   

= 71.39 lb1/ 2 / ft = 4.59 N 1/ 2 / cm



(9.4)

K 3 = L[C12 + 2(C22 + C32 + C42 + C52 ) + C62 +

C1C2 + C2C3 + C3C4 + C4C5 + C5C6 ] / 30



(9.5)

Equations 9.2 through 9.5 form the CRASH3 crush energy algorithm. Figure 9.3 shows a collision between two cars where the collinear arrows represent the equal and opposite impulses of the forces acting over the common intervehicular crush surface during the collision. This line of action of the impulse measured with respect to the heading of each vehicle is generally referred to as the principal direction of force (PDOF). The resultant impulse is the temporal and spatial average of the intervehicular force, varying in magnitude and direction as a function of time that exists across the contact surface while the vehicles are in contact. It has a long history of being called a

228

Crush Energy and DV

Figure 9.3 Collision orientation with arrows representing the equal and opposite impulse (PDOF) and distances h1 and h2.

direction of force. The figure also shows distances, h1 and h2, each measured from the center of gravity of the vehicle perpendicularly to the line of action of the PDOF. In the original version of CRASH [9.5], a somewhat different model was used where the intervehicular crush force per unit width is assumed to have the form

FC / w = A + BC

(9.6)

where the crush, C = C(x), was taken to vary piece-wise linearly with the measurement positions, x, along the crush surface. The total crush energy, EC , is computed by integrating the energy per unit width, w, over the entire crush width, L, to find the work of FC . The result is

EC / w = AC + BC 2 / 2 + A2 / 2 B

(9.7)

This produces a model where A and B are the crush stiffness coefficients and by using a set of equations different from Eqs. 9.2 to 9.5, the crush energy can be calculated from residual crush measurements. Prasad [9.16] showed that the A and B model (Eq. 9.6) and the d0 and d1 model (Eq. 9.1) are equivalent (compare coefficients of Eq. 9.1 and Eq. 9.7) where

d1 = B

(9.8)

d0 = A / B

(9.9)

and

229

Chapter 9

Equations 9.6 and 9.7 are given for information but are not used here. To complete the CRASH3 model, crush energy is related to the velocity change, ΔVi, i= 1, 2, of each vehicle using basic impact mechanics (see Chapters 6 and 7). In the CRASH3 ΔV model, four assumptions are usually made to simplify the results. These assumptions are: 1. The crush energy, EC , is equal to the impact kinetic energy loss, TL . 2. The collision is perfectly inelastic; i.e., there is no rebound or restitution of the vehicles at and perpendicular to the crush surface (e = 0). 3. Relative sliding velocity of the vehicles along (tangent to) the crush surface ends (becomes zero) before or at the time the vehicles separate (μ = μ0). 4. Forces external to the colliding vehicles (including tire-ground forces) are negligibly small compared to the level of the force acting on the intervehicular crush surface. Together, assumptions 2 and 3 are often referred to as the common-velocity conditions and often are valid assumptions except for low-speed collisions (where significant restitution can be present) and/or sideswipe collisions (where relative tangential sliding continues through separation). Assumption 4 ensures that momentum is conserved. Assumptions 2 and 3 are not always exactly true, even for high-speed collisions. Consequently, the following equation for ΔVi is not as general as is needed in many crash reconstructions. This is discussed in more detail below. According to the original CRASH3 formulation, and for the four assumptions above, momentum is conserved, leading to

mi DVi = 2me EC , i = 1, 2



(9.10)

where in this equation, EC is the total crush energy of both vehicles. The quantity i = 1 represents Vehicle 1, and i = 2 represents Vehicle 2; also,



me =

γ 1m1γ 2 m2 γ 1m1 + γ 2 m2

(9.11)

ki2 ki2 + hi2

(9.12)

where

γi =

The quantity ki is the yaw radius of gyration of vehicle i, and hi is the perpendicular distance illustrated in Fig. 9.3 for Vehicles 1 and 2. Note that radius of gyration is the square root of the mass moment of inertia divided by mass; that is,

ki =

230

Ii mi



(9.13)

Crush Energy and DV

where Ii is the mass moment of inertia about the yaw axis of vehicle i and mi is the mass of vehicle i. Equation 9.10 calculates ΔV directly from the residual crush energy, EC (and the mass properties of the vehicle and the PDOF). This method is based on the residual crush measured perpendicular to the undeformed surfaces and the common-velocity conditions. It does not take into account the energy loss associated with the change in relative tangential velocities. That is, the energy dissipated due to sliding of one vehicle relative to the other vehicle and due to tangential deformation and entanglement over the crush surfaces of the vehicle is not included in the above equations for the energy loss, EC . In some collisions, this energy loss can be significant and must not be ignored. The original CRASH3 method suggests the use of a correction factor to address the energy loss associated with tangential effects. However, as it is defined, the correction is not theoretically sound, and its application is subjective. Moreover, this aspect of the method has never been validated. A more accurate and rigorous assessment of the energy loss associated with tangential impulses is covered in Chapters 6 and 7 through the use of the impulse ratio and its critical value. In particular, compare Eq. 9.10 above with the more rigorous planar impact mechanics version, Eq. 6.65. There, it is shown that impulse and momentum techniques can be used to independently estimate the energy dissipated by the normal crush (the same energy as determined above with the CRASH3 method) and the additional energy dissipated by tangential effects. Another advantage of using the methods of Chapters 6 and 7 for calculating ΔV from crush energy is that, in the CRASH3 approach, the direction of the PDOF (and the location of application of the impulse) must be visually estimated from the damaged surfaces (to determine h1 and h2). Without a great deal of experience and a knowledge of impulse mechanics, this task can be difficult. On the other hand, the PDOF (direction of impulse) is automatically calculated using methods covered in Chapters 6 and 7 and need not be estimated in advance. The selection of the location of application of the impulse in the use of the CRASH3 method is replaced by the selection of the impact center, C, in planar impact mechanics. Due to these limitations of the CRASH3 method in calculating ΔV, it is recommended that the energy associated with the normal crush should be calculated using Eq. 9.2, but the ΔV of each vehicle should be computed not with Eq. 9.10, but with planar impact mechanics due to its rigor in handling the energy loss. There is another significant advantage to using this approach to reconstruct the ΔV of the vehicles. Note that the CRASH3 method is based on the common velocity conditions (e = 0 and μ = μ0). These conditions come into play only in the computation of the ΔV, Eq. 9.10 (not in the computation of EC). The coefficient of restitution may not be zero (one of the two common velocity conditions) for some collisions (see Tables 6.1 and 6.2). By using planar impact mechanics to calculate the ΔV of the vehicles, nonzero restitution can easily be accommodated. In addition, the tangential common velocity condition assumption does not apply to sideswipe collisions (μ < μ0). When there is measurable crush, yet the physical evidence suggests that relative sliding in the tangential direction along the

231

Chapter 9

crush surface continued through separation, planar impact mechanics can be used, whereas the CRASH3 method cannot. The calculations carried out in the original CRASH method and commercial versions of CRASH31 are much more extensive than the equations presented above, because they also include a postimpact spinout model. That is, the energy loss due to the motion of the vehicles from impact to rest is estimated using accident data when available. It is strongly recommended that vehicle dynamic simulation methods (see Chapter 13) should be used to reconstruct postimpact vehicle motion. These simulations methods are much more accurate, particularly under special conditions such as a single locked wheel of the vehicle, motion across surfaces each with different frictional drag characteristics, etc. In summary, the equations for calculating the crush energy loss from residual crush (uncorrected for tangential effects), Eqs. 9.2 to Eq. 9.5, are algebraic and can be calculated conveniently using a spreadsheet. Once the normal crush energy loss is calculated, the ΔV for each vehicle can be calculated using planar impact mechanics, which automatically incorporates the tangential energy loss and can handle nonzero restitution. This reconstruction method is illustrated with an example. Example 9.2 Assume a collision of two vehicles with a configuration as shown in Figure 9.4. Vehicle

1 has a weight at impact of W1 = 3696 lb (16.4 kN) and Vehicle 2 has a weight of W2 = 3413 lb (15.18 kN). Crush measurements, crush widths, crush stiffness coefficients for these vehicles and other data are given in Table 9.1. It is assumed for this example that the common velocity conditions (e = 0 and μ = μ0) apply.

Figure 9.4 Collision geometry for Example 9.2.

1. Commercially available versions of CRASH3 include EDCRASH, Engineering Dynamics Corporation, 8625 SW Cascade Blvd., Suite 200, Beaverton, OR 97008-7100, www.edccorp.com; CRASHEX, Fonda Engineering Associates, 649 S Henderson Rd., Suite C307, King of Prussia PA 19406, www.crashex.com; M-CRASH, McHenry Software, PO Box 5694, Cary, NC 27512, www. mchenrysoftware.com.

232

Crush Energy and DV

Table 9.1

Input Data for Example 9.2. Vehicle 1

Vehicle 2

Weight: 3696 lb (16.44 kN)

Weight: 3413 lb (15.18 kN)

Yaw Moment of Inertia: 2437 ft-lb-s2 (3304 kg-m2)

Yaw Moment of Inertia: 2165 ft-lb-s2 (2935 kg-m2)

Wheelbase: 8.9 ft (2.7 m)

Wheelbase: 8.6 ft (2.6 m)

h1 = 2.39 ft (0.73 m) C1

C2

C3

h2 = 1.80 ft (0.55 m) C4

C5

C6

C1

C2

C3

C4

C5

C6

21.3

20.3

13.8

11.3

5.3

4.8, in.

3.0

15.8

13.5

6.7

3.3

5.0, in.

54.1

51.6

35.1

28.7

13.5

12.2, cm

7.6

40.1

34.3

17.0

8.4

12.7, cm

Crush Width: L = 65 in. (1.65 m)

Crush Width: L = 84 in. (2.13 m)

Crush Stiffness Coefficients:   A = 587.4 N/cm, B = 63.3 N/cm2   d0 = 35 lb½, d1 = 115 lb½ ∕ft

Crush Stiffness Coefficients:   A = 339.3 N/cm, B = 28.8 N/cm2   d0 = 30 lb½, d1 = 77.5 lb½ ∕ft

Solution  A spreadsheet used to calculate the crush energy and CRASH3 ΔV values for these conditions and data is shown in Fig. 9.5. It shows that (without correction for the tangential impulse) the total crush energy of both vehicles is EC = 103864 ft-lb (140.8 kJ). It also gives a ΔV value for each vehicle calculated with Eq. 9.10. Tangential effects can be taken into account using planar impact mechanics covered in Chapters 6 and 7. For this collision, the impact parameters for Vehicle 1 are d1 = 6.3 ft (1.9 m), θ1 = 0°, and φ1 = -22.4° and for Vehicle 2 are d2 = 3.8 ft (1.2 m), θ2 = 67°, and φ2 = -39.0°. The impact surface is oriented along the passenger side of Vehicle 2, so Γ = 67°. The commonvelocity conditions are applied by setting e = 0 and μ = μ0. Additional information available from an investigation of this accident shows that Vehicle 2 had stopped at a stop sign and accelerated to a speed of approximately 10 mph in the direction of its preimpact heading. Thus, v2x = 5.73 ft/s (1.75 m/s) and v2y = 13.50 ft/s (4.12 m/s). Also, it is assumed that both vehicles had negligible preimpact angular velocities; that is, ω1 = ω2 = 0. Finally, with these conditions, planar impact mechanics (in spreadsheet form) can be used in an iterative fashion (or using the optimization feature of the spreadsheet), changing the preimpact speed of Vehicle 1 to achieve the objective of reaching a value of normal (crush) energy loss, EC = 103864 ft-lb (140.8 kJ). The results of the planar impact mechanics analysis are given in Fig. 9.6 with a preimpact speed for Vehicle 2 of 10 mph. It shows that for the preimpact speed of Vehicle 1 of 48.5 mph (78.1 km/h), the crush energy (for both vehicles) is 103864 ft-lb (140.8 kJ). In addition, it shows that the total energy loss is approximately TL = 134403 ft-lb (182.3 kJ). The difference in these energy losses is the energy lost due to tangential effects, nearly 23% of the total energy loss. Finally, the velocity changes of the two vehicles in this collision are ΔV1 = 20.7 mph (33.3 km/h) and ΔV2 = 22.4 mph (36.0 km/h). These are greater than the uncorrected ΔV values of ΔV1 = 18.3 mph (29.5 km/h) and ΔV2 = 19.8 mph (31.9 km/h) given directly from the CRASH3 crush energy equations (see Fig 9.5).

233

Chapter 9

Figure 9.5 Results of crush energy calculation using CRASH3 algorithm for Example 9.2.

234

Crush Energy and DV

Figure 9.6 Results of planar impact mechanics calculation using Example 9.2, with the same normal crush energy as in Fig. 9.5.

235

Chapter 9

9.2 Crush Stiffness Coefficients Based on Average Crush from Rigid Barrier Tests One of the most common test methods used to determine the frontal crush stiffness coefficients, d0 and d1, is from residual crush measurements of a vehicle crashed into a fixed rigid barrier. This section shows how the crush stiffness coefficients can be determined from six measurements of the residual crush. Example 1 of this chapter lays a foundation for the method. The first task is to establish a threshold speed below which no measurable residual crush occurs; that is, a speed for which C = 0. Note that this does not necessarily mean no visible damage, but rather no significant residual crush. Often this speed is chosen to lie in the range of 5 to 7 mph (8.0 to 11.3 km/h). Once the kinetic energy and the crush width, w, are known, then with C set equal to zero (for no residual crush), Eq. 9.1 is used to solve for d0. The coefficient d1 can then be calculated from experimental barrier data (known energy loss, crush width, and six measurements of the residual crush) using a modified form of Eq. 9.1:

d1 =

1 Cavg

 2 EC  − d 0    w 

(9.14)

where Cavg is computed with the six equally spaced measurements of the residual crush, using the weighted average (weighted according to Eq. 9.4):

γi =

ki2 ki2 + hi2



(9.15)

Example 9.3 Experimental data [9.17] gives frontal crush values of C1 = 17.8 in. (45.2 cm), C2 = 18.7

in. (47.5 cm), C3 = 21.1 in. (53.6 cm), C4 = 20.6 in. (52.3 cm), C5 = 18.2 in. (46.2), C6 = 17.4 in. (44.2 cm), for a 1997 Honda Accord with a test weight of 3293 lb (14,648 N) and a test speed of 35.0 mph (56.3 km/h) into a fixed, rigid barrier. The crush width is w = 65.6 in. (1.67 m). Determine the crush stiffness coefficients, d0 and d1. Solution  The corresponding energy loss, assuming zero restitution, is the initial kinetic energy of the vehicle, EC = 134,868 ft-lb (182.9 kJ). Using Eq. 9.15 gives Cavg = 19.2 in. (48.8 cm). For C = 0 and a zero crush speed of 5 mph (8.1 km/h), Eq. 9.1 gives d0 = 31.7 lb½ (66.9 N½). Equation 9.12 gives d1 = 119.0 lb½ ∕ft (7.6 N½ ∕cm).

To check if these values of crush stiffness coefficients are correct, a spreadsheet set up to calculate crush energy loss can be used to determine the energy loss for the measured crush values and the computed d0 and d1. (To represent a rigid barrier, the parameters of Vehicle 2 are given exaggerated values such as a crush width of w = 0 and a weight of 999,999 lb.) Placing d0, d1, w, and C1 through C6 into the crush energy spreadsheet gives an energy loss of 135700 ft-lb (184.0 kJ). This is a difference of approximately 0.6 % from EC = 134868 ft-lb (182.9 kJ) and validates the values of d0 and d1 as representative

236

Crush Energy and DV

of a 1997 Honda Accord for a frontal crush analysis. If desired, the value of d1 can be changed to achieve an exact match with the initial energy of the vehicle. Rigid barrier tests of vehicles at high speed do not always have a coefficient of restitution exactly equal to zero. If a nonzero test value is known, the energy loss is reduced. This effect can be taken into account when calculating the stiffness coefficients. If, for the above example, it is known that during the barrier test, the Honda Accord rebounded due to a value of e = 0.1, then the energy lost in the impact is 90% of the initial kinetic energy of the vehicle. The crush stiffness coefficients would be d0 = 31.7 lb½ (66.9 N½) and d1 = 111.9 lb½ ∕ft (7.2 N½ ∕cm). (Note that d0 does not change because its value is independent of restitution.) Additionally, if multiple barrier tests are available for a given model of vehicle, the crush stiffness coefficients, d0 and d1, from these multiple tests can be statistically averaged to obtain information about the variation of these parameters. This information can be used (with Monte Carlo analysis, for example) to evaluate the uncertainty of the energy loss and ΔV for each of the vehicles. Another example, presented here, illustrates the method outlined in this chapter that uses the crush energy determined from the residual crush measured on a vehicle to reconstruct the speed of a vehicle at the time of impact. The example includes the use of a non-zero coefficient of restitution based on information from experimental data. A 2002 Chevrolet Cavalier four-door sedan with a weight of 3279 lb (14591 N) slides off Example 9.4 an icy roadway and strikes the side of a concrete building (adjacent to rigid protrusion of the building) in the orientation shown in Fig. 9.7 with negligible preimpact rotational velocity. Crush measurements, crush widths, and crush stiffness coefficients for the vehicle are given in Table 9.2. Solution  The spreadsheet used to calculate the crush energy and CRASH3 ΔV values for these values, conditions, and data is shown in Fig. 9.8. It shows (without correction for the tangential impulse) that the total crush energy of Input Data for Example 9.4. Table 9.2 the vehicle is EC = 124976.4 ft-lb (169.4 kJ). It also gives Vehicle 1—Chevrolet Cavalier a ΔV value for the vehicle, Weight: 3279 lb (14.59 kN) calculated with Eq. 9.10, of Yaw Moment of Inertia, 1489.51 ft-lb-s2 (2.03kN-m-s2) 27.2 mph (42.8 km/h). Wheelbase: 8.7 ft (2.7 m) The tangential effects, and the reconstruction of the speed of the vehicle, are taken into account using planar impact mechanics. For this collision, the impact parameters for Vehicle 1 are d1 = 6.29 ft (1.9 m), θ1 = -60°,

h1 = 3.14 ft (0.96 m) C1

C2

C3

C4

C5

C6

8.3

13.7

16.3

15.0

11.6

5.0, in.

21.1

34.8

41.4

38.1

29.5

12.7, cm

Crush Width: L = 60 in. (1.52 m) Crush Stiffness Coefficients:   A = 1146.91 N/cm, B = 126.93 N/cm2   d0 = 48.27 lb½, d1 = 162.82 lb½ ∕ft

237

Chapter 9

Figure 9.7 Collision geometry for Example 9.4.

and φ1 = 10.0°. Vehicle 2 is used to represent the rigid wall and is given an exaggerated mass of 99,999,999 lb-s2/ft such that the ΔV of Vehicle 2 is zero. The impact surface is oriented along the front of Vehicle 1, so Γ = -60E. In this case the presence of the protrusion of the building restricts the motion of the front of the vehicle along the face of the building such that relative sliding ends before separation. Thus, it is known that for this set of circumstances, μ = μ0. The restitution for the barrier test used to calculate the crush stiffness parameters, d0 and d1, for Vehicle 1 was found to be e = 0.1. Therefore, this value is used here. With these conditions, planar impact mechanics (in spreadsheet form) can be used in an iterative fashion (or using the optimization feature of the spreadsheet) by changing the preimpact speed of Vehicle 1 to achieve the objective of the Normal Crush Energy loss to be EC = 124976.4 ft-lb (169.4 kJ). The results of the planar impact mechanics analysis are given in Fig. 9.9 for a preimpact speed of Vehicle 1 of 40.9 mph (66.0 km/h) for which the work of the normal impulse matches the CRASH3 crush energy. In addition, it shows that the total energy loss is TL = 130219.7 ft-lb (176.6 kJ). The difference in these energy losses is the energy lost due to tangential effects, relatively small in this situation, contributing 2.9% of the total energy loss of 71.0%. Finally, the velocity change of Vehicle 1 is ΔV1 = 35.8 mph (57.7 km/h). This is greater than the uncorrected ΔV value of ΔV1 =

238

Crush Energy and DV

Figure 9.8 Result of crush energy calculation using CRASH3 algorithm for Example 9.4.

239

Chapter 9

Figure 9.9 Results of planar impact mechanics calculation using Example 9.4 with the same normal crush energy as in Fig. 9.8.

240

Crush Energy and DV

27.2 mph (43.8 km/h) given directly from the CRASH3 crush energy equations (see Fig 9.8) by 32%. Note that, in addition to accommodating non-zero restitution (CRASH3 assumes e = 0), the planar impact mechanics solution also provides the postimpact angular velocity of the vehicle(s). For this set of circumstances, the angular velocity after impact of Vehicle 1 was nearly 250˚/s. An angular velocity this large is expected due to the geometry and initial conditions of the collision, the preimpact speed of the vehicle, and the fact that the other “vehicle” is an immoveable barrier. Uncertainty in the preimpact speed of Vehicle 1 in this reconstruction might involve variation of the crush energy calculated using the CRASH3 algorithm. This variation would be due to factors such as variation in the d0 and d1 coefficients and variation in the measurements of the crush. Note also that, had EDR data been available from the accident, the data could have been used in the reconstruction of the preimpact speed. For example, had the longitudinal ΔV of the vehicle been available, the optimization in planar impact mechanics could have been set up to simultaneously meet both the longitudinal ΔV and the energy loss criteria (or perhaps a range of energy loss). The circumstances of this example are similar to the conditions of a series of fixed barrier tests used to validate the reconstruction method presented here, which uses residual crush of a vehicle to determine the normal crush energy dissipated in the collision, and then planar impact mechanics to reconstruct the speed of the vehicle at the time of impact based on that energy loss. In addition, these tests validated the concept and use of the critical impulse ratio, μ0. These validation studies are presented elsewhere [9.18].

9.3 Application Issues 9.3.1 Crush Stiffness Coefficients from Vehicle-to-Vehicle Collisions Sometimes the crush stiffness coefficients of a vehicle are obtained from a test where a striking vehicle is crashed into a struck vehicle. Example collision geometries are shown in Fig. 9.10. This section covers a method for determining the crush stiffness coefficients from such a test. An important condition is that a reasonably accurate measurement of the impact energy can be made. Any energy dissipated by tire forces, if significant, must not be included as crush energy. (If the striking vehicle is a moving rigid barrier, then its residual crush and its crush energy both are zero, and the methods of the previous section can be used.) Another condition is that the damage widths, w1 and w2, be equal. This condition rules out the use of oblique collisions and also means that any induced damage must be negligible. A problem that arises with two-vehicle tests is that the total kinetic energy loss (of both vehicles) of the collision can be determined easily, but the crush energy for each individual vehicle is unknown. In fact, if the struck vehicle is initially at rest, it ends up gaining kinetic energy as a result of the collision, even though its crush contributes to the energy loss. Fortunately, this problem can be overcome. For a central collision, the total collision energy loss, TL , is given by Eq. 6.22. If the collision energy loss, TL , is entirely converted to crush energy, then TL = EC and so

241

Chapter 9

Figure 9.10 Two example collision geometries of controlled vehicle-to-vehicle crashes.



EC =

1 m(1 − e 2 )(v2 − v1 ) 2 2

(9.16)

where m is given by Eq. 6.17. For each vehicle, Eq. 9.1 relates crush energy loss and the residual crush as





2 E1 / w = d 01 + d11C1avg 2 E2 / w = d 02 + d12C2 avg



(9.17)



(9.18)

where the average values of crush are computed according to Eq. 9.15. Note that the speed at which negligible residual crush occurs can be used with each of Eqs. 9.17 and 9.18 directly, with an average crush of zero to calculate d01 and d02 . This means that they can be treated as known, and d11 and d12 are the remaining unknowns (as are E1 and E2). Newton’s third law requires that the forces between two objects in contact must be equal in magnitude. Equating Eq. 9.6 for each vehicle and using Eq. 9.8 and Eq. 9.9 gives



d11 (d 01 + d11C1avg ) = d12 (d 02 + d12C2 avg )



(9.19)

Taking a ratio of the square of Eqs. 9.17 and 9.18 for each vehicle gives an energy ratio, and using Eq. 9.19:



242

E1  d 01 + d11C1avg = E2  d 02 + d12C2 avg

2

2

  d12   =     d11 

(9.20)

Crush Energy and DV

Since , Eq. 9.20 gives EC = E1 + E2 ,

E1 =



EC 1 + (d11 / d12 ) 2

(9.21)







(9.22)

Equations 9.17, 9.18, 9.21, and 9.22 form a set of four nonlinear equations with the unknowns of E1, E2, d11, and d12 . These equations can be solved conveniently using various methods.

9.3.2 Damage to One Vehicle Unknown It sometimes occurs that one of two vehicles involved in a crash is not available for crush measurements. Following certain assumptions, a method has been developed [9.19, 9.20] that can be used to estimate the ΔV’s of the two vehicles without the residual crush values of one. The reader is referred to those references. 9.3.3 Side Crush Stiffness Coefficients, Two-Vehicle, Front-to-Side Crash Tests Crush stiffness values can be obtained from test data for additional collision configurations. Sometimes, the front of a vehicle (striking vehicle) with known crush stiffness coefficients can be crashed into the side of a vehicle (struck vehicle) with unknown crush stiffness coefficients. A method has been developed [9.21] that allows determination of accident-specific crush stiffness coefficients for the target vehicle. The reader is referred to that reference. 9.3.4 Nonlinear Models of Crush Equation 9.1 is a linear relationship between the square root of energy and residual crush. Nonlinear models have been proposed but are not covered here. Two notable approaches are available. Woolley [9.11] extends the model covered here to nonlinear ranges. This extension allows more accurate application over a wider range of collision speeds. Of course, the coefficients of the nonlinear model must be determined experimentally for the vehicles being analyzed. Wood, et al. [9.22] derived a model, basically different from Equation 9.1, which is nonlinear and follows a power law. They show the model’s agreement with crash test data using European vehicles. 9.3.5 Arbitrary Number of Crush Measurements The original CRASH3 method was developed for two, four, or six crush measurements at equidistant intervals. Singh [9.23] developed equations for computing crush energy for an arbitrary number of crush measurements.

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CHAPTER

10 Frontal VehiclePedestrian Collisions

10.1 Introduction The reconstruction of collisions between vehicles and pedestrians requires a different approach than the reconstruction of collisions between vehicles. Preimpact motion of pedestrians often is unimportant because their speeds at impact typically are much lower than the speed of the vehicle. Also, the large mass difference and the friction developed between the vehicle and pedestrian at impact will result in the pedestrian moving predominantly in the direction of the vehicle. The postimpact motion of a pedestrian frequently includes a trajectory through the air followed by an impact with the ground and a tumble and/or slide to rest. The speed and frontal geometry of the striking vehicle can significantly affect the nature of the postimpact motion of the pedestrian. Physical evidence such as the location and pattern of vehicle deformation, type and locations of injuries, vehicle and pedestrian impact and rest positions, ground and roadway markings, etc. can be used to establish the nature and details of a vehicle-pedestrian collision and lead to a successful and reasonably accurate reconstruction. Thus, when the motion and damage of a vehicle, and the motion and injuries to a person, animal, or bicycle rider fit those of the types of collisions discussed in this chapter, reliable and reasonably accurate reconstructions can be carried out. Vehicle-pedestrian collisions often are categorized according to the type of vehiclepedestrian contact interaction and the pedestrian postimpact motion [10.1]. These categories are neither mutually exclusive nor exhaustive but serve as descriptive guides. The most common ones are forward projection, wrap, carry, roof vault, and fender vault. A forward projection collision is where the frontal surface of the vehicle is essentially vertical and flat with respect to the pedestrian and extends above and below the pedestrian’s center of gravity. Examples are buses and cab-over trucks with adult pedestrians, as

245

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in Fig. 10.1, and pickup trucks and children. A forward projection collision drives the pedestrian straight forward relative to the vehicle. A wrap collision occurs when, because of a low frontal contact surface of the vehicle relative to the pedestrian’s center of gravity as well as the pedestrian’s freedom to rotate and body flexibility, the pedestrian’s body wraps back onto the hood following initial contact. An example illustration is in Fig. 10.1. If the speed of the vehicle is high enough, the pedestrian impacts the vehicle again at the windshield, a windshield pillar, or roof. Following this secondary impact, the pedestrian often is thrown or projected forward at some angle relative to the ground in a trajectory through the air. If the pedestrian wraps back onto a hood and remains on the hood or fender for a period of time while the vehicle moves forward, this is the carry category. For a carry, if the pedestrian and/or clothing is not snagged on a part of the vehicle, the pedestrian eventually slides forward from the hood or fender as the vehicle decelerates. A roof vault begins as a wrap collision but, usually because of the vehicle’s high speed, the pedestrian continues up onto the roof of the vehicle. In such cases, the pedestrian continues rearward relative to the vehicle and may hit a rear surface such as a trunk lid before falling to the ground. A fender vault collision occurs when the pedestrian is struck by the front of a vehicle near one of its sides, wraps up onto the fender, and slides off to the side of the moving vehicle without a significant secondary impact with a pillar or windshield. Acceleration or deceleration of the vehicle after initial impact can play a role in a pedestrian collision. In a carry collision, the pedestrian remains on the vehicle’s hood for a period of time. If the vehicle continues to accelerate, this period of time will be increased; if the vehicle begins to decelerate, the pedestrian will tend to slide forward relative to the vehicle. Other factors, such as the slope of the hood, may play a role. In a roof vault, an accelerating vehicle will tend to move out from underneath the pedestrian, while a decelerating vehicle will tend to remain below the pedestrian. In a wrap collision, vehicle deceleration allows the pedestrian to move forward without sustained interactive contact with the vehicle.

Figure 10.1 Examples of forward projection collision (left) and wrap collision (right).

246

Frontal Vehicle-Pedestrian Collisions

Different approaches may be taken to reconstruct a vehicle-pedestrian accident. The approach, to a large extent, depends on the goals of the reconstruction and the available information. A distinction is made between speed reconstruction and reconstruction of vehicle-pedestrian interaction dynamics. The former is the determination of vehicle speed from the physical evidence from the accident. Examples of physical evidence used for speed reconstruction are throw distance and distance between vehicle and pedestrian rest positions. Throw distance is defined as the distance between the position of the initial pedestrian contact and the final uncontrolled rest position; that is, the distance the pedestrian moves between initial contact and rest. The latter, vehicle-pedestrian interaction dynamics, is aimed substantially at determining detailed pedestrian-vehicle interaction motion and forces to assess a specific injury or to assess effects of changes in vehicle geometry. This analysis typically is done using relatively sophisticated, multi-body dynamics computer software; see, for example, van Wijk, et al. [10.2] and Moser, et al. [10.3]. If such detailed information is not particularly important, a speed reconstruction can be done using relatively simple equations. Speed reconstruction is the topic covered in this chapter. Several approaches are available for carrying out speed reconstructions. They are distinguished by the source of the equations. One approach is to use empirical equations, developed from vehicle-pedestrian experimental data. These empirical models are based on a wide range of tests, some involving reconstructed collisions, tests with dummies, tests with cadavers, etc. Another approach is to use equations derived from principles of mechanics that have been compared favorably to experiments. A third is a hybrid approach where equations are based both on physical models and experiments. Empirical equations have proven to be a practical means of speed reconstruction because a strong relationship has been found to exist between throw distance, sp, and the square of the vehicle speed, vc0 [10.4, 10.5, 10.6]. An electronic compendium of formulas exists [10.7]. Each of the speed reconstruction approaches has advantages and disadvantages. Empirical equations are simple and easy to apply. Although empirical models are usually presented with variations, these variations are due primarily to the differences in test conditions and represent experimental error. These variations do not represent reconstruction uncertainty. Theoretical models based on mechanics and that contain physical parameters have a distinct advantage. Variations in these models are based on changes in the physical parameters, so the physical models can be used not only for reconstruction, but also to estimate uncertainty of a specific reconstruction. Hybrid models combine analytical and empirical equations and have advantages and disadvantages of both approaches. A hybrid approach [10.8] is first presented for wrap collisions followed by an experimental model [10.9] for forward projection collisions. Then a mechanical model is developed, based on the work of Han and Brach [10.5], which can be used to analyze and reconstruct forward projection and wrap collisions.

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10.2 General and Supplementary Information A great deal of literature has developed over the years related to vehicle-pedestrian collisions. Well over 400 technical articles have been published on various aspects of the subject. A summary of some pertinent papers is contained in Backaitis [10.11]. The collision and postimpact motion of the vehicle and pedestrian often are only one part of a pedestrian accident. Events prior to impact often play an important role. In many cases it is necessary to take into account factors such as expectation, visibility (daytime/nighttime), driver inattention, and the direction and speed of the pedestrian (bicycle rider, animal, etc.). These aspects of a reconstruction are not explicitly treated here. Various sources exist that contain information and data on walking, jogging, and running speeds as well as acceleration capability of adults and children. Some of this information is contained in references such as Eubanks [10.1], Fugger, et al. [10.12], Toor, et al. [10.6], and Vaughan and Bain [10.13, 10.14].

10.3 Hybrid Wrap Model A single-segment physical model of a pedestrian and its impact with a vehicle was developed by Wood and Simms [10.8]. It is a relatively sophisticated model of vehiclepedestrian impact that takes into account possible multiple impacts between the pedestrian and vehicle, head contact position, head velocities, time of wrap and time of head contact, among other considerations. In a later work, Wood and Walsh [10.9] developed a hybrid model based on the single-segment physical model. This model is also based on a number of vehicle-pedestrian tests. The results comprise a simple algebraic relationship between the initial vehicle speed, vc0, and the throw distance, sp:

vc 0 = cW s p , m / s



(10.1)

where three different values of cW provide estimates of the mean and the experimental uncertainty. In this equation, velocity has units of m/s and pedestrian throw distance has units of meters. The three values of the constant are

cW = 2.5, 3.6, 4.5

(10.2)

where cW = 3.6 is a mean value and the other values provide lower and upper bounds. Wood’s hybrid model is intended primarily for wrap and forward projection collisions.

10.4 Forward Projection Model Wood and Walsh [10.9] present a model specifically intended for forward projection vehicle-pedestrian collisions. Results are given as bounds with minimum and maximum speeds as functions of throw distance for children and adults. For a child,

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Frontal Vehicle-Pedestrian Collisions



vc 0 = cc s p



(10.3)

where a minimum value of the initial speed is given for cc = 2.03 and a maximum value for cc = 3.90. For adults the equation is written as



vc 0 = ca s p



(10.4)

where ca = 1.95 gives a minimum value or lower bound and ca = 3.77 gives the maximum or upper bound.

10.5 Analysis Model Figure 10.2 shows various vehicle and pedestrian positions corresponding to vehiclepedestrian collision events and indicates various variables that enter into the model. Initial contact is at time τ = 0 with a velocity of the vehicle of vc0. The vehicle and pedestrian move forward at different speeds until time τ = τ0, when a secondary impact occurs. Between τ = 0 and τ = τ0, the center of gravity of the pedestrian has moved forward a distance, xL , and is at a height, h, above the ground. For a forward projection collision, xL is zero and there is no secondary impact. As a result of the impact at τ = τ0, the pedestrian instantaneously rebounds and is launched in an airborne trajectory with a velocity of vp0 and at an angle θ. The trajectory has a range (distance parallel to the ground) of R and reaches the ground at time τp1. At that time, the pedestrian impacts the ground with a vertical (downward) velocity component determined by the trajectory. From the point of ground impact to the point of rest, a distance s, the pedestrian is assumed to be uniformly decelerated with a frictional drag factor, fp. The motion over the distance, s, typically consists of rolling, sliding, and tumbling. The total distance from initial impact to rest is the pedestrian throw distance, sp. It is assumed that from the time of the secondary impact, τ0, to the time of rest, τp, that the pedestrian motion is independent of the vehicle. To provide a degree of generality for reconstructions, the vehicle motion is given an arbitrary travel distance, s1, from τ = τ0 to τ = τc1 over which the velocity remains constant. Following this, the vehicle decelerates to rest over a distance, s2, with drag factor, a2 . With this information, equations can be derived to develop a mechanics/analysis model. See Table 10.1 for a list of notations and definition of variables.

10.5.1 Pedestrian Motion Based on the above and on Fig. 10.2, the throw distance is

s p = xL + R + s



(10.5)

The mass of the pedestrian, mp, is typically negligible in comparison to the mass of the vehicle, mc. However, cases where the momentum loss is important (such as an impact

249

Chapter 10

Figure 10.2 Diagram showing coordinates and variables associated with a vehicle-pedestrian collision model.

with a large animal), allowance is made for a momentum loss of the vehicle due to the collision with the pedestrian. This gives

vc′ 0 =

mc vc 0 mc + m p



(10.6)

Again for generality, a factor, α, is used to relate the pedestrian throw velocity to the vehicle velocity such that

v p 0 = α vc′ 0





(10.7)

It is assumed that aerodynamic drag is negligible as the pedestrian undergoes the trajectory through the air. Under such a condition, the trajectory is parabolic with zero horizontal acceleration and uniform vertical acceleration due to gravity. This leads to the following equations for the range, R, and trajectory time, τR:

1 R = v p 0 cos θ τ R − g sin ϕ τ R2 2





τR =

v p 0 sin θ g cos ϕ

+

(10.8)

v 2p 0 sin 2 θ + 2 gh cos ϕ g cos ϕ



The trajectory velocity components at the instant before ground impact are

250

(10.9)

Frontal Vehicle-Pedestrian Collisions

Notation and list of variables for vehicle-pedestrian throw model.





a2

deceleration of vehicle over distance s2

d

distance between rest positions of vehicle and pedestrian

fp

drag resistance coefficient of pedestrian over distance s

g

acceleration of gravity

h

height of pedestrian center of gravity at launch, τ0

mc

mass of vehicle, weight /g

mp

mass of pedestrian, weight /g

R

range of pedestrian throw, launch to ground impact

s

pedestrian ground contact distance, impact to rest

s0

distance of travel of vehicle with pedestrian contact

s1

distance of travel of vehicle at uniform speed

s2

distance of travel of vehicle with uniform deceleration, a2

sp

pedestrian throw distance; total distance, initial contact to rest

τc

vehicle travel time, initial contact to rest

τc1

time of travel of vehicle at steady speed

τp

total time of travel of pedestrian, initial contact to rest

τp1

time from impact to pedestrian initial contact with ground

v’c0

velocity of vehicle after impact with pedestrian

vc0

initial speed of vehicle

vp0

initial speed of pedestrian

xL

xdistance of pedestrian from initial contact to launch

α

ratio of pedestrian speed to vehicle speed at time of launch

θ

angle of launch of pedestrian relative to x axis

μ

impulse ratio for pedestrianground impact

φ

road grade angle

v pRx = v p 0 cos θ − gτ R sin ϕ v pRy = v p 0 sin θ − gτ R cos ϕ



(10.10)



(10.11)

Table 10.1

The normal component of the impact of the pedestrian with the ground is assumed to be perfectly inelastic. That is, there is a single impact with the ground with no vertical bounce or rebound. According to the planar impact theory developed in Chapter 6, this defines a vertical impulse, Pn = -mp vpRy. A corresponding tangential impulse, Pt = μPn, develops. Because the pedestrian slides throughout the ground impact, μ = -fp. This leads to a velocity at the beginning of the slide distance, s, of

v′pRx = v pRx + µ v pRy



(10.12)

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Chapter 10

This allows determination of the distance, s, using Eqs. 3.6 and 3.9:

s=

(

( v′ )

2

pRx

2 g f p cos ϕ + sin ϕ

)

(10.13)

10.5.2 Vehicle Motion Except for the momentum change of the vehicle due to impact with the pedestrian, the motions of the vehicle and pedestrian are assumed to be independent of each other for τ > τ0. Any incidental contact during the early part of the trajectory is neglected. Over the distance s0 and time τ0, the speed of the vehicle is vc0. Over the distance s1 and time v' interval (τc1 - τ0), the vehicle travels with speed c 0 . Uniform acceleration of -a2 occurs over the distance s2 . The difference, d, between the total travel distance of the vehicle and the pedestrian throw distance is



d = vc 0t0 + vc′ 0 (tc1 − t0 ) +

a2 (tc − tc1 ) 2 − s p 2



(10.14)

This distance can be important in a speed reconstruction because after a vehiclepedestrian accident, the point of impact often is unknown, but the distance, d, is known because both rest positions are known. In such cases, it sometimes is possible to reconstruct the vehicle speed using d. This technique is discussed by Evans and Smith [10.15] and is illustrated in a later example.

10.6 Values of Physical Variables If all of the physical information is known or can be determined, the throw distance of the pedestrian and vehicle travel distance can be calculated from the above equations, which are called the analysis, or mechanics, model. The model results can be compared to experimental data. This was done by Han and Brach [10.5] for more than fourteen selected sets of data corresponding to wrap and forward projection collisions. Results showed excellent agreement with the analysis model and the data trends. The comparisons provide information about realistic ranges of values for some of the physical variables. Information concerning some of the variables, such as α, θ, R, and s, could not be determined directly from experimental data because in many of the experiments these variables were not measured. Some information was inferred from the comparison process. Information about the pedestrian drag factor, fp, has been determined from measurements and reports of others and is covered shortly. By definition, the value for the launch angle for forward projection collisions must be θ = 0 because the pedestrian is projected directly forward. This allowed fitting of the analysis model equations to the experimental data and determination of values of the constant α. As defined in Eq. 10.7, α corresponds loosely to a coefficient of restitution. If the pedestrian were to be caught or snagged on some part of the vehicle, there would be

252

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no rebound, and α = 0. Although of academic interest, there would be no trajectory, and so this case has little practical application to reconstructions. On the other hand, a value of α > 1 has the implication that the impact between the pedestrian and vehicle has an elastic component; that is, the pedestrian rebounds and is projected forward at a speed higher than the vehicle. Values of α as high as 1.2 and 1.3 were found by Han and Brach. However, they caution that values this high need to be substantiated by additional tests, possibly where α is measured directly. Generally, a value of α = 1 is expected for wrap collisions. For forward projection collisions, Wood and Walsh [10.9] also noted values of α greater than 1, although smaller than Brach and Han’s values. Wood and Walsh also noted a speed dependence of the value of α. Existing pedestrian collision experiments appear to have never directly measured the launch angle, θ. From comparison to adult-reconstructed and adult-dummy wrap experimental collisions, typical values of θ from about 4° to 13° were found by Han and Brach in the fitting of the model to experimental data. Some evidence of higher values, up to θ = 35°, were found. Although physically possible, these occurred in only one or two groups of experiments and are believed not to represent common conditions. A range of 0° ≤ θ ≤ 15° should be considered for reconstruction of typical wrap collisions.

10.6.1 Pedestrian-Ground Drag Coefficient One of the most important variables in determination of throw distance is the pedestrianground frictional drag coefficient, fp. Han and Brach [10.5] and Wood and Simms [10.16] analyzed the data from a group of measurements by Hill [10.17]. Hill’s experiments were conducted using dummies dressed in various types of clothing that were dropped from a moving vehicle onto an asphalt-type pavement (described as an old airfield tarmac). Wood and Simms determined average values of 0.70 and 0.73 from Hill’s data. Han and Brach concluded that for application to the above vehicle-pedestrian analysis model, values in the range of 0.7 ≤ fp ≤ 0.8 are appropriate for dry pavements. However, dummies in nylon clothing produced a value near f p = 0.6. This exception indicates that the clothing can affect fp but that there is not a high sensitivity to the clothing because none of the other four types of clothing made a significant difference. Roadway pavements such as rough concrete and ground conditions such as grass, ice, etc. are expected to require different values. Unfortunately, it appears that no comprehensive experiments have been conducted under such widely different conditions. Happer, et al. [10.4] present a table of values of f P for different conditions. Funk, et al. [10.18] give values taken from analysis of passengers thrown from rollover vehicles. Based on experiments inclusive of Hill’s, Wood and Simms [10.16] also report a mean value of fp = 0.58 with a standard deviation of sf = 0.1, with upper and lower bounds at ± 3sf. These data may differ from above because it was fit to a different analytical model. A vehicle-pedestrian wrap collision occurs under the conditions listed in Table 10.2 Example 10.1 representing a vehicle with a low frontal geometry traveling at 30 mph (48 km/h) and where the pedestrian has a secondary impact with the windshield. Calculate the throw

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Table 10.2

Vehicle-pedestrian wrap collision conditions, Example 10.1. a2

0.90

deceleration of vehicle over distance s2

fp

0.80

drag resistance coefficient of pedestrian over distance s

h

4.00

ft

height of pedestrian center of gravity at launch, τ0

s1

0.00

ft

distance of travel of vehicle at uniform speed

vc0

44.00

ft/s

initial speed of vehicle

30

mph

initial speed of vehicle

xL

2.00

α

1.00

xdistance of pedestrian from initial contact to launch

θ

5.00

˚

angle of launch of pedestrian relative to x axis

φ

0.00

˚

road grade angle

μ

0.80

mc

93.24

lbs2/ft

mass of vehicle, weight ⁄g

mp

5.44

lbs /ft

mass of pedestrian, weight ⁄g

ft

ratio of pedestrian speed to vehicle speed at time of launch

impulse ratio for pedestrianground impact 2

distance, sp. Determine the effect on the throw distance of varying the launch angle, θ, from θ = 0° to θ = 10°. Solution  The pedestrian throw equations can be set up for a solution in a spreadsheet. Such a solution using the values from Table10.2 produces the results in Fig. 10.3. There it is seen that the throw distance is 43.35 ft (13.2 m). Additional solutions for the launch angle values of θ = 0° and θ = 10° show that the throw distance has corresponding values of sp = 38.78 ft (11.8 m) to sp = 47.60 ft (14.5 m), respectively. Note that in all cases the pedestrian travels farther than the vehicle and the vehicle comes to rest quicker than the pedestrian, resulting in a negative value of d. Example 10.2 Consider the same collision as given by the data in Table 10.2 of Example 10.1 with the

speed of the vehicle unknown. Consequently, both the throw distance and the vehicle speed are unknown. However, the distance traveled by the pedestrian was measured and found to exceed that of the vehicle by 20 ft (6.1 m). Determine the speed of the vehicle.

Solution  The spreadsheet can again be used here, but because one of the input values is unknown, an optimization feature of the spreadsheet can be exploited. Figure 10.4 shows that for a distance of d = -20 ft (-6.1 m) stated as a goal, the initial speed of the vehicle is vc0 = 66.66 ft/s (20.32 m/s). This reconstruction indicates that point of impact is about 73 ft (22.3 m) back from the vehicle rest position.

10.7 Reconstruction Model Although the analysis model above can be used for reconstruction purposes, as seen in Example 10.2, sometimes it is better to calculate initial vehicle speed directly from pedestrian throw distance. An important example of this is for estimation of the uncertainty of a reconstruction. If the area where the vehicle-pedestrian collision occurs

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Frontal Vehicle-Pedestrian Collisions

Figure 10.3 Results of calculations of a wrap vehicle-pedestrian collision using the pedestrian throw model.

255

Chapter 10

Figure 10.4 Results of calculations of a wrap vehiclepedestrian collision using the pedestrian throw model with a spreadsheet search procedure.

is level; that is, the grade angle, ϕ = 0°, then the equations can be inverted, solving for vp0 in terms of sp. This gives

vc 0 = A s p − B





(10.15)

α mc

(10.16)

where

A = Ap



Ap =

256

mc + m p

2 fpg f p2 sin 2 θ + f p sin 2θ + cos 2 θ



(10.17)

Frontal Vehicle-Pedestrian Collisions

and

B = xL + f p h



(10.18)

Consider again the collision whose conditions are given in Table 10.2, except the throw Example 10.3 distance is known from physical evidence to be 43.35 ft (13.2 m). The initial speed of the car is unknown and is to be reconstructed. In addition, suppose that to provide an estimate of the uncertainty of the reconstruction and to examine its sensitivity to different input values, the launch angle and pedestrian frictional drag coefficient are given uniform statistical distributions as

θ = u (0,10)

f p = u (0.7, 0.8)



(10.19)

where the notation is that u(min, max) represents a uniform statistical distribution with min and max variable values. Estimate the reconstruction uncertainty as represented by the distribution of the initial vehicle speed obtained using Eq. 10.15. Solution  Equation 10.15 is nonlinear. According to Equation 1.20, the mean vehicle speed is approximately the value from Eq. 10.15 evaluated at the mean values of fp and θ. These are fp = 0.75 and θ = 5° and give A = 6.93, B = 5.00, and an initial vehicle speed of vc0 = 42.9 ft/s (29 mph, 46.8 km/h). The distributions given by Eq. 10.19 both are uniform, so the upper and lower limits of the vehicle speed can be obtained by solving Equation 10.15 for all combinations of the minimum and maximum values of the distributions. These solutions provide the range of initial vehicle speeds of 39.86 ≤ vc0 ≤ 46.90 ft/s (27 ≤ vc0 ≤ 32 mph) and (43.5 ≤ vc0 ≤ 51.6 km/h) for the conditions given. Although the vehicle speed has been bounded and the mean or average value determined according to the material in Chapter 1, the distribution is unknown. Other methods must be used to obtain the distribution; in this example, the Monte Carlo method is used. These calculations were performed using commercial software [10.19] and are not presented here. The distribution of the initial vehicle speed is shown in Fig. 10.5. Although the input distributions are uniform, the distribution of the vehicle speed is not. In fact, it is not too far from a normal distribution. This is not unusual and the reader is encouraged to consult a statistics reference to explore this further. Wood’s hybrid model as well as Han and Brach’s analy­sis and reconstruction models have been presented in this chapter. The analysis and reconstruction models include the most important parameters and physical variables that play a role in planar vehicle-pedestrian collisions. In specific recon­structions, reasonably accurate values of these variables must be used. Variations of the variables should be used to estimate sensitivity and uncertainty of reconstructions. It is wise to also compare results using different models to develop confidence in a given reconstruction.

257

Chapter 10

Figure 10.5 Statistical distribution of the vehicle speed from a Monte Carlo analysis of the reconstruction model equations.

258

CHAPTER

11 Photogrammetry for Accident Reconstruction

11.1 Introduction Photogrammetry is the process of obtaining quantitative dimensional information about physical objects through the process of recording, interpreting, and measuring photographic images [11.1]. It has been used extensively with aerial photographs for the purposes of mapping ground terrain. Numerous books have been written on this subject including Hallert [11.2], Moffitt and Mikhail [11.3], and Slama [11.1]. These books describe the mathematical foundation of this topic and present many applications of the theory. In general, there are two types of photogrammetry: aerial photogrammetry, which involves two cameras with parallel view lines at a fixed, known distance apart, and close-range photogrammetry, which involves two or more camera positions with large differences in view angles and variable separation distance from the object being measured. Close-range photogrammetry is the specific type of photogrammetry that is applicable in the field of automotive accident reconstruction. The title “close-range photogrammetry” is appropriate in the application of these techniques to automotive accident investigation and reconstruction due to the fact that the camera-to-subject distance is short when compared to aerial photogrammetry. Moreover, the specific needs of accident reconstructionists have led to the use of two variations of the classical application of photogrammetry. Some of the details and technical background of these variations are presented in other references [11.4, 11.5]. The former considers photogrammetry for use in accident reconstruction and approaches the topic as the use of photographs for constructing diagrams (a scaled chart depicting local attributes and physical characteristics) of

259

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accident scenes and sites. In the latter reference, the presentation is narrowed to the uses of photogrammetry in the field of accident reconstruction, discussing two techniques used to obtain accurate, quantitative information from a single photograph. Another technique, finding wider using in the field, uses multiple photographs of a single subject (e.g., an accident vehicle or an accident scene or site) for acquiring dimensional information. The first of the two single-photograph methods, reverse projection, or camera reverse projection, is empirical, and is quite useful for extracting information about the location of physical evidence from a single photograph. The second method, referred to alternatively as numerical rectification, four-point transformation, or planar photogrammetry, is based on a mathematical transformation [11.6] between the film plane, or the plane of a charge coupled device (CCD) in the case of a digital camera, and a physical, geometric plane visible in the photograph (i.e. a roadway surface). This mathematical transformation enables quantitative information to be determined from a single photograph (provided certain conditions are met). This method will be referred to here as planar photogrammetry. The third method presented involves a more general application of the mathematical equations of photogrammetry. This method permits the general determination of coordinates in three-dimensional space from a series of photographs of a single object. The complete mathematical detail involved in the derivation of the equations is presented by other authors [11.2, 11.3]. The last section of this chapter presents an overview of the equations and considers the application of the method to accident reconstruction. During the time between the first edition of this book and the second, digital cameras have virtually replaced the use of film cameras in the field of accident reconstruction. This transition has greatly facilitated the use of photogrammetry in that field. In general, photogrammetric analysis is done using computer software, which necessarily requires that the photographic images be available on the computer. The digital camera simplifies the process of loading the images onto the computer. The material presented herein considers these photogrammetric processes from the point of view that the image may come from either a film camera or a digital camera. Readers can use and/or adapt the processes to their specific circumstances.

11.2 Reverse Projection Photogrammetry The reverse projection photogrammetric method is useful for extracting information about the location or size of an object that is no longer visible at the accident site when the object is contained on a single scene photograph. The photograph is usually taken with a camera whose properties are typically unknown to the analyst, and the information shown in the photograph, such as tire marks on a roadway worn away through time, a snow pile that has since melted, and markings or characteristics that have been lost due to road improvements or modifications, has often changed or been removed from the accident site. Frequently, this information pertains to the position of a vehicle, vehicle component, or mark made by a vehicle.

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11.2.1 Overview and Requirements of the Reverse Projection Process The reverse projection process is empirical in nature and application of the process is relatively straightforward. In classical photogrammetry, with a photograph that contains three or more objects with known three-dimensional coordinates, known principal distance of the camera (focal length), and known principal point of the image (essentially the geometric center of the image), the location of the lens and the orientation of the camera can be determined relative to a coordinate system. This process is known as a spatial resection or simply as a resection. If the principal distance and principal point location are not known, five or more points with known spatial coordinates are required to perform the resection analytically. Reverse projection photogrammetry is the analog process of performing this resection. Once the analog resection is completed, the method uses the camera to project information onto the site visible to a viewer looking through the camera. The typical role of the camera of gathering and recording information (light) is reversed in this process. Rather than using the traditional use of gathering the light rays from the environment and focusing them onto the film plane or the CCD, the image and camera are used to project, or place, information onto the environment. Due to this reversal in the role of the camera from capturing information to projecting information, the method has become known as reverse projection. It has also been referred to as camera reverse projection. Various techniques have been presented in the literature [11.4], but the procedure described here employs digital technology for image handling and is based loosely on the technique presented previously in the literature [11.7]. Previous versions of this photogrammetric method used techniques considered more analog in nature [11.4]. Both techniques work adequately, but the newer, computer-based image handling methods make the process much simpler to implement. The process described below outlines the steps required for analysis of one photograph, but this process can be performed sequentially or simultaneously with more than one photograph of the same scene, yielding similar, related, or complementary information. This multi-image analysis is a means for refinement of the location or dimension of the attribute or characteristic of interest, or to obtain additional information about the object or characteristic of interest. An example presented later shows application of the reverse projection process using two photographs simultaneously. The recognition of a good photograph depicting the desired (transient) information is the prerequisite for the application of the camera reverse projection method. The next step is to obtain a first-generation, full-frame print of the photograph of interest if taken with a film camera, if possible, or a copy of the original electronic file of the image if taken with a digital camera. A full-frame, high-resolution digital scan of the negative for the print(s) of interest is a superior alternative to obtaining a full-frame photograph. A full-frame photograph is one in which the entire exposed portion of the negative is visible. The best way to ensure that the full frame is available is to request that the shop printing the photograph or scanning the negative include the sprocket holes on the top and bottom and a portion of the unexposed negative on the sides of the image in the

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printing or scanning process. Typical prints from the local pharmacy, in addition to not including the sprocket holes, routinely crop the negative when printing. Negatives of accident photographs are not always obtainable, but some law enforcement officials preserve their negatives. If a digital camera was used to capture the image, the original, unmodified version of the image data should be requested. An unmodified version of the original electronic data file for a digital photograph is by definition full frame for a digital camera. When using an image from a film camera, full-frame photographs are important for maximum accuracy in that the specific orientation of the camera is discernable because the “frame” of the full-frame photograph is determined by the limits of the light path dictated by the camera body and lens combination. This frame will assist in the positioning of the image in the camera in an attempt to place the principal point of the image at the location of the principal ray in the camera used in the method. If the principal distance (focal length) of the camera is not known, a zoom lens can be used to enable the determination of this parameter. In addition to the characteristic or object of interest, a photograph useful for reverse projection must also include several fixed landmarks (street signs, road markings, light/ utility poles, roadway cracks, lines, sewer grates, manhole covers, fence lines, etc.). These landmarks must still be present at the accident site with their position and character essentially unaltered. They will be used to complete the analog resection process. The importance of these landmarks should not be underestimated. They will be used in the positioning of the camera at the site to obtain a match between the information captured in the photograph and its counterpart at the site. Once the analog resection is completed, the camera is positioned in approximately the same location as the camera used to capture the original image at the time the photograph was taken. Moreover, a full-frame photograph (or original digital image) includes the entire exposed portion of the negative (or CCD) and may contain additional landmarks not visible in a cropped print. The absence of a full-frame photograph does not preclude the use of reverse projection photogrammetric analysis. Use of a cropped photograph may reduce the accuracy of the determination of the position of the camera at the scene at the time the original image was captured. However, for some analyses this may not be significant. Use of a conventional cropped photograph for reverse projection photogrammetry will preserve the relative location of the characteristics and attributes depicted in an image, and can therefore produce useful results. Convergence of the image with the actual site may be more difficult with an image that is less than full frame. If possible, the make and model of the camera and lens used to take the photograph(s) at the accident scene should be determined. The focal length of the lens on the camera at the time the photograph was taken is of particular importance. In the case of a digital JPEG (Joint Photographic Experts Group) image, the EXIF (Exchangeable Image File Format) data stored with the file of the original electronic image contains the camera model and

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the focal length of the camera at the time the image was captured (as well as other data). In the case of film cameras, knowledge of the camera body permits assessment of the percentage of the full image viewable through the viewfinder. While using the same model camera in the photogrammetric analysis as the one used to create the image may enhance credibility if the results are to be used in court, it is not required to successfully conduct a reverse projection photogrammetric reconstruction. As will be shown later, the attributes of certain single-lens reflex (SLR) film format cameras simplify the process significantly.

11.2.2 Reverse Projection Procedure The reverse projection process begins by obtaining a digital version of the image. In the case of an image captured with a film camera, digitally scanning the image in the (full-frame) photograph at a high resolution is recommended. Scanning the negative rather than the photograph produces better results but is not essential. Once the digital image is obtained, it is imported into a CAD package. It is convenient, for reasons to be described later, to have the lower left corner of the photograph located at the origin of the coordinate system of the CAD drawing. The main features of the photograph are then outlined onto a layer of the drawing separate from the image using the drawing capabilities of the CAD package. The outlines must include the information of interest that no longer exists at the site and the landmarks expected to be at the site when the camera positioning is done. In addition to all the major features, the four corners of the frame should be carefully marked. The layer with the outlines is printed onto transparent material at a size prescribed by the physical requirements of the camera to be used for the reverse projection. The image printed on the transparent material will be referred to below as an overlay. The details of each step in the process are included in an example presented later in this section. An analog approach to create the overlay can also be used. In this approach, a transparent film is affixed in place over the print such that there is no movement between the overlay and the print. The main features of the photograph are then traced by hand with a thin indelible marker in a manner similar to the outlining that is done with the CAD program described above. The overlay is then placed on a flatbed scanner and scanned back into the computer. This image can then be printed onto transparent film, at the size appropriate for the camera, to be used for the positioning exercise. Regardless of whether the analog or digital method is used to capture the necessary information from the photograph, a reduced-size overlay displaying the outlines of that information can be created. The overlay is then fit into the camera in such a manner that the image is centered at the visible portion of the viewfinder. This locates the principal point of the overlay such that the principal ray of the lens travels through it. The principal ray is the light ray not refracted by the lens. For practical purposes, the principal ray of the lens can be considered the ray that passes through the geometric center of the lens. Some careful trimming of the overlay may be necessary to complete this step in the process. Alignment

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braces located equidistant from the four corners and a border outline of the exposed region of the photograph, both shown later in an example, are particularly useful for this purpose. These alignment braces provide a visual assessment of the position of the image in the viewfinder. It is important that the overlay not move while positioned in the camera. Typically, the means of securing the focusing screen to the camera will hold the overlay in place. Careful evaluation of the overlay and the camera should be done prior to taking the camera with the overlay into the field where it will be subjected to forces due to handling. The size of the image on the overlay and the trimming requirements will depend on the specific camera being used for the analysis. Some 35-mm single-lens reflex cameras, such as the Canon F-1 and the Nikon F-3, permit the complete and convenient removal of the focusing screen by removal of the prism or eyepiece (see Fig. 11.1 for this feature for the Canon F-1). This capability greatly simplifies the process for installation and removal of the overlay. This may be of particular importance if the intentions of the photogrammetrist are to conduct more than one reverse projection analysis during a single inspection. (More than one camera could be outfitted with an overlay in the office, and multiple cameras from different viewpoints can be used in the field. However, the typical situation would be a single camera and multiple transparencies, necessitating a transfer of the overlay in the field.) Removal and installation of an overlay in the field can be handled much quicker with a camera equipped with a removable eyepiece and focusing screen. Figure 11.1 Photographs showing the removable focusing screen for the Canon F-1.

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After the overlay is correctly installed in the camera, the image on the overlay is continuously visible in the eyepiece of the camera. Because the overlay is positioned above the mirror, it will not appear in any photographs taken with the camera. (In a strict sense, the camera need not be capable of collecting an image to perform its function in the reverse projection process.) A photograph taken with the camera in the final adjusted position can be useful for comparison with the original photo. Similarly, the authors have heard of, but have not yet employed, the use of a video camera “looking” through the viewfinder of the camera to capture the view of the camera with the overlay in place. Interested readers can experiment with this technique. The process described above can be modified to accommodate the large viewing screens of current digital cameras. The overlay, formerly sized to fit under the focusing screen of a 35-mm camera, can be sized to fit directly over the viewing screen on the rear of a digital SLR at a size corresponding to the portion of the viewing screen used in the live preview mode of the camera. An experimental evaluation of this process by the authors showed that the steps outlined above using a film camera in the process can be easily modified to accommodate the new location of the overlay on the viewing screen of a digital camera. However, locating the overlay over the viewing screen can present difficulties in adequately observing the marks necessary to position the camera, particularly in a sunny environment. Improvised hoods, commercially available viewing loupes, and other similar devices had limited effect. Readers wishing to use this method are encouraged to evaluate the process prior to using it in the field. As mentioned previously, the traced image of the original photograph is printed at a reduced size. The percent reduction is dictated by the characteristics of the viewing area vs. the exposure area for the camera from which the image is to be viewed. For example, as with most 35-mm film-format cameras, the size of the exposed portion of a 35-mm negative is approximately 36 × 24 mm. The viewfinder of the camera typically has reduced coverage of the light that is passed to the negative when the shutter is released. For example, for the Nikon N90 35-mm camera, this reduction is 8%, or the image visible in the viewfinder is 92% of the exposed area of the negative when the photograph is taken. Therefore, in recreating the 36 × 24 mm image, the image visible in the viewfinder of the N90 does not necessarily represent what the original photographer saw in the viewfinder of the camera at the time the photo was taken, unless of course the original photograph was also taken with an N90. Because the task in reverse projection is to use the image as viewed through the viewfinder of the camera to recreate the position of the camera at the time the photograph was taken, and thereby the position of the missing characteristics, this reduction in the visible portion of the image should be taken into account in setting up the reverse projection project. To modify the image appropriately in the case of the use of the N90 for the reverse projection camera, the reduction in area is 8%. To account for this reduction, the dimensions of the smaller visible image must be computed. Figure 11.2 shows the size of the original image and the visible area depicted by the inside rectangle. To determine the size of the viewable area, shown by the crosshatched inside rectangle, assume that

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the long side of the actual negative is length B and its reduced counterpart is b, and the shorter side of the actual negative is length A with its reduced counterpart being A a a. Assuming that the areas have the same aspect ratio; i.e., B = b and that the camera specifications in this example are 8% reduction in visible area; i.e., 0.92 AB = ab, then the following calculations give the differences in the dimensions of the two regions:



0.92 AB = ab A 0.92 AB = b2 B 2 0.92B = b2 b = 0.959B

(11.1)

Using this to determine the spacing between the sides and maintaining the position of the center of the two areas, we get the following:



B − b B − 0.959 B = 2 2 B −b = 0.021B 2 B −b = 0.738 mm 2

(11.2)

A−a = 0.504 mm 2

(11.3)

Similarly,



Figure 11.2 shows the relationship between the two areas. The area visible through the viewfinder is indicated with the hatch. After these calculations are completed, the size of the visible region of the camera to be used in the analysis can be included on the tracing before printing of the overlay. This permits the accurate location of the reduced overlay to be set visually through the viewfinder of the camera. This is needed because objects near the perimeter of the exposed image may not be visible when the small overlay is located in the view of the camera. In fact, it has proven useful to put locating indicators at an even further reduced location for better locating of the image in the viewfinder. This allows for easier assessment of the symmetric positioning of the overlay in the camera. As mentioned previously, the location of these indicators on the image inside the CAD program is simpler to address numerically if the lower left corner of the image is located at a convenient coordinate such as the origin. In the case of the overlay affixed to the viewfinder on the rear of a

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Figure 11.2 Relation between the visible area and the exposed area. Visible area shown hatched.

digital camera, the viewfinder very likely has 100% coverage of the image collected by the CCD. In this situation, no allowance is needed for a reduction in visible area of the viewfinder versus that of the CCD. After the overlay has been placed into (or onto) the camera, the next step in the process is establishing the location and orientation of the camera at the site. This is done by locating and adjusting the orientation of the camera to match the conditions at the time the original image was taken. An iterative method is used by locating the camera equipped with the overlay at the approximate location of where the photographer was standing when the original photograph was taken. This is done by looking through the eyepiece and adjusting the various degrees of freedom until a match is obtained between the feature outlines on the overlay and the corresponding features in the image visible through the camera lens. A list of the degrees of freedom of movement of the camera available to the analyst in this process is presented in Table 11.1.

List of the seven degrees of freedom for the reverse projection method.

Table 11.1

Description of the Degrees of Freedom of the Camera in Reverse Projection Photogrammetry Translation parallel to the ground plane (two degrees of freedom) Translation perpendicular to the ground plane (may not be needed if the height of the photographer is known with reasonable accuracy and the photographer was standing when the image was captured) Rotation about the normal to the film plane Rotation about one axis in the film plane Rotation about the other axis in the film plane Zoom lens should be used if the focal length of the original camera is not known

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Systematic approach techniques can be used to change the camera position to obtain the location and orientation of the camera at the time the original photograph was taken (and the focal length if not known). After an initial estimate has been made of the location of the camera, it is beneficial to try to locate objects in the background of the image that have been included on the overlay, such as distant utility poles, signs, tree lines, the horizon, etc. Once the background objects have been appropriately matched, landmarks in the foreground that have been marked can be matched to the overlay. A final fit can then be made using small adjustments of the location and orientation of the camera. Several additional techniques can simplify this adjustment process: • Accurate information about the eye height of the photographer and his/her position at the time the image was captured eliminates one of the degrees of freedom in the camera positioning. • Information about the focal length of the camera and lens that was used to capture the original image also eliminates another degree of freedom. If the focal length is known, this degree of freedom is eliminated. If the focal length of the original camera cannot be determined, a zoom lens can be used during the trial and error locating process to vary that degree of freedom, but this adds complexity to the operation. • Use of a tripod is indispensable in this process because locating and adjusting the camera is time-consuming, and holding the camera steady and at a fixed location will become increasingly difficult. An adjustable pistol grip attachment on the head of the tripod, available at most full-service camera stores, simplifies the small adjustments that are needed for a good match between the landmarks at the site and the outlines on the overlay. Once the process of adjusting the camera position and orientation to match the overlay image with the site landmarks has been competed, the process of locating the features of interest in such a way that they can be measured by conventional means can be done. The marks can be preserved using any means such as spray paint, staking, cones, etc. Prior to presenting an example that shows how the location of a snow pile and its height were determined years after the snow had melted and disappeared, a list of the main steps used in the reverse projection process is given.

11.2.3 Summary of the Main Steps in the Reverse Projection Photogrammetry Process 1. Obtain a photograph (negative, print, or digital file) in which the characteristic of interest appears along with various permanent landmarks and features. 2. Using a CAD program, trace the outline of the landmarks and the required information that appear in the photograph. 3. Prepare an image of the traced information as an overlay and fit it into (or onto) a camera such that the image is centered in the viewfinder.

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4. At the site, visually compare the overlay image characteristics in the viewfinder with the actual landmarks, making adjustments to the camera location and orientation until a match between the two is obtained. 5. Use the information visible in the camera viewfinder to place marks at the locations required by the analysis. In this project, good photographs were available from the investigating officer. However, Example 11.1 the photographs were taken at night, making the project slightly more difficult. The main difficulty resulting from the darkness was the fewer visible fixed landmarks available to set the position of the camera, particularly landmarks in the distance. Ultimately, a sufficient number of signs, streetlights, and other landmarks were available to make the project tractable. In this case, the local law enforcement officials allowed high-resolution full-frame color scans of the negatives to be made. Two photographs were selected for the purposes of reverse projection photogrammetry. Digital scans were imported into a CAD program, and outlines of the landmarks were traced onto a separate layer of the drawing. Figures 11.3 and 11.4 show the original full-frame image as scanned and the image with the landmark feature outlines displayed, respectively, for one of the photographs chosen for analysis. The original images were obtained in color but are shown here in black and white with no loss in generality. Figure 11.3 Image as obtained from a scan of the negative from local law enforcement officials.

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Figure 11.4 Image with the major features traced onto a layer separate from the photograph; both layers are shown.

Note that on the image with the outlines, shown in Fig. 11.4, the exposed region of the negative has been blocked-in along with two sets of corner braces. These braces will appear at the corners of the visible region of the viewfinder. The outer corner braces correspond to the reduction in visible area compared to the exposed area. The inside corner braces correspond to an arbitrary but consistent area-reduction location inward from the corners but located completely in the visible area of the view finder. This is done for purposes of visually locating the overlay symmetrically within the viewfinder. If the overlay is located properly, these braces will appear equidistant from the corners of the limits of the viewfinder. Figure 11.5 depicts an image of the tracing that appears on the overlay separate from the photograph, which was shown in Fig. 11.4. The image shown in Fig. 11.5 was printed onto transparent material at a size determined by the characteristics of a Canon F-1. The overlay was trimmed and placed into the camera under the focusing screen. The camera was taken to the accident site for the resection. Using the landmark information on the overlay, the camera was positioned in the approximate location and orientation at the time the original image was captured. The information on the overlay associated with the pile of snow was then used by the analyst in directing a colleague to mark points on the roadway where the snow pile ended. The profile of the snow pile, including its height, was needed to assess its effect on the visibility of drivers approaching the intersection. The analysis determined that the photographer was standing in the turn lane used by traffic approaching the snow pile.

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Figure 11.5 Transparency created from the photograph shown in Fig. 11.3.

The height of the snow pile along a section line was determined by placing a telescoping measuring rod at intervals along the section line and noting the height of the intersection of the line on the overlay with the rod. The locations of the measuring rod along the section line were noted and surveyed along with the rest of the accident site using a total station. The heights of the snow pile at the various locations were incorporated into a CAD drawing and used to create the profile of the pile of snow that would have been seen by a driver approaching the intersection. Reverse projection projects can present practical difficulties in following the theory directly. In the majority of projects, the negative (or a digital file) of the photograph is not available and the photograph itself must be digitally scanned. The photograph is unlikely to be full-frame. Although these factors require the analyst to deviate from the ideal situation, acceptable results can be obtained from reverse projection projects under these circumstances. Useful information has been obtained using reverse projection photogrammetry under extreme circumstances. For example, the careful use of the reverse projection method with a video image has yielded useful data. The use of the method under less than ideal circumstances leads to the question of the accuracy of the method. The empirical nature of this method makes evaluation of accuracy difficult, and only two references were found in the literature that addresses the accuracy of the method [11.7, 11.8]. In the first reference, a quantitative comparison of the reverse projection method used to quantify the residual crush of a vehicle is made with direct measurements of the vehicle. The second reference considers uncertainty of the reverse projection photogrammetry process through two examples in which errors

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in the process are identified. A formal, quantitative study of the uncertainty associated with the reverse projection process does not appear to have been undertaken. In addition to discussing accuracy of reverse projection photogrammetry, Wooley, et al. [11.7] present a variation of the method that uses two cameras performing reverse projection simultaneously. In that application, both cameras use transparencies created from photographs of the same object taken from different positions to determine the crush profile of a vehicle. The second reverse projection example uses this same technique to determine the location of a tire mark on a roadway that has been resurfaced. In a typical application of the reverse projection method, the analyst uses the camera as a means to project the information contained on the overlay out into the object space. An assistant is typically guided to mark one or more locations in object space at the intersection of a light ray with some feature in the object space. This is usually done by locating the intersection of the light ray with a surface, often the surface of a roadway. In circumstances when the object of interest does not lie on a surface or the surface is inaccessible, a surface in the object environment or some other means must be defined by the analyst to determine the location of interest on the ray projected from the camera. In the previous example with the upper profile of the snow pile, a plane was defined along a section line through the pile prior to conducting the analysis. The intersection of the projected ray with the plane was used to delineate the outline of the snow pile. When the definition of a plane in the object space by the analyst is not possible, or the location of the plane is physically inaccessible, two cameras can be used, and the intersection of the two light rays from the two cameras through the same point can be used to locate the point of interest on the rays. An application of this technique is presented next. Example 11.2 In this application, two police photographs were available in which several tire marks

of interest appeared. Between the time of the accident and the time of the inspection, the road at the accident site had been stripped and resurfaced, thereby eliminating any possibility of directly examining the evidence. Moreover, the new road surface was higher than the old surface, making the plane on which the mark appeared in the scene photographs completely inaccessible. Under these circumstances, two cameras were required to conduct the reverse projection analysis. Figures 11.6 and 11.7 show the view of the site with the new roadway surface through the lens of the cameras used in the analysis at their respectively reconstructed camera locations. The markings from the overlay and their corresponding accident site counterparts are visible in the photographs. Because the location of the intersection of the two light rays from the two cameras was below the surface of the new roadway, no physical means was available at the site to locate the point in three-dimensional space. A diagram of this situation is shown in Fig. 11.8. Therefore, a total station was used to document the location and orientation of each camera and the location of the intersection point of both light rays from each camera with the new road surface for a given feature. These latter two points are indicated in Fig. 11.8 by the cross-hairs. Using the location of the cameras and the intersection of their respective light rays with the new, higher road surface, two lines in three-dimensional

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Figure 11.6 Photograph of site through the eyepiece of one camera.

Figure 11.7 Photograph of site through the eyepiece of a second camera.

space were defined. These lines were subsequently extended in a CAD program to locate their intersection point. A series of points located using this technique was used to define the length of the tire mark. An alternative approach to the reverse projection photogrammetry process described above has been developed [11.9, 11.10]. Under the alternative approach, the manual resection is done on the computer rather than at the accident site. The accident site is represented by a detailed total station survey and is imported, along with the photograph

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Figure 11.8 Diagram showing the use of two cameras, each implementing reverse projection for the same tire mark, which is located below the new road surface. The portion of the ray from each camera between the cross-hairs and the tire mark is the portion of the ray below the surface of the new roadway.

of interest, into a three-dimensional software program such as 3D Studio Max [11.11]. Programs of this type allow for a viewpoint (essentially a camera location, orientation, and focal length) to be defined within the three-dimensional environment defined by the survey. The resulting view is visible on the computer monitor. The viewpoint location, orientation and focal length are changed iteratively until the various features of the survey match their counterparts on the photograph. Once a successful resection is completed, the transient information from the photograph is projected into the threedimensional survey environment. One benefit of this method is that if the location of the camera happens to be on a busy roadway, the resection process is significantly easier to handle on the computer rather than out on the actual roadway.

11.3 Planar Photogrammetry There are times when it is appropriate (and convenient) to use a different photogrammetric tool to determine the location of a mark (or similar physical evidence) that has been captured on a photograph taken at an accident scene. When the mark lies on a surface that can be reasonably approximated by a geometric plane, usually the surface of a roadway, a one-to-one mathematical transformation between a planar coordinate system affixed to the plane of the CCD (or the film plane) and a planar coordinate system on the roadway can be established, provided certain information about the road surface is available. A generalized development of the coordinate transformation between the planar coordinate system in the CCD plane (also called the image space) and the planar coordinate system on the roadway surface (also called the object space) can be found elsewhere [11.2]. The Appendix to this chapter presents the salient points of this derivation. The results of the derivation are two equations that relate points in the two coordinate systems:

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xm =

ym =

c1 + c2 x p + c3 y p c4 x p + c5 y p +1



(11.4a)

c6 + c7 x p + c8 y p c4 x p + c5 y p + 1



(11.4b)

In Eqs. 11.4, the m-subscript refers to the coordinate system corresponding to the roadway plane and has traditionally been referred to as the map coordinate system, giving the m-subscript. The p-subscript refers to the coordinate system on the photograph (the CCD or film plane). The quantities, ci, i = 1 to 8, represent eight unique constants involved in the transformation. In these equations, eight constants appear and must be determined to yield a useful transformation between the two planar coordinate systems. To determine the eight unknown constants, eight equations are required. Four reference points are needed in which the x-y coordinate pairs in both CCD plane and roadway plane, the p-coordinates and the m-coordinates, respectively, are known. No more than two of the four reference points can be collinear. The eight equations are linear in the constants c1 through c8 , and any suitable numerical method that can solve linear equations can be used to obtain a solution. Once the eight coefficients are known, the same equations can then be used to determine the map coordinates, (xm, ym), of any other point visible in the photograph that lies on the roadway plane. Utilities to solve sets of linear equations are generally available in spreadsheet programs, sometimes with the assistance of macro programming. This technique, used for the examples in this book, can be used to locate a single point on a roadway, or repetitive calculations can be used create a piecewise linear approximation of characteristics or objects such as tire marks located on a roadway. The assumption made in the development of this method is that both the captured image and the (desired) physical characteristics lie on a plane. In the case of the captured image, the film itself is contained securely in the “film plane” of the camera and inherently meets this requirement. The same is true of a digital camera that uses a CCD to capture the image. The CCD, and the image it captures, satisfies the assumption of being a plane. The assumption that the physical objects that appear in the image lie on a plane is usually just that, an assumption. In the case of accident investigation, the physical planar object is typically a roadway. Roadways may have a curvature, a crown, or both, and it may be difficult to assess the planar condition quantitatively. Any nonplanarity of the physical object captured in the photograph introduces error in the results of the analysis. The first example presented in this section, in addition to demonstrating the method, considers the magnitude of the error associated with a nonplanar condition and some of the factors that influence it. A simple and practical way to obtain the coordinates from the photograph is to digitize the photo (or the negative if it is available) and import the image into a CAD program. (Of

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course, a digital image can be directly imported into the CAD program.) A convenient point of the image should be designated as the origin of the film plane coordinate system. Using the lower left corner of the image as the origin makes all the points on the image have positive coordinates. As with the technique described in the section on reverse projection, it may also be useful to draw lines, points, etc. on the photograph in the CAD program as a means to document and create an accounting of the points. The image in Fig. 11.9 was created this way. With the attributes marked on the photo, the coordinates associated with them can be obtained easily and directly from the CAD program. The quality of the photograph can have a large influence on the accuracy of the results of this analysis. For this reason, it is important to obtain the best-generation photograph possible at the start of the process. A high-resolution scan of the negative or use of a digital photograph can produce better results than a scan of a photograph from a flatbed scanner. Care should still be taken with digital photographs. The mere existence of a given digital file of a photograph does not necessarily preclude the existence of another digital file of higher resolution, as images can be downsampled; i.e., a decrease in the number of pixels in the image. Example 11.3 The image shown in Fig. 11.9 was acquired using a digital camera (3.1 megapixel), and

the (reasonable) assumption is that the tile floor is a close approximation to a geometric plane. High-contrast markers placed on the floor were used to designate the four corners of a square that is, for the purposes of this example, exactly 10 ft on a side. The corner of the square at the bottom of the photograph serves as the origin of the object-space coordinate system, (x0, y0), to be used in the analysis that follows. Additionally, the positions of the four corners are known in both the map and photograph coordinate systems and can be used as the reference points used to determine the eight coefficients. The coordinates of other points that lie on the plane (whether within the square or outside it) can then be obtained from the transformation.

Figure 11.9 Photograph prepared for four-point transformation analysis.

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This example was staged to evaluate the accuracy of the planar photogrammetry process. It is not meant to be a demonstration of the technique as it applies to a real situation. An additional example will be presented that addresses a practical application of the method. Every reasonable attempt was made in this example to control the parameters involved. The centers of the high-contrast corner markers were placed at the intersection lines of the tile floor; each tile is a square with each side having a length of one foot. The photograph was intentionally taken with a lens aperture yielding sufficient depth of field to ensure proper focus of both the foreground and background. In the coordinate system on the floor plane, the corner closest to the camera is assigned as the origin (0, 0), with the corner up and to the right in the photograph assigned (10, 0). Figure 11.10 shows a drawing that depicts this coordinate system in a plan view. A machined V-block with precise dimensions (see Fig. 11.9) was also placed on the floor at a specific known location and will be used later to assess the error for an “out of plane” point. For reference, the dimensions of the V-block are given in Fig. 11.11. For example, in an actual setting, a nonplanar condition can occur when a crown in a road is present.

Figure 11.10 Coordinates assigned to the floor plane (plan view).

The position of other points measured in the photograph coordinate system can be used for computation of their corresponding coordinates on the floor plane. Table 11.2 lists the points that were transformed and the corresponding actual coordinates. Another source of error in the planar photogrammetry process comes from the accuracy of the coordinates associated with the four points used to determine the constants in the transformation. Using the photograph presented in Example 11.3 and allowing for

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Figure 11.11 Detail of the V-block used in the evaluation.

Table 11.2

Comparison of computed vs. actual coordinate values (in feet). Description

Computed x

Computed y

Actual x

Actual y

Center of the square

4.999

5.004

5.000

5.000

Corner of the V-block at (5,6) but 3.125” out of plane

5.341

6.313

5.000

6.000

Base of the V in the V-block, 2.125” above the floor plane

5.079

6.372

4.843

6.147

inaccuracies in the selection of the four base points on the photograph, the magnitude associated with base-point error can be investigated. Example 11.4 In this example an error is introduced into the reference point measurement by

intentionally introducing a five-pixel error in the selection of the base point. Pixels were used to define the error, as opposed to the photograph coordinate system units or absolute distances in the plane, as it is this resolution that will be the likely limitation in the process of selecting these points. The error of absolute distance per pixel is greater the further the point is from the camera that captured the image. In general, a higherresolution image can help mitigate this type of error. The direction of the error from the actual point will have an effect on the magnitude of the error. Therefore the error was selected systematically as follows: • The errors of the points (10, 0) and (10, 10) were selected to be to the right in the image. • The errors of the points (0, 10) and (0, 0) were selected to be to the left in the image. The net effect of the errors is an enlarging and a slight skewing of the square. Using the dimensions associated with these new reference point dimensions from the photograph,

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the analysis can be performed. A comparison can be made between the center of the square (5, 5) and the right lower corner of the V-block (5, 6). Table 11.3 presents the results for comparison. Comparison of computed and actual locations with the computed values determined from base-point locations with introduced error. Computed x

Computed y

Actual x

Actual y

Center of the Square

Description

4.995

5.023

5.000

5.000

Lower right corner of the V-block

4.985

6.025

5.000

6.000

Table 11.3

Note that in Examples 11.3 and 11.4, the plane used in the object space, the tile floor with the high-contrast markers, creates an almost ideal situation for the application of planar photogrammetry. Applications of this method in the field will certainly be done with photographs that do not have high-contrast markers at the reference points, and the points of interest will not be as easily determined. Errors will increase with the lessaccurate information. The following example illustrates a practical application of planar photogrammetry. Having considered the development and accuracy of planar photogrammetry, a practical Example 11.5 field example is presented. In this situation, a vehicle is traveling north along a two-lane rural road and makes a sudden move to the left across the oncoming lane. Figure 11.12 shows a plan view diagram of the accident site. Figure 11.13 shows a photograph to be used with planar photogrammetry to determine the location and characteristics of the tire marks. As a result of the sudden steer input to the left, the vehicle begins to yaw in a counterclockwise direction. It crosses the southbound lane and leaves the roadway at an angle of about 50° counterclockwise from north and comes to rest on a parallel frontage road. During the yaw maneuver, the right side tires leave visible tire marks on the roadway. It is desired to determine the speed of the vehicle on the roadway. The critical speed formula (see Chapter 4) can be used to estimate the speed of the vehicle if the radius of the tire marks can be determined. Planar photogrammetry is used to reconstruct the tire marks. Although the details of the application have not yet been described, the tire marks have already been added to Fig 11.12 for simplicity. The tire marks were not measured by the police and were no longer visible during a site inspection. The police took photographs of the accident scene, and one of the photos, shown in Fig. 11.13, was chosen for analysis. The assumption that all the points of interest lie on a plane is reasonable, as the road is essentially flat over the region of interest. This is assessed visually from the scene photograph. In this instance, the known position of four landmark points required for implementation of the method and associated with the physical roadway could not be established in

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the scene photograph. An alternative was needed. The intersection of the utility pole shadows across the roadway with the east and west edges of the roadway presented a convenient solution to the dilemma. This is a somewhat unconventional approach to solve the problem but illustrates the flexibility that can be offered by this photogrammetric analysis technique. Figure 11.12 Plan view of the accident scene.

The information required to locate the intersection points between the poles and the east and west edges of the highway is the positions of the poles relative to the highway, this distance between the poles (which had not changed), and the angles of the shadows relative to the highway. The distance between the poles and from the poles to the roadway were measured at the site during an inspection. A software program [11.12] was used to determine the angle between the pole shadows and the highway. (Solar position information can now be readily found on the internet.) At the time and location of the accident, this angle was 14°. The locations of the four intersection points of the pole shadows with the two edges of the highway were used for the four reference points needed to determine the eight constants in the transformation. Figure 11.13 Police photograph (looking north) showing the tire marks and the utility pole shadows.

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The planar photogrammetry equations, Eq. 11.4, were used to establish the location of three points along the mark left by the leading front tire. The radius of curvature of the mark can be determined from these three points, and the critical speed formula can be used to reconstruct the speed of the vehicle. In this case, it was determined that the vehicle was exceeding the speed limit posted for this section of road. In practical applications of planar photogrammetry, it not always possible to estimate the error in the analysis. However, one or more additional points with known coordinates in both the CCD/film plane and object plane coordinate systems can be used to assess the accuracy of the analysis. After the transformation has been determined, the CCD/ film plane coordinates of the additional point are used to calculate the coordinates of the point in the object space. These calculated coordinates are then compared to the measured coordinates to assess the accuracy of the analysis. In this example, an additional point on the photograph with known object plane coordinates was available to evaluate the accuracy of the analysis. A third utility pole located north of Poles 1 and 2 (not shown in Fig. 11.12) and its corresponding shadow can be seen in the photograph (Fig. 11.13). Close inspection of a first-generation photograph indicated that the shadow crosses the edge line on the east side of the road just south of the northernmost northbound vehicle shown in the photograph. This point permits the comparison of the value calculated using the transformation to the value determined via the CAD drawing and the known position of the sun. The location of Pole 3 was also measured during the site inspection. With the location of Pole 3 and the angle of the sun known, the location of the intersection of the shadow of the pole with the east edge of the northbound lane was determined from the site diagram. It was found to be 453.7 ft (138.3 m) north of the intersection of the shadow from Pole 1 along the west edge of the roadway. The corresponding location of the same intersection point, calculated using planar photogrammetry, was found to be 436.4 ft (133.0 m). Assuming the location of the point calculated from the site diagram based on the location of the pole and sun is the true value, calculation yields an accuracy of (453.7 - 436.4)/453.7 × 100 = 3.8%. This low level of error indicates that reasonable accuracy for the radius of the tire mark was achieved in this analysis.

11.4 Three-Dimensional Photogrammetry The photogrammetric techniques presented in the previous two sections are commonly used in accident reconstruction. They address specific needs related to obtaining quantitative information of objects from a single photograph. These methods have been the subject of numerous technical papers over the last 30 or 40 years and have found acceptance in the accident reconstruction community. Other methods must be used in applications where three-dimensional measurements are required. A mathematical foundation has been developed for three-dimensional photogrammetry [11.1]. This foundation, and the equations that accompany it, are complicated, and the use of threedimensional photogrammetry has traditionally been beyond the reach of most potential users, including automotive accident reconstructionists. However, developments in the

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personal computer hardware and software arenas over the last eight to ten years have changed this situation dramatically. New hardware, software, and image capturing and handling methodologies have put the use of three-dimensional photogrammetry into the hands of accident reconstructionists as well as others [11.13]. This type of photogrammetry, where the distance between the camera and the subject is small compared to aerial photogrammetry, is referred to as close-range photogrammetry. This section presents the fundamentals of the mathematical basis for the method. The mathematical basis for transforming coordinates on a photograph into threedimensional coordinates is presented. Readers interested in the method for program development or a deeper understanding of the theory are referred to books with more detail [11.2, 11.3]. This chapter also includes some practical insights about the application of the technique to automotive accident reconstruction. For a given set of inputs, the mathematics produces a one-to-one transformation between points in two coordinate systems. However, the accuracy of the process of threedimensional photogrammetry is difficult to characterize. There has been information published about the accuracy of the method [11.2]. However, software that performs close-range photogrammetry typically does not advertise the accuracy of the results, as accuracy in the practical application of threedimensional photogrammetry depends heavily on factors related to the inputs to the software program. These factors include the quality of the photographs, the accuracy of the measurements used to control the scale of the photographs, as well as the number of the points marked in the photographs used in the analysis and the quality with which they have been marked by the analyst. This inability to advertise accuracy is in contrast to the other instruments that the accident reconstructionist community uses wherein the accuracy of the instrument does not usually depend on how the instrument is used. Accuracy is frequently a performance specification provided by the manufacturer. This is true for instruments such as total stations and coordinate measuring machines. In three-dimensional photogrammetry the accuracy of the output dimensions from the analysis cannot be more accurate than the input dimensions that control the scale of the analysis. Hence care must be exercised when using this method. However, comparison of the method has shown that three-dimensional photogrammetry produces results that are acceptable for use in accident reconstruction applications. Beyond some of the concerns that exist, there are times when three-dimensional photogrammetry offers the only means for the extraction of useful dimensional information from an accident site. Accident reconstructionists routinely encounter the situation where the accident site has changed prior to an inspection, information such as the height of an object is no longer at the accident site, or an unmeasured rest position of a vehicle shown in photographs is required for reconstruction purposes, and the reverse projection or planar photogrammetry methods cannot be applied. One useful characteristic of photogrammetry is that it allows measurements to be made without necessarily contacting the measurement point. In certain situations,

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this characteristic can be an advantage over other measurement methods. An example would be when the items to be measured are in a location where it is dangerous to make direct measurements such as a roadway where traffic cannot be conveniently stopped to gain access for direct measurement. Other environmental conditions may render information needed from an accident site accessible only from a distance by taking photographs. An example of this is the height of a street light above the roadway for visibility assessment pertaining to a night accident. Often, photogrammetric analysis can be used to acquire the needed dimensional information. Moreover, under certain circumstances, the remote, nonintrusive nature of the method may be an advantage as access to the site or specimen under study may be restricted to a noncontacting level. The nature of a typical photogrammetry project consists of preparing for and taking the necessary photographs and using appropriate software for performing the analysis to obtain the desired dimensional information. Two options related to this process can make photogrammetry an alternative to other measurement methods such as a total station or hand measurements (tapes and measuring wheels). The first is that the necessary photographs for a successful photogrammetric analysis can be taken during a site or vehicle inspection, typically with little additional time investment. The time and expense associated with the photogrammetric analysis can be deferred until it is decided that the unknown site or vehicle dimensions are actually needed. If the dimensions are never needed, then the money for the analysis is not spent. Another advantage is that after the analysis is performed and the required dimensional information is obtained, the same photographs can frequently be used at a later time to obtain additional dimensional information with minimal time and expense and no need to visit the accident site again. These options are illustrated using the example of measuring the residual crush profile of a vehicle using photogrammetry. For this task, the vehicle can be set up for the photographic session and the appropriate photographs taken. The analysis can be deferred until such time that the profile of the crush is explicitly needed. If, after an initial analysis is performed, additional profile or dimensional information is needed, such as the location of a displaced front axle, the same set of photographs can be used to determine this information with little effort. This analysis can be performed long after the vehicle has been salvaged.

11.4.1 The Fundamental Information Related to Three-Dimensional Photogrammetry Three-dimensional photogrammetry, and photogrammetry in general, uses a certain vocabulary. This section introduces the technical basis for the method and familiarizes the reader with some of the terminology. In addition, information is presented that provides the users of photogrammetric software with insight into the mathematical process that takes place when the analysis is carried out by the computer. An understanding of the mathematics assists in the proper taking of the photographs and assessment of the suitability of photographs taken by another photographer for photogrammetric analysis. This view will become clearer as the section progresses.

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11.4.2 Mathematical Basis of Three-Dimensional Photogrammetry Photogrammetry, in its most basic sense, is a series of coordinate point transformations. Three coordinate systems are used in photogrammetry [11.15, 11.16]. The coordinate systems are associated with the space in which they exist. Object space is the threedimensional coordinate space that we operate in daily. Image space is the coordinate system associated with the camera at the moment that the image is captured. This coordinate system can be considered to be fixed to the camera and defined in location and orientation by the action of capturing the image. The third coordinate system resides in image space and is referred to as the coordinate system of the measuring device. It is a coordinate system that exists in the plane of the film or on the face of the CCD in the camera. There are certain attributes of these coordinate systems that serve to relate them. The first is that a unique line exists along which the light that enters the camera through the lens glass is not refracted. This line is called the principal ray. The point where the principal ray intersects the image capturing device (film or CCD) is called the principal point. This point can be considered to be the geometric center of the image and is typically designated as the origin of the image-space coordinate system. The x-axis of this system is in the plane of the film in the camera, intersects the principal point, and is usually parallel to the base of the camera. The y-axis of the image-space coordinate system lies in the plane of the film, passes through the principal point, and is perpendicular to the x-axis. The z-axis of the image-space coordinate system is perpendicular to the x- and y-axes and runs along the principal ray. The lens axis lies along the principal ray, and a certain location along this axis is designated as a single point, usually denoted O. This point is also frequently referred to as the lens nodal point. Although in reality complex lens arrangements don’t follow this approximation strictly, particularly zoom lenses, this approximation will suffice here. The distance between the lens location, O, and the principal point is the effective focal length of the camera-lens combination. For a fixed-focal-length lens, this value is provided with the lens, but for a zoom lens, this value can change. When this value is known, it can be provided as input to the analysis routine, and when it is unknown, it is calculated as part of the analysis. Ultimately, as part of the analysis required for the determination of measurements from the photographs, points will be located on the photographic images. The location of these points will be defined in another coordinate system. This system is referred to as the measuring instrument coordinate system. In the analysis, these coordinates need to be transformed into the image space and then to the object-space coordinates. Once these transformations exist, they can be performed in either direction. Typically, we are looking for the three-dimensional object-space coordinates associated with known coordinates from the measuring instrument coordinate system. The following transformation equations are from object space to image space and are called the projection equations.

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11.4.3 Projection Equations The coordinate transformation from object-space coordinates to image-space coordinates is comprised of a coordinate translation and coordinate rotation. If the coordinates in the object-space are given as {X Y Z}T, then the translation to the camera position at the nodal point is:



 X '  X − X O       Y '  =  Y − YO  Z '  Z − Z  O    



(11.5)

where the O subscript refers to the nodal point of the lens, and {XO YO ZO}T are the coordinates of the nodal point of the lens in the object-space coordinate system. At this point, the coordinate system has been translated to point O, but it needs to be rotated to the orientation of the camera/image-space system. A three-dimensional coordinate rotation is needed. This transformation takes the form:

X ′′ = m11 ( X − X O ) + m12 (Y − YO ) + m13 ( Z − Z O )

Y ′′ = m21 ( X − X O ) + m22 (Y − YO ) + m23 ( Z − Z O )

(11.6)

Z ′′ = m31 ( X − X O ) + m32 (Y − YO ) + m33 ( Z − Z O ) where



 m11 m  21  m31

m12 m22 m32

m13  m23  m33 



(11.7)

is called the rotation matrix. Note that the form of Eq. 11.6 incorporates both the coordinate rotation and coordinate transformation. Another transformation relates the coordinate system as defined in Eq. 11.6, which is at the nodal point of the lens, to the plane of the film. In making this second and last transformation, it is instructional to first consider what is referred to as the thin lens approximation of a single-axis projection. Referring to Fig. 11.14, the quantity f is the effective focal length of the camera-lens combination. For this geometry and the use of similar triangles,



f D = h H

(11.8)

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Figure 11.14 Single-axis projection.

Using this relationship, the equations for the projection of the coordinate system located at the lens nodal point to the plane of the film can be written. Because this is a projection of the coordinate system onto a plane, there are only two equations. In keeping with the development by Townes and Williamson [11.16], the resulting coordinates are assigned lowercase letters:

x = − f X ''



y =− f



Z ''

(11.9a)

Y '' Z ''

(11.9b)

Using the relationships in Eq. 11.6, a single transformation for each coordinate can be written between the object-space coordinates and the image-space film plane coordinate space. These are:





x − xP = − f

m11 ( X − X O ) + m12 (Y − YO ) + m13 ( Z − Z O ) m31 ( X − X O ) + m32 (Y − YO ) + m33 ( Z − Z O )

(11.10a)

y − yP = − f

m21 ( X − X O ) + m22 (Y − YO ) + m23 ( Z − Z O ) m31 ( X − X O ) + m32 (Y − YO ) + m33 ( Z − Z O )

(11.10b)

In these equations, the values of xp and yp are the distances from the principal point to the origin of any arbitrary film-plane based coordinate system and have been introduced for

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generality. For background on the m-coefficients, interested readers can refer to many advanced mathematics or engineering texts; for example, [11.17]. If all of the quantities on the right side of Eq. 11.10 are known, then the object-space coordinates (X, Y, Z) can be transformed to image-space coordinates (x, y).

11.4.4 Collinearity Equations At the start of a typical photogrammetry session, the parameters described above that appear on the right side of Eq. 11.10 are not known. The m-coefficients depend on knowing the orientation of the camera at the time the image was captured, and the coordinates (XO, YO, ZO) specify the lens nodal point location at the same time. These six parameters—three orientation angles and three location coordinates—along with the focal length, f, provide for seven parameters that need to be determined. The process used to determine these parameters is called a resection. This process of determining the seven parameters is accomplished by using Eq. 11.10 with four points, where both the image-space coordinates and the object-space coordinates are known. This creates a set of eight equations and seven unknowns. This is an overspecified system, and a solution is possible using iterative methods. A solution is still possible if more than four known points are available, but four points is the minimum number. 11.4.5 Coplanarity Equations After the parameters are determined, Eq. 11.10 can now be used to transform twodimensional coordinates in image-space to three-dimensional coordinates in object space. In accident reconstruction this is typically done for points in the photographs that are of particular interest, such as points that define the residual crush of a vehicle, a vehicle location in photographs of the accident scene, or any object visible in the photograph for which location is desired. 11.4.6 Multiple Image Considerations The description above is the photogrammetric process as it applies to a single image. Once the seven parameters for the transformation are computed and the image-space coordinates for a point of interest are determined, the only unknowns are the three object-space coordinates for the point of interest. Because one photograph provides only two equations, those listed in Eq. 11.10, more than one photograph containing the point of interest must be available to determine all three unknown coordinates. If one of the three coordinates is available by some other means, then the solution for the other two coordinates can be obtained using one photograph. For the general case where all three coordinates need to be determined, another photograph taken from a location different than the first photograph must be used to provide for another set of two equations. The two photographs together yield four equations for three unknowns, an overspecified system. In practice, a single point whose object-space coordinates are needed can be located on more than two photographs. The solution process remains the same because the system under all situations is overspecified.

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11.4.7 Considerations of the Use of Three-Dimensional Photogrammetry in Practice Now that the analytical treatment of photogrammetry has been presented, the practice of multi-image photogrammetry, particularly as it applies to accident investigation and reconstruction, can be addressed. First and foremost is that the process of threedimensional photogrammetric analysis takes practice to become proficient. This fact is true independent of the software used to perform the analysis. It cannot be overstated that there is a ramp-up time associated with acquiring the tools and the eye for the process. It is recommended that prospective users who are serious about including photogrammetry in their repertoire of accident investigation measurement tools perform several practice projects before trying to apply the method in an actual project. In addition to the tutorials that typically accompany photogrammetric software, the user should set up several projects with relatively simple geometries to develop the skills associated with a successful project. Examples of simple photogrammetry projects appropriate for practice are • one or two sides of a building with regular patterns associated with windows, bricks, roof line, etc. • the front profile of an undamaged vehicle • part of the interior of a large room such as a gymnasium that has brick work, windows, and other regular patterns as part of the wall Because the scale of the objects does not matter, it may be simpler to perform the practice analysis on a scale model vehicle. This can make the session easier to stage and easier to introduce scaled items into the photographs. For instance, the scale model can be placed on a large sheet of graph paper, thereby establishing an object-space coordinate system while providing a simple means to select points in the foreground and background. Three-dimensional photogrammetry will not likely replace any of the other measurement tools or techniques that are currently in use in the accident investigation community. However, it does offer an alternative that, as with the other measurement techniques and tools, has advantages and disadvantages. Frequently these various measurement techniques need to be used together to produce the desired results. The practice that is required to develop proficiency in this method should commence with the analyst taking a series of photographs specifically for photogrammetric analysis. In addition to traditional knowledge about photographic topics such as shutter speed, aperture, ISO, and CCD resolution, this section lists several guidelines about photography for three-dimensional photogrammetric purposes: • Photographs should be taken with the largest possible depth of field (smallest physical aperture size, highest aperture number). A tripod should be used, as a smaller physical aperture will typically require longer shutter speeds for correct exposure. These longer shutter speeds are frequently at, or beyond, the limit of handheld steadiness for a camera. The larger the depth of field, the better the opportunity to use points in the foreground and distant background of the

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photograph that are in focus. The use of these points enhances the accuracy of the process. • High-contrast markers should be used and placed at the points of interest in the scene when possible. These markers include florescent color stickers or reflective markers. They are essential when using photogrammetry for crush measurements, as selecting the same point on a damaged vehicle from photographs taken from different camera angles can be difficult. Reflective roadway and driveway markers can be introduced into a site. These markers can be either free-standing or affixed to a pole. These have a dual function as they appear prominently in the photograph, and most total stations will shoot the marker directly, thereby simplifying the use of this point as a control dimension. (Some of the photogrammetry software packages now provide dedicated targets to facilitate automated marking and referencing of points in a photograph. This capability eases the use of the software and increases productivity.) • The introduction of high-contrast markers assists in creating common points between photographs. An example of this technique is placing small cones on top of a crushed vehicle to serve as common control points in the photographs taken for the purpose of measuring crush profile. If composed properly, the top points of some or all of the cones will be visible. This simple technique serves to relate photographs taken on one side of the vehicle to those taken on the other. • The location of the same point in different photographs can be visualized as that point at the intersection of two light rays, each from a different photograph, that intersect the same point in object space. Photographs taken from the same location and “nearly” the same angle can create numerical problems for the solution routines of photogrammetry software. The same would be true for camera angles that are approximately 180° apart. Proper camera angle separation is important in the accuracy of the solution. In photogrammetry applied to automotive accident reconstruction, it is convenient to consider the possible camera positions for a photograph session of an object of interest described loosely as cameras positioned on a hemisphere. Typically, photographs from inspections related to accident investigation are captured by a camera position that encompasses an approximate circle about 5½ ft to 6 ft (1.7 m to 1.8 m) off the ground (the eye height of a standing adult). This should be modified to include camera positions higher (and lower) on the hemisphere, giving angular separation by vertical camera motion rather than horizontal camera separation only. The use of a step-ladder, squatting, or using a monopod with the camera held in the air to take a picture facilitates camera angle separation. Accuracy of the three-dimensional photogrammetry session is influenced by many aspects of the process. Several of these are • The ability to adequately mark the points in the photographs used for control. This requires the analyst to compose the photographs ahead of time to determine whether adequate object-space features exist to provide sufficient control to the project.

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• The measurements of the control points. This is critical to the process in that the accuracy of the measurements made using the photogrammetric process can be no more accurate than the measurements of the control points. It should be pointed out that measurements of the control points made using pacing, visual estimates, and tape measures are not acceptable for the use of establishing control points for photogrammetric analysis. The preferred method is to use a laser measuring device to establish the three-dimensional points on immovable objects such as points on fire hydrants, guard rail components, markings on utility poles, features associated with structures (corners of windows, peaks of roofs, bridges, etc.), road markings, features of traffic signs, and others. • The quality of the photographs can affect the overall quality of the photogrammetry process. When possible, the negatives or earliest generation of the required photographs should be used. This is true for digital photographs as well as film format. Digital photographs can be resampled for a reduction in resolution. Inferior photographs can be avoided when the analyst is also the photographer. • Calibration of the camera and lens combination is typically required for the photogrammetric analysis. Therefore, as a practical matter, photographers should use the set of photographs as a means to document the process to avoid confusion during the analysis. For example, a documentation sheet should be photographed at the beginning of the photography session that lists the camera and lens combination used to take the photos. If the lens on the camera is a zoom lens, the sheet should indicate the focal length setting that was used for the photographs (typically the shortest). For practical purposes the focal length of a zoom lens used for photogrammetric purposes should be set only at either extreme focal length for the entire set of photographs. The software used to perform the calculations generally does not provide a direct means to assess the accuracy of the results of the process. However, there are certain relatively simple means to internally check the accuracy of a three-dimensional photogrammetric project. In the scope of this discussion, the accurate control points obtained from a survey of the object space are used to establish the transformation needed to calculate the desired coordinates of unknown features of interest. The process typically requires that there be four control points per photograph. An indication of the accuracy of the project can be easily assessed by obtaining more than the required control points, but leaving one or more of the control points as desired information to be calculated in the session. After the project is complete, the computed location of these “pseudo-control” points can be compared to the known measured location. With a number of these pseudocontrol points located throughout the object space (foreground, background, and close to the object of interest), a quantitative assessment of the accuracy of the overall project can be obtained. In a similar vein, if the analyst is also the photographer, various devices of known, accurate dimension can be introduced into different locations of the object space prior to the photographs being taken. In the analysis of the project, the location of points

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defining these dimensions can be calculated, and the computed dimension can be compared to the known quantity. Several of these devices located throughout the object space can provide a quantitative assessment of the accuracy of the other dimensions calculated in the project for which measurements do not exist. This example demonstrates the use of three-dimensional photogrammetry for Example 11.6 determining the residual crush of a vehicle involved in an accident. The photographs used in the analysis were taken with a 35-mm film format camera. The negatives of the photographs were scanned (3072 H 2048 pixels, 6 megapixels) to enable importation into the analysis software. The software used for the photogrammetric analysis was Photomodeler [11.18], although an alternative computer program is available that is suitable for use in reconstruction applications [11.19]. The fiducial marks shown on the left side of the photographs are created by a fiducial insert provided with the software. These fiducial marks are needed for film format cameras and are not required for digital cameras. The residual crush was the result of a staged accident. For the staged accident, the vehicle was marked in various locations with red and white high-contrast circular markers and checked tape. Some of these markers were used in the photogrammetric analysis. Additional markers were added to the vehicle prior to taking the photographs, as described later. Six photographs, shown in Figs. 11.15 through 11.20, were used in the photogrammetric analysis. In this situation, the photographs were taken with the express purpose of determining the residual crush of the vehicle. Therefore, items visible in the photographs were included to facilitate the photogrammetric analysis. The unusual-looking chart situated on the floor in front of the vehicle was used to set the scale of the analysis. The distance between the dots at the center of the two lobes on either end of the device was measured and recorded at the start of the inspection. This distance is used later as an input during the analysis. The circular format of the chart was selected to enhance visibility, even at great distances, but also to take advantage of the automatic targetmarking utility in the software that can locate the center of circular targets. As with hand measurements of the residual crush of vehicles, the location of an undamaged part of the vehicle is required as a reference. The location of the rear axle is typically used for this purpose. In this situation, the damage to the vehicle was so severe that the rear axle may have moved due to the impact. Therefore, the rear face of the rear bumper was used as the reference. This requires that the photographs of the undamaged rear end of the vehicle be directly related in some fashion to the photographs of the damaged front end of the vehicle. Direct referencing of points on the photographs of the rear of the vehicle with points that appear in the photographs of the front of the vehicle is accomplished using several small cones placed on top of the vehicle. These cones appear in both the photographs of the front and the rear of the vehicle. The blunt hemispherical tips of the cones were made sharper by piercing the tips from the underside with a large screw. This simplifies the marking of the very tip of the cone during the computer analysis. Several of the cones appear in most of the six photographs used in the analysis, thereby creating a set of common points in all of the photographs. Additional points

291

Chapter 11

marked on the sides of the vehicle link the photographs of the front of the vehicle with those of the rear of the vehicle through intermediate photographs. Direct referencing between photographs enhances the accuracy of the analysis, but use of both methods of referencing between points on opposite sides of the vehicle should be used. Figure 11.15 Photo used in photogrammetric analysis.

Figure 11.16 Photo used in photogrammetric analysis.

Measurement of the deformation to the front end of the vehicle is the task to be performed. As such, the front end of the vehicle is marked using circular adhesive dots of various colors demarcating the deformation at different positions on the front end. Dots of different colors were used to mark the deformation along the upper radiator support, the middle of the radiator, and the center and bottom of the bumper. Additional dots were placed at several locations that were deemed to be of interest for possible later

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Photogrammetry for Accident Reconstruction

Figure 11.17 Photo used in photogrammetric analysis.

Figure 11.18 Photo used in photogrammetric analysis.

reference. These locations include the ends of the frame rails, and a point at the base of the driver side of the windshield. Dots made of florescent colors or even dots made of retro-reflective material are typically easier to distinguish than dots of other colors, although both types can be used. Also note that a fill flash should be used in these situations to prevent a marker from being hidden in a shadow in one or more of the photographs. This is a common problem when photographing a damaged vehicle.

293

Chapter 11

Figure 11.19 Photo used in photogrammetric analysis.

Figure 11.20 Photo used in photogrammetric analysis

It is important to note that dimensional information about the residual crush of the vehicle is obtained only at the points marked in the analysis. In this example, the dots are these locations. If dimensional information of a particular feature or characteristic of the vehicle is needed, a marker of some sort attached to the feature is recommended. It is possible that the feature may be prominent enough that a marker is not needed. An example of this type of “natural” marker would be center of the circular head of a dark bolt located on a light background. Once the photographs are available digitally, the analysis is basically a three-step process:

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Photogrammetry for Accident Reconstruction

1. Calibrate the camera to be used. This calibration can be done before or after the photographs of the subject are taken (as long as the fiducial insert is used during the collection of the images for a film format camera). The camera calibration requires that the camera and lens combination be used to take a series of nine photographs of a calibration chart provided (electronically) with the photogrammetry software and run a separate calibration program using the nine photographs. 2. Load the photographs to be used in the analysis into the software. This requires the photos to be available digitally either through the use of a digital camera or through a digital scanning process. Higher-resolution photographs make the accurate marking of the individual points a simpler task. A six-megapixel image for the type of analysis performed in this example is usually sufficient, provided the photographs are taken with sufficient lighting and depth of field. 3. Mark the points of interest and reference the same points between photographs. These points include those associated with the deformation on the front of the vehicle, the points along the undeformed rear of the vehicle, the intermediate points for referencing, and the points for the scaling of the photograph. The markings placed on the image in the marking process in Photomodeler are shown in Fig. 11.21. The markings on the other five photographs look similar. Once all the points are marked and referenced between photographs, the photogrammetric analysis can be performed. The process of marking the photographs and referencing the points between the photographs requires practice to acquire proficiency. Most software

Figure 11.21 Photograph showing the markings associated with point designations in Photomodeler.

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Chapter 11

packages provide tools to assist in this process, and some automatic marking capability is a feature now included in the software. After a successful analysis, the dimensional information of the residual crush can be obtained in a number of ways. The simplest means for the users of Photomodeler to access the dimensional data is to export it to a DXF file (Drawing Exchange Format) and read it into a CAD program. The data can then be oriented to a plan view and compared to a scale drawing of an undeformed vehicle to obtain the dimensions required for use in the speed reconstruction. Figure 11.22 shows the data from this analysis of the residual crush located on a plan view of the perimeter of an undeformed vehicle. Figure 11.22 Diagram showing the plan view of the damaged vehicle profile and the undamaged vehicle profile.

The traditional application of three-dimensional photo­grammetry presented above is flexible enough to permit special uses. As an example, the method described above can be augmented with additional techniques to determine the three-dimensional coordinates associated with a point in the engine compartment that is not visible to the camera. In the photographs of the front of the vehicle, a straight, slender, white rod is protruding from the engine compartment. The end of the rod that is not visible is resting on a component whose location is desired (perhaps an engine mount that broke, fuel hose, etc.). This rod is of known length, and the visible end has been sharpened to a point and marked with a black marker. A black stripe has been made on the rod one foot below the tip. The coordinates of these two points on the rod can be located through the photogrammetric analysis just presented. These two points define a line in three-dimensional space, and the location of the other end of the rod (of known length) can be determined using geometry or through

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Photogrammetry for Accident Reconstruction

the use of the CAD package. In this way, the location of individual points not visible in the photogrammetric analysis can be determined.

11.5 Appendix: Projective Relation for Planar Photogrammetry The following is development of the equations for the four-point photogrammetric transformation that are presented in Hallert [11.2]. In Fig. 11A.1, the image-space and object-space planes are parallel. The origins of the coordinate systems in the two planes are located at N’ and N, respectively. The following relationships can be written for the coordinates between the points P and P’:

x=

y=

x 'h f

(11A.1)

y 'h f

(11A.2)

For an image that was captured when the camera was at the same location as shown in Fig 11A.1, but that has been rotated about the xr, yr, and zr axes by θ, ϕ, and κ, respectively, as shown, and using the transformation for a general rotation about three axes, the Eqs. 11A.1 and 11A.2 can be rewritten as x=

y=

h{ x '(cos ϕ cos κ − sin ϕ sin θ sin κ ) + y '(cos ϕ sin κ + sin ϕ sin θ cos κ ) + f sin ϕ cos θ } − x '(sin ϕ cos κ + cos ϕ sin θ sin κ ) + y '(cos ϕ sin θ cos κ − sin ϕ sin κ ) + f sin ϕ cos θ

(11A.3)

h ( − x ' cos θ sin κ + y ' cos θ cos κ − f sin θ ) − x '(sin ϕ cos κ + cos ϕ sin θ sin κ ) + y '(cos ϕ sin θ cos κ − sin ϕ sin κ ) + f sin ϕ cos θ

(11A.4)

These equations express the relationship between the image coordinates x’, y’, and z = -f on a plane that has been rotated through the angles θ, ϕ, and κ and the object coordinates x, y, and z = -h. Note that these equations can be rewritten in the form presented in Eq. 11.4 with some algebraic manipulation and letting x’ = xp , y’ = yp, xm = x, and ym = y:



xm =

ym =

c1 + c2 x p + c3 y p c4 x p + c5 y p +1



(11A.5a)



(11A.5b)

c6 + c7 x p + c8 y p c4 x p + c5 x p + 1

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Chapter 11

Figure 11A.1 Image and object planes for projective relation.

298

CHAPTER

12 Railroad Grade Crossing and Road Intersection Conflicts

12.1 Introduction This chapter contains methods of analysis and reconstruction for situations related to vehicle conflicts at railroad grade crossings and at road intersections. The methods (mathematical models) presented here deal primarily with potential conflicts when a train and road vehicle simultaneously approach a railroad grade crossing but apply equally well to two road vehicles simultaneously approaching an intersection of two roads. Parallel descriptions of the two situations (train-vehicle and vehicle-vehicle) are presented and illustrated. The methods address situations when a driver needs to see along the tracks in the presence of a visual obstruction to determine if a train is approaching and makes a decision to proceed through and clear the crossing. Analysis of this situation involves the concepts of sight distance and driver perception-decisionreaction time. A second situation occurs when a vehicle is moving toward a grade crossing or intersection, the driver needs to establish sight in the presence of a visual obstacle of an approaching train or vehicle, decides to stop, and needs enough time and distance to stop short of the crossing or intersection. Equations and their solutions are presented for modeling each of these two situations. The equations take into account the geometry of oblique crossings and intersections, different-length vehicles, and an arbitrary position of the edge of an obstruction to the driver’s line of sight. Another, separate set of equations is presented. These are equations and diagrams for stopped and moving vehicles approaching railroad grade crossings and are adopted from the U.S. Federal Highway Administration (FHWA) Grade Crossing Handbook [12.1, 12.2], covering situations similar to those above.

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Except when exempted, federal regulations require the sounding of locomotive horns in the vicinity of a crossing [12.3]. An issue that frequently arises at grade-crossing accidents is the sound level of a locomotive train horn at the location of a vehicle that is approaching the crossing. A method for calculating the sound pressure level at and inside the vehicle is presented based on principles of acoustics.

12.2 Clearing a Crossing or Intersection Using a Sight Triangle Figure 12.1 shows a diagram of a grade-crossing incident with images of an approaching locomotive on the tracks and an approaching vehicle on the road. Both are assumed to approach the intersection with constant velocities: VT for the locomotive and VV for the vehicle. The line of sight from the driver to the front of the locomotive is blocked by a fixed sight obstruction. The figure shows the line of sight of the locomotive to the driver that passes through point O (x0, y0) of the obstruction. (Figure 12.2 shows a similar situation for vehicles approaching an intersection of two roads with a potential conflict in the intersection.) In the following development, equations, variables, and descriptions of the events are presented using the railroad grade crossing terminology, but the same equations and concepts apply to the two-vehicle intersection conflict.

Figure 12.1 Diagram of a vehicle and train approaching a railroad crossing with a sight obstruction, where the vehicle driver expects to clear the crossing before the train arrives.

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Railroad Grade Crossing and Road Intersection Conflicts

Figure 12.2 Diagram of two vehicles approaching a road intersection with a sight obstruction, where the vehicle driver expects to clear the crossing before the crossing vehicle arrives.

Assumptions, or conditions, used in the development of the equations include constant velocities for the train and vehicle, a perception-decision-reaction time for the driver of the vehicle, a fixed sight obstruction, an unhindered line of sight after the obstruction is passed, a straight road, and a straight track. For the x, y coordinate system shown, the equation of the sight line from the driver to the front of the locomotive can be written as

y − y1 =



y2 − y1 ( x − x1 ) x2 − x1

(12.1)

where point 1 is the driver’s head position in the vehicle: (x1, y1) = (dH + dR, 0). The distance dH + dR is the distance the vehicle travels from the beginning of the open line of sight to the intersection. The driver’s “reaction” distance (perception-decision-reaction distance) is dPDR = dR = V V × τPDR , where τPDR is the driver’s perception-decision-reaction time. Let point 2 of Eq. 12.1 be the corner, O, of the obstruction: (x2, y2) = (x0, y0). This gives

y=

y0 (x − dH − dR ) x0 − d H − d R

(12.2)

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The initial view of the train occurs when the front of the locomotive is on the same line. Thus Eq. 12.2 is evaluated at the center of the front of the locomotive such that (x, y) = (xL , yL) = (dT cos γ, dT sin γ):



dT sin γ =

y0 (dT cos γ − d H − d R ) x0 − d H − d R

(12.3)

where dT is the distance the locomotive must travel to reach the crossing. The distance the vehicle needs to travel to completely clear the intersection is

dCL = d H + d R + D + L

(12.4)

The corresponding clearance time is

τ CL =



dCL VV

(12.5)

It is assumed that there is simultaneity between the clearance of the vehicle and the arrival of the locomotive to the crossing. Therefore, the clearance time of the vehicle is also the arrival time of the locomotive:

dT VT

(12.6)

VV dT VT

(12.7)

τ CL =

Equating Eqs. 12.5 and 12.6 gives

dCL =



Using Eq. 12.4 to solve for dH, and using dCL from Eq. 12.7 gives



dH =

VV dT − ( D + L ) − d R VT

(12.8)

Equation 12.8 is used to eliminate dH from Eq. 12.3. This results in a quadratic equation for the distance dT :

V    V dT sin γ  V dT − x0 − ( D + L)  + y0  dT cos γ − V dT + ( D + L)  = 0 VT  VT   

302

(12.9)

Railroad Grade Crossing and Road Intersection Conflicts

From the standard form of a quadratic equation:



ax 2 + bx + c = 0

(12.10)

The solution is



−b ± b 2 − 4ac 2a

x=

(12.11)

Comparison of Eqs. 12.9 and 12.10 gives the coefficients of Eq. 12.10 as

a=



b = y0 (cos γ −

VV sin γ VT

VV ) − sin γ  x0 + ( D + L) VT

c = y0 ( D + L)

(12.12)

(12.13) (12.14)

The solution, for x = dT, is presented in this form for use with computerized evaluations. Once dT is known, dH can be found from Eq. 12.8, where dR = dPDR = V V × τPDR . A locomotive pulling a single car is moving northbound at 20 mph (32.2 km/h) toward Example 12.1 a crossing while a sedan is headed westbound at 45 mph (72.4 km/h) toward the same crossing. A large building blocks the view of the train from the driver of the sedan. The track makes an angle of γ = 266° with the highway. This geometry is illustrated in the drawing of Fig. 12.3. The northwest corner of the building has coordinates (x0, y0) = (27, -102 ft) = (8.2, -31.1 m), relative to the (x, y) coordinate system on the drawing. The distance, D, from the track (see Fig. 12.1) is 34 ft (10.4 m), and the distance from the driver’s head position to the rear of the sedan is L = 12 ft (3.7 m). If the driver’s perception-decision-reaction time is 1.5 s for the conditions of the event, determine the clearance distance, dCL , and the distance, dH, traveled to the crossing by the car following the reaction time. Solution  The perception-decision-reaction distance is dR = 1.5 × 66 = 99 ft (30.2 m). Being a quadratic equation, Eq. 12.9 has two solutions, one for the plus sign in Eq. 12.11 and another for the minus sign. The solution of Eq. 12.9 gives dT(+) = 17.4 ft (5.3 m) and dT(-) = 120.5 ft (36.7 m). These, in turn, give dCL(+) = 39.0 ft (11.9 m) and dCL(-) = 271.2 ft (82.7 m). The latter value, dCL(-) = 271.2 ft (82.7 m), is the appropriate distance for this example. This means that the sedan must be dH + dR = 225.17 ft (68.7 m), or less, from the crossing center to clear the tracks before the front of the locomotive reaches the crossing.

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Figure 12.3 Railroad crossing geometry for Example 12.1.

12.3 Sight Distance for Stopping Before a Crossing or Intersection Figures 12.4 and 12.5 illustrate a situation similar to above except that here it is assumed that the driver of the vehicle is expected to stop, or decides to stop, before entering the crossing or intersection. As above, the description of this scenario is given in terms of a railroad grade crossing with the understanding that the results apply as well to a road intersection conflict of two vehicles approaching a road intersection. The equation of the sight line from the driver to the front of the locomotive is



y − y1 =

y2 − y1 ( x − x1 ) x2 − x1

(12.15)

where point 1 is the driver’s head position in the vehicle: (x1, y1) = (dH + dR + D, 0). The driver’s “reaction” distance (perception-decision-reaction distance) is dPDR = dR = V V × τPDR . Let point 2 be the fixed corner, O (x0, y0), of the obstruction so that (x2, y2) = (x0, y0). Then

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Railroad Grade Crossing and Road Intersection Conflicts

Figure 12.4 Diagram of a vehicle and train approaching a railroad crossing with a sight obstruction, where the vehicle driver expects to stop before the crossing.

Figure 12.5 Diagram of two vehicles approaching a road intersection with a sight obstruction where the vehicle driver expects to stop before the intersection.

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Chapter 12

y=

y0  x − ( d + d + D )   H R x0 − ( d H + d R + D )



(12.16)

The initial view of the train occurs when the front of the locomotive is on the same line. Thus, the equation is evaluated at the center of the front of the locomotive, (xL , yL) = (dT cos γ, dT sin γ) so that



dT sin γ =

y0 (dT cos γ − d H − d R ) x0 − d H − d R

(12.17)

In this case, the driver slows to a stop before the crossing with constant deceleration (negative acceleration), aV. The distance traveled during deceleration is



1 d H = VVτ H + aVτ H2 2

(12.18)

where τH is the time taken to cover the stopping distance, dH. The vehicle comes to a stop at a distance D from the geometric center of the crossing; therefore VH = 0 = VV + aV τH. Solving for τH gives



τH = −

VV aV

(12.19)

If the train travels the distance dT when the vehicle comes to rest, then

dT = VT (τ R + τ H )

(12.20)

or



τH =

dT −τ R VT

(12.21)

From Eqs. 12.19 and Eq.12.20,

aV = −

VV dT −τ R VT

(12.22)

Using this to eliminate aV from Eq. 12.18 gives



306

d 1 d H = VV ( T − τ R ) 2 VT

(12.23)

Railroad Grade Crossing and Road Intersection Conflicts

Finally, eliminate dH from Eq. 12.17 using Eq. 12.23 to get



  d 1 dT sin γ  x0 − d R − D − VV ( T − τ R )  2 VT     d 1 − y0  dT cos γ − d R − D − VV ( T − τ R )  = 0 2 VT  

(12.24)

This is a quadratic equation with the unknown as the distance dT with coefficients (Eq. 12.10), and where dR = V V × τR:

a=−

1 VV sin γ 2 VT

(12.25)



1 1V   b = sin γ  x0 − D − VVτ R  − y0 (cos γ − V ) 2 2 VT  

(12.26)



1 c = y0 ( VVτ R + D) 2

(12.27)

As before, the solution for dT is left in the form of the coefficients of a quadratic equation for use with Eq. 12.11 and computerized solutions. Note that the condition was invoked that the vehicle comes to a stop at the time the locomotive reaches the path of the vehicle. This condition of simultaneity was used to provide a complete algebraic formulation of the problem. Once the solution is found for the given condition, the acceleration, aV, of the vehicle can be calculated. If the magnitude of this acceleration exceeds the capability of the vehicle (such as for conditions of a locked-wheel skid on a low-frictional-drag-coefficient roadway), then this implies that the driver cannot stop the vehicle to avoid a collision/conflict for the given conditions. On the other hand, if the acceleration, aV, is small compared to the road conditions, this implies that the driver can stop the vehicle before entering the crossing. This example is similar to Example 12.1 except that in this case, the driver of the sedan Example 12.2 makes a decision to stop before entering the crossing because of the presence of the approaching train. A locomotive pulling a single car is moving northbound at 20 mph (32.2 km/h) toward a crossing, while a sedan is headed westbound at 45 mph (72.4 km/h) toward the same crossing. A large building blocks the view of the train to the driver of the sedan. The track makes an angle, γ = 266°, with the highway; see Fig. 12.6. The northwest corner of the building has coordinates (x0, y0) = (27 ft, -102 ft) = (8.2 m,

307

Chapter 12

-31.1 m), relative to the (x, y) coordinate system on the drawing. The distance, D, from the track (see Fig. 12.4) is 34 ft (10.4 m). If the driver’s perception-decision-reaction time is 1.5 s for the conditions of the event, determine the distance of travel of the train, dT, and the distance, dR + dH, traveled to the crossing by the sedan needed to bring the vehicle to a safe stop. Solution  The perception-decision-reaction distance is dR = 1.5 × 66 = 99 ft (30.2 m). Being a quadratic equation, Eq. 12.24 has two solutions, one for the plus sign in Eq. 12.11 and another for the minus sign. The solution of Eq. 12.24 gives dT(+) = 121.06 ft (36.9 m) and dT(-) = -62.69 ft (-19.1 m). The solution for the positive sign is appropriate. This, in turn, gives dH + dR = 254.32 ft (77.5 m), or less, from the driver’s head position to its position in the stopped vehicle at the stop bar, which coincides with the front of the locomotive crossing the path of the vehicle. The corresponding value of acceleration to make this stop is aV = -0.78 g. Note that the acceleration/deceleration, aV = -0.78, needed to stop the vehicle over the distance dH before the crossing is relatively large and would likely be considered an emergency stop for a dry pavement. Depending on existing roadway frictional drag Figure 12.6 Railroad crossing geometry for Example 12.2.

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Railroad Grade Crossing and Road Intersection Conflicts

coefficients, such a stop may not be possible. A reconstruction of such conditions would require consideration of a variety of factors such as pavement conditions, the speed limit on the road, awareness of the driver, and crossing design.

12.4 FHWA Grade Crossing Equations The U.S. Federal Highway Administration (FHWA) publishes a Railroad Highway Grade Crossing Handbook [12.1, 12.2]. In this handbook, equations and diagrams are presented for three cases of a vehicle in proximity of a railroad grade crossing. The approach taken for these cases is different than the above derivations. In the FHWA handbook, the first case considers a vehicle approaching a crossing from a distance, and the driver needs to determine if a train is already occupying the crossing and/or if there is an active traffic control device indicating the approach or presence of a train. The second case is similar to the crossing clearance problem above, where a fixed sight obstruction is present and a sight triangle between the driver, crossing, and train exists. The FHWA handbook provides equations for computation of the distance, dT, of the approaching train. The third case considers a vehicle already stopped at a crossing (such as for a stop sign). Under these circumstances the driver needs to see both directions along the track to determine if a train is approaching, estimate the train’s speed, and decide if sufficient time to accelerate and reach the other side of the crossing exists (even though the train might come into view as the driver begins the crossing process). The FHWA handbook provides formulas for calculating the distance, dT, of the front of the train from the crossing under such conditions. These equations are presented here in identical form as in the handbook. It should be noted that these equations have been developed mainly for the physical and operational improvements of crossing-geometry design or evaluation of crossing design, not necessarily for reconstruction of specific accidents.

12.4.1 Stopping Distance A driver approaching a grade crossing needs to determine from a distance if a train is occupying the crossing, if there is an active traffic control device indicating the approach or presence of a train, or if there is an approaching train. In all cases the driver must be able to stop, when necessary, before entering the crossing. According to the FHWA handbook [12.1, 12.2], the relationship between vehicle speed and sight distance is covered by the following formula:



d H = AVV t +

BVV2 + D + de a

(12.28)

Table 12.1 contains a listing of the definitions of the variables in Eq. 12.28 provided by the Highway Grade Crossing Handbook. Figure 12.7 shows the conditions for this case.

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Figure 12.7 Diagram from the FHWA handbook [12.1, 12.2] for the track sight distance (Eq. 12.28).

310

Railroad Grade Crossing and Road Intersection Conflicts

Table 12.1

Definition of variables in Eq. 12.28. dH

sight distance measured along highway from the nearest rail to the driver of the vehicle, which allows the vehicle to be safely stopped without encroachment of the crossing area, ft (US units), m (SI units)

A

constant, 1.47 (US units), 0.278 (SI units)

B

constant, 1.075 (US units), 0.039 (SI units)

VV

velocity of the vehicle, mph (US units), km/h (SI units)

t

perception-reaction time, seconds (s), assumed to be 2.5 s

a

driver (vehicle) deceleration, assumed to be 11.2 ft/s2 (US units), 3.4 m/s2 (SI units)

D

distance from the stop line or front of vehicle to the nearest rail, assumed to be 15 ft (US units), 2.4 m (SI units)

de

distance from the driver to the front of the vehicle, assumed to be: 8 ft (US units), 1.5 m (SI units)

12.4.2 Stopping Sight Distance The second FHWA case uses a so-called “sight triangle” in the quadrant on the vehicle approach side of the track. These triangles are illustrated in Fig. 12.7 and are formed by • the distance dH from the front of the vehicle to the track (nearest rail) • the distance dT of the front of the train from the crossing • the unobstructed sight line from the driver to the front of the train The corresponding FHWA Handbook formula is intended to determine the minimum safe sight distance, dT, along the track for selected vehicle and train speeds. The formula is given as



 BVV2 VT  dT =  AVV t + + 2D + L + W  VV  a 

(12. 29)

The definition of the variables used in Eq. 12.29 is given in Table 12.2. The FHWA Handbook provides a table of typical values for several selected highway speeds and train speeds.

12.4.3 Clearing Sight Distance In the case of a vehicle stopped at a crossing, the driver needs to see both directions along the track to determine if a train is approaching and to estimate its speed. The driver needs to have a sight distance along the tracks that will permit sighting the train early enough to allow sufficient time to accelerate and clear the crossing prior to arrival of the train, even though the train might come into view as the vehicle is beginning its departure. This situation is illustrated in Fig. 12.8. The corresponding formula from the FHWA Handbook is

311

Chapter 12

Table 12.2

Definition of variables in Eq. 12.29. dT

sight distance measured along the railroad tracks to permit the vehicle to cross and be clear of the crossing upon the arrival of the train, ft (US units), m (SI units)

A

constant, 1.47 (SI units), 0.278 (SI units)

B

constant, 1.075 (US units), 0.039 (SI units)

VV

velocity of the vehicle, mph (US units), km/h (SI units)

VT

velocity of the train, mph (US units), km/h (SI units)

t

perception-reaction time, seconds (s), assumed to be 2.5 s

a

driver (vehicle) deceleration, assumed to be 11.2 ft/s2 (US units), 3.4 m/s2 (SI units)

D

distance from the stop line or front of vehicle to the near rail, assumed to be 15 ft (US units), 4.5 m (SI units)

L

length of vehicle, assumed to be 65 ft (US units), 20 m (SI units)

W

distance between outer rails; for a single track this value is 5 ft (US units), 1.5 m (SI units)

V  L + 2D + W − da dT = 1.47VT  G + +J VG  a1 



( 12.30)

The definition of the variables used in Eq. 12.30 is given in Table 12.3 and where

da =



Table 12.3

312

VG2 8.82 or = 26.4 ft 2a1 (2)(1.47)

(12.31 )

Definition of variables in Eq. 12.30. VG

maximum speed of vehicle in selected starting gear, assumed to be 8.8 ft/s

VT

velocity of the train, mph

a1

acceleration of vehicle in starting gear, assumed to be 1.47 ft/s2

dT

sight distance measured along the railroad tracks to permit the vehicle to cross and be clear of the crossing upon the arrival of the train, ft

da

distance the vehicle travels while accelerating to maximum speed in first gear, given by Eq. 12.31

J

sum of perception time and the time required to activate the clutch or an automatic shift, assumed to be 2 s

D

distance from the stop line or front of vehicle to the near rail, assumed to be 15 ft

L

length of vehicle, assumed to be 65 ft

W

distance between outer rails (for a single track this value is 5 ft)

Railroad Grade Crossing and Road Intersection Conflicts

12.5 Locomotive Horn Sound Levels at Railroad Grade Crossings

Figure 12.8

Safety studies as well as analysis and reconstruction of accidents that occur between trains and road vehicles at highway-rail grade crossings frequently require estimation of the sound level of the train horn both outside and inside an approaching vehicle. A method is presented here that focuses on the sound transmission from the train horn to the exterior of an automobile. The method includes the prediction of the attenuation of the sound level over the path from the train horn to the vehicle and is based on the classical sound decay equation (including variable directivity and theoretical 6-dB drop-off per doubling of distance) but is modified to accommodate different drop-off rates, including experimentally measured values. In addition, insertion loss values for typical vehicles are presented that permit the estimation of the drop in sound level across the body of the vehicle, thereby allowing the estimation of the train horn sound level inside the vehicle. Related topics such as psychoacoustics of vehicle drivers, awareness, audibility, detectability, and signal-to-noise ratios are not discussed [12.4].

Diagram from the FHWA Handbook [12.1, 12.2] for the track sight distance (Eq. 12.30).

12.5.1 Computation of Horn Sound Levels at a Distance from a Point Source The classical equation for computing the sound pressure level, LP, at a distance, r, and direction, θ, from a point source is [12.5]



LP = LW + 10 log

Q(θ ) 4π r 2

(12.32)

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where LW is the sound power level of the source and Q(θ) is the directivity function; see Fig. 12.9. Sound pressure level in decibels (dB) is defined as [12.6]

LP = 10 log

p2 2 pref



(12.33)

where p is the rms (root-mean-square) sound pressure in N/m2, and the international standard reference pressure is pref = 2 × 10-5 N/m2. Sound power level in decibels (dB) is defined as

LW = 10 log

W Wref



(12.34)

where W is the sound power in watts and the international standard reference power is Wref = 10-12 watts. Because the reference values for LP and LW are in metric units, the distance r in Eq. 12.32 must be expressed in units of meters. The use of Eq.12.32 permits the calculation of the sound pressure level, LP, at a receiver position, R, at a direct-line distance, r, from a point Figure 12.9 Diagram illustrating railroad crossing geometry and variables for Eq. 12.32.

314

Railroad Grade Crossing and Road Intersection Conflicts

source, S. The following coverage is intended for application to locomotive horn sources, but Eq. 12.32 can be used for any point sources such as car horns. Eq. 12.32 should not be applied to environments with numerous nearby large acoustically reflective surfaces (such as in large midtown cityscapes). This approach, along with experimental measurements, can be made more useful for prediction of the sound level at various relative positions of a source and receiver. Consider the railroad crossing illustrated in Fig. 12.9 with a locomotive horn as a source and a road vehicle as a receiver. The angles α and γ define the roadway and track geometry, and dL and dV are the respective frontal distances of the locomotive and vehicle from the center of the crossing center, C. The distance hS is the horn setback, the distance from the front of the locomotive to the horn position. The directivity Q(θ) in Eq.12.32 represents the change in the sound power of the source as the angle, θ, varies. This permits taking into account changing of the sound radiation of the source as the receiver is located at different angles relative to the heading of the source (locomotive); i.e., as θ changes. For an omnidirectional source, Q(θ) = 1 [12.5], Eq. 12.32 predicts a drop-off based on spherical propagation of 6 dB per doubling of distance for all angles, θ. Measurements have indicated that, in practice, the level of the sound pressure can drop off at rates different from 6 dB per doubling of distance, depending on the environment between the source and receiver. Using repeated measurements, Seshagiri and Stewart [12.7] found values over flat, snow-covered ground to vary from 4.3 dB to 8.9 dB. In their work, Rapoza and Fleming [12.8] found that the drop-off rate for train horns varied from 5.7 to 8.4 dB, was inversely proportional to the height of the source above the ground, and was approximately 6 dB for a horn height of 16 ft (4.9 m). In order to model different empirical drop-off rates, Eq.12.32 can be modified such that



 Q(θ )  LP = LW + 10 log  2k   4π r 

(12.35)

where k is introduced to allow different drop-off rates. For distances r and 2r, Eq.12.35 can be used to solve for k, giving

k=

LP 2 r 20 log 2

(12.36)

where LP2r = LP (2r) - LP (r) is the drop-off rate per doubling of distance in decibels. For example, a drop-off rate of 5 dB per doubling of distance needs a value of k = 0.830. For a drop-off rate of 8 dB per doubling of distance, k = 1.329. Drop-off rates computed using Eq. 12.36 are independent of the source power level, LW, and directivity, Q(θ). Figure 12.10 shows differences in sound propagation for three values of k. Equations 12.35 and 12.36 can be used for computing individual frequency band values when the drop-off rate is a function of frequency. This would be the case, for example,

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Figure 12.10 Illustration of the effect of the drop-off factor k on the sound level.

when atmospheric attenuation or diffraction differ significantly with frequency. Once the spectral values are computed, the overall level can be computed by summation. Example 12.3 Suppose a locomotive horn satisfies current Federal Railway Administration regulations

[12.9] and produces a sound level of 96 dBA at 100 ft (30.5 m) directly in front of the locomotive. For the locomotive height of 15 ft (4.6 m), horn setback of 30 ft (9.1 m), and a measurement height of 4 ft (1.2 m), this corresponds to a sound power level of LW = 139 dBA. Assume the train horn is omnidirectional, Q(θ) = 1, and the drop-off rate calculated from measurements at the crossing site is k = 1.18. The crossing and vehicle conditions are as follows: perpendicular crossing (α = 0°, γ = 0°), the speed of the train is vT = 50 mph (80.5 km/h), and the speed of the vehicle is 30 mph (48.3 km/h). Table 12.4 gives three times and positions of the vehicles as they approach the crossing, along with the corresponding sound pressure levels using Eq. 12.33. The third time and distance, dV, listed in Table 12.4, τ = 3.45 s, corresponds to the stopping time [12.10] for a vehicle on a pavement with a frictional drag coefficient of f = 0.7 and a driver with a perceptiondecision-reaction time of 1.5 s. Knowing the sound levels at R, the appropriate insertion loss can be used to determine the estimated sound pressure level inside the vehicle. For example, if the driver’s window is fully opened, an insertion loss of approximately 4 ± 2 dBA can be subtracted from the values of LP at R given in Table 12.5.

12.5.2 Insertion Loss of Light Vehicles Insertion loss is defined [12.5] as the difference, in decibels, between two (overall) sound pressure levels (or power levels or intensity levels) which are measured at the same location in space before and after an acoustical device is inserted between the measurement location and the sound source. Here, the “measurement location” is at the receiver and the “acoustic device” is a vehicle body. Insertion loss measurements were made [12.10] for seven light vehicles. Results are given in Table 12.5. Measurements

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Railroad Grade Crossing and Road Intersection Conflicts

Sound levels at R, dBA, Example 12.3. correspond to levels at the driver’s head position with the vehicle Time to Distance to Crossing, engine running, no interior vehicle Crossing, s ft accessories operating, and for three dV dL LP at R conditions: 0, all windows fully closed 10.0 440 733 71 in vehicle R; 1, the driver’s window 5.0 220 367 78 open approximately 1 in.; and 2, the 3.45 152 253 81 driver’s window fully opened (all other windows fully closed). Insertion loss is often measured or computed as a function of frequency and presented as spectra [12.8, 12.10]. Here only overall levels are presented.

Table 12.4

Measurements of insertion loss of road vehicles have been made by others. Dolan and Rainey [12.11] made dynamic measurements of the insertion loss over a 35-s duration of moving train horn sounds for three vehicles with closed windows, a 1989 Toyota pickup truck, a 1999 Toyota 4-Runner SUV, and a 2001 Pontiac Grand Prix sedan. The overall vehicle insertion loss values were 25.4, 27.1, and 28.0 for the pickup, SUV, and sedan, respectively. Rapoza, et al. [12.12] measured insertion loss for three different sound incidence angles relative to vehicle heading and found that it did not vary significantly between the angles. An NTSB study [12.13] reports on measurements of the insertion loss of a variety of vehicle types. The sound source was a train horn at a distance of 96 ft (29.3 m) from the vehicle. The insertion loss levels are given in Table 12.6. Measurements comparable to

Measured sound levels and insertion loss, dBA. Vehicle

Interior level, dBA

Table 12.5

Insertion loss (IL), dBA

0

1

2

0

1

2

‘02 Honda Civic

64.2

84.8

‘01 Honda CRV

67.2

82.6

94.3

33.9

13.3

3.8

94.4

30.9

15.5

3.7

‘03 Honda Odyssey

63.2

‘03 Ford Windstar

61.2

85.9

94.2

34.9

12.2

3.9

83.5

96.7

36.9

14.6

1.4

‘98 Ford Windstar

61.4

81.6

93.7

36.7

16.5

4.4

‘07 Ford Focus

64.2

83.9

91.4

33.9

14.2

6.7

‘98 Chrysler T & C

67.1

85.1

93.6

31.0

13.0

4.5

35

14

4

Average Insertion Loss, dBA Standard Deviation (IL), dBA Range (IL), dBA

2.4

1.5

1.6

31 to 37

12 to 17

1 to 7

0 - window fully closed, 1 - window opened 1 in., 2 - window fully opened

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those reported here are those of the light vehicles, the last five vehicles listed in Table 12.6 which arithmetically average to 30 dBA. Table 12.6

Insertion loss, dBA [12.13]. Vehicle 1986 Freightliner cab over truck tractor

Insertion Loss, dBA 17

1996 Freightliner conventional truck tractor

18

1996 Thomas/International school bus

21

American La France fire truck

21

1990 Ford F-350 ambulance

27

1997 Thomas/Ford school bus

27

1978 TMC Crusader coach bus

28

1994 Dodge Ram 1500 pickup truck

28

1996 Ford F-250 pickup truck

28

1987 Mercedes 300 SDL turbo

29

1995 Oldsmobile Achieva

32

1986 Chevrolet Corvette

33

Example 12.4 This example uses the conditions depicted in Fig. 12.3. The speed of the locomotive is

20 mph (32.2 km/h) and the speed of the car is 45 mph (72.4 km/h). When the car is 198 ft (60.4 m) from the crossing, the front of the locomotive is 88 ft (26.8 m) from the crossing. The horn is set back 18 ft (5.5 m) from the front of the locomotive and is 15 ft (4.6 m) above the road level. The horn sound power level is LW = 140 dBA, has uniform directivity (Q(θ) = 1), and the environment has a drop-off factor of k = 1.2. The driver’s window is open and has an insertion loss of 4.0 dBA. Calculate the sound pressure levels outside the vehicle and at the driver’s position inside the vehicle. Solution  The source-to-receiver distance is r = 66.8 m (219.1 ft). (Eqs.12.32 and 12.35 require the distance to be expressed in SI units.) Figure 12.11 shows the results of calculations using Eq. 12.35, estimating the sound pressure level as 85.2 dBA outside the driver’s window. With an insertion loss of 4.0 dBA, the level should be 81.2 dBA inside the car.

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Railroad Grade Crossing and Road Intersection Conflicts

Figure 12.11 Results of the calculation of sound pressure level for Example 12.4.

319

CHAPTER

13 Vehicle Dynamic Simulation

13.1 Introduction In engineering, the word simulation has the meaning of a mathematical model that relates the variables and parameters of some physical process or system in a way that the model mimics the behavior of the process or system. Typically, the model is formed using differential equations and is solved numerically using a computer. The results produce the system behavior as a function of time for given conditions of the system. In this chapter a simulation is presented. It is a simulation of the dynamic motion of a vehicle, with or without a semitrailer, as it is braked, or accelerated, steered, and/or moves freely over a roadway. This is referred to as a vehicle dynamic simulation. In the context of accident reconstruction, a simulation can be used to examine preimpact and postimpact motion of a single vehicle, either by itself or when pulling a semitrailer. Vehicle motion under various conditions can be simulated such as during a lane-change maneuver, sudden-avoidance maneuver, braking over dual-coefficient roadway surfaces, etc. Not only is the motion produced, but tire forces are modeled and calculated as well. For purposes of this book, a distinction is made between vehicle dynamic simulation and the topic of vehicle dynamics. By vehicle dynamic simulation is meant the capability to predict motion of a vehicle with specified characteristics (such as mass, wheelbase, track, weight, yaw inertia, and tire properties) and under specified conditions (initial conditions, brake application, driver steer input, etc.). The term vehicle dynamics refers to the study of vehicle motion behavior (such as handling and stability) based on its subsystem characteristics (tire characteristics, springs, shock absorbers, suspension geometry, aerodynamic shape, etc.). Vehicle dynamics is typically concerned with vehicle design. Vehicle dynamics is not covered in this book; however, excellent books exist on

321

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the subject, many of which are listed in a bibliography at the end of the references to Chapter 13.

13.2 Planar Vehicle Dynamics Simulation When carrying out an accident reconstruction, it is often useful to determine the motion of a vehicle under various conditions and combinations of acceleration, braking, and steering. For example, it may be desirable to determine what steering input at the front wheels of an automobile is necessary to complete a lane-change maneuver in 4 s at 97 km/h (60 mph) and to find out what frictional forces are developed between the tires and pavement during the maneuver. Such information can be obtained experimentally by carrying out the maneuver with an instrumented vehicle. The information can also be obtained using a vehicle dynamic simulation. Such simulation software exists with a wide range of capabilities. Some are three-dimensional such as Highway Vehicle Obstacle Simulation Model, HVOSM [13.1, 13.2], and Engineering Dynamics Vehicle Simulation Model, EDVSM [13.3], for vehicles and vehicle-barrier interactions. Some, such as Simulation Model for Automobile Collisions, SMAC [13.4, 13.5], not only simulate vehicle motion but model collision deformation as well. Some vehicle simulation software programs, such as Vehicle Dynamics Analysis Nonlinear, VDANL [13.6], and Vehicle Dynamics Models for Roadway Analysis and Design, VDM RoAD (University of Michigan Transportation Research Institute) [13.6], include a vehicle suspension system model. Others, such as VdynVB [13.7] and EDSVS [13.8], use a rigid suspension. VdynVB and EDVTS [13.8] include a semitrailer model. Driving simulators need an underlying vehicle dynamics simulation [13.9]. More sophisticated models that include simulation of crash deformation continue to be developed [13.10]. A simulation program that incorporates planar impact mechanics for complete reconstructions is PCCrash [13.11]. Another, an accumulation of reconstruction spreadsheets with a vehicle dynamics simulation, is VCRware [13.12]. One of the most important parts of accurately modeling vehicle motion over a road is the tire model. In the large majority of applications, the most important forces that control vehicle motion are generated by the tires. In some cases, aerodynamic forces can be significant, but these are not discussed here. Tire forces used in the simulations to follow are covered in Chapter 2 and are discussed only briefly here. There are two important aspects to tire models. The first is the individual characteristics of the tire’s traction (longitudinal) force and the lateral (transverse, cornering, and steering) force. The second is the way in which these characteristics are combined for simultaneous braking and steering. One effective tire model for traction and cornering forces is attributed to Fiala [13.13]. Another model for traction and cornering forces, presented by Allen, et al. [13.14], is used for both on-road and off-road applications. The model used here is a combination of the BNP tire characteristics [13.15] and the Modified Nicolas-Comstock model [13.16], both covered in Chapter 2.

322

Vehicle Dynamic Simulation

Compared to other existing vehicle dynamic simulation models, the one presented here is relatively versatile yet not overly complicated. In accident reconstruction, it is rare to have access to accurate values of vehicle physical parameters corresponding to the specific vehicle and its components at the time of an accident. This includes component parameters such as nonlinear suspension system stiffness, damping and limiting coefficients, tire condition and inflation pressure, and transient braking or acceleration information. For that reason, a reasonably versatile model is presented that requires a minimal number of vehicle and road parameters. The simulation model presented here is developed for the following conditions (see Fig. 13.1): 1. A flat, level road surface 2. A two-axle, four-wheeled rigid-body vehicle, C (car or cab), with mass mC and yaw moment of inertia JC 3. An optional rigid-body semitrailer, T, with mass mT and yaw moment of inertia JT, pinned to mC at pin P, and with a single axle 4. A rigid suspension system (no roll or pitch motion of either the vehicle or semitrailer) Simulations such as this are based on differential equations that are solved using numerical integration on a computer. Simulations in the examples that follow were solved using a spreadsheet format by coupling the spreadsheet to a macro. The macro uses a computer program to solve the equations and transmits the results back to a spreadsheet. The model here is derived by applying Newton’s second law for planar motion of each body and by applying the two constraint equations of the pinned connection for a semitrailer. For planar motion, the variables are the position coordinates of the center of gravity of each body, xC , yC , and θC of the car (or cab) and xT, yT, and θT of the semitrailer. There are three equations of motion for each body and two constraint equations at the pin, leaving four, second-order differential equations that can be integrated numerically. The equations and their integration are referred to as a time-forward simulation, because an integration process determines the positions and velocities of the vehicles as functions of time. Specifically, integration of the four differential equations produces the values of xC , yC , θC , and θT and their velocities as functions of time starting with a set of initial conditions. The positions xT and yT and their velocities xT and yT can be found from the kinematic equations after integration. The model equations are presented in an Appendix to this chapter. In the examples described here, the planar equations of motion are solved using Runge-Kutta-Gill numerical integration [13.17].

323

Chapter 13

Figure 13.1 Vehicle configuration and variable names.

The simulation based on the equations presented here is set up with various features, including 1. The vehicles move over a flat, level surface with tire-roadway frictional drag coefficients f R for a roadway bounded by 0 ≤ Y ≤ RW, and with a coefficient f B to represent a berm, or shoulder, outside this region (Y < 0 and Y > RW) 2. Three modes of steering: a. Preprogrammed, sinusoidal-steer, lane-change maneuver over a given time duration b. Braking with all wheels locked c. An arbitrary, tabular front-wheel steer input, δ(τ), where τ is time

324

Vehicle Dynamic Simulation

3. Arbitrary braking wheel-slip values, si, for each wheel, or arbitrary acceleration with traction coefficients at each wheel for the lane-change and tabular-steer modes 4. Arbitrary wheel locations, with position dimensions Wi and Li, i = 1,2, .... ,6, (individualized to permit relocation of damaged wheel positions) as shown in Fig. 13.1 5. Arbitrary initial conditions for the displacements xC , yC , θC , and θT and their velocities 6. Calculation of individual wheel normal forces based on quasi-static vehicle lateral accelerations for center of gravity heights, hC and hT, of the car/cab and semitrailer, respectively 7. Selectable integration time interval and print intervals A common way of choosing the integration interval is to start very small and increase it until the results begin to change significantly, then go back and use the last interval before the significant change occurred. An interval of 0.005 s should be satisfactory for most cases. The dimensions of the vehicles (track widths, wheelbase lengths, etc.) used as input to the software are illustrated in Fig. 13.1. This figure also is reproduced as the first page of a spreadsheet solution (Fig. 13.2). Particular note should be taken of the lateral pin locations on the tow vehicle and trailer, which are referenced to wheel 3 of the car/cab and wheel 5 of the semitrailer. When the pin location is behind the rear axle of the tow vehicle car (cab), a weight-equilibrating hitch can be chosen that causes the static frontto-rear weight distribution of the tow vehicle to remain the same, as if no trailer exists. Otherwise, the hitch is treated as a simple pin joint.

13.3 Tire Side-Force Stiffness Coefficients Values of the tire stiffness coefficients Cs and Cα, are needed for the use of dynamic simulation for accident reconstructions. The discussion of tire coefficients is split into two categories, light vehicles and heavy vehicles. In reality, a continuum of tire sizes exists, but for reconstruction, these categories are the ones typically encountered.

13.3.1 Light-Vehicle Side-Force Coefficients Unfortunately, few collections of experimental data exist that contain the characteristics of light vehicle tires, particularly values of stiffness coefficients Cs and Cα, ordinarily needed for carrying out simulations. The tire industry and vehicle manufacturers are protective about the characteristics of the tires they produce and use. Limited data can be found in some publications [13.18, 13.19, 13.20, 13.21, 13.22, 13.23] but data for a specific tire (or tires) involved in a particular accident are rarely available. Thus the reconstructionist must resort to other methods such as estimating the coefficients and managing the uncertainty.

325

Chapter 13

13.3.2 Heavy-Vehicle Side-Force Coefficients Equations are available [13.24] for estimating truck tire longitudinal and lateral/ cornering coefficients, Cs and Cα, used in simulations of heavy-truck dynamics. The equation that gives the longitudinal coefficient, Cs, of a single wheel as a function of the wheel’s normal force, Fz, is

Cs = 10 Fz − Fz2 / 3000



(13.1)

in units of lb per unit slip. A second equation gives a value of the corresponding lateral stiffness, Cα:

Cα = 0.9Cs



(13.2)

in units of lb/radian. Because most heavy trucks have dual wheels and tandem axles at the respective wheel locations, equivalent values for the wheel sets have to be developed. These values are then assigned to a single equivalent wheel, located at the geometric center of the tandem/ dual combination, which represents the performance characteristics of the two- or fourwheel combination. Such a process can be carried out in the following way using Eq. 1 and Eq. 2. 1. Calculate Cs for each individual wheel of the tandem/dual configuration, with normal force Fz on a per-wheel basis using Eq. 13.1. 2. Combine the Cs values for the number of wheels by addition. For a dual configuration with two identical wheels, the equivalent Cs is Cseq = 2×Cs, and for four tandem wheels it is Cseq = 4×Cs. 3. The equivalent lateral stiffness, Cαeq, for each wheel is found using Eq. 13.2; Cαeq = 0.9 × Cseq. Combine the Cα values for the number of wheels by addition. For a dual configuration with two identical wheels, the equivalent Cα is Cαeq = 2×Cα, and for four tandem wheels it is Cseq = 4×Cα. 4. The single wheel as defined by Cseq and Cαeq is located at the geometric center of the wheel assembly.

13.4 Examples Because of the large number of input combinations and versatility of vehicle dynamic simulations, it is impractical to present an exhaustive number of examples. The following examples are representative of applications to the field of accident reconstruction. For convenience, the first four examples use the same rear-wheel-drive vehicle. The input parameters are given in Table 13.1.

326

Vehicle Dynamic Simulation

Consider a single, four-wheeled vehicle with no trailer and with conditions listed in Example 13.1 Table 13.1. For straight-line motion under freewheeling conditions (no braking and no rolling resistance) at the front wheels, s1 = s2 = 0, determine the rear-wheel brake slip values, s3 and s4 , that cause a drag on the rear wheels to equal about 15% of the normal force at the rear wheels. Common vehicle parameters for simulation, Examples 13.1 through 13.4.

Table 13.1

Weight, WC, 3000 lb (13.34 kN) Moment of Inertia, JC, 1900 ft-lb-s2 (2.58 m-kN-s2) Lengths, L1 = L2 = 3.75 ft (1.14 m), L3 = L4 = 4.58 ft (1.40 m) Widths, W1 = W2 = W3 = W4 = 2.50 ft (0.76 m) Height of cg, hC = 1.0 ft (0.3 m) Tire Lateral Steer Coefficients:   Cα1 = Cα2 = 8800 lb/rad (39.14 kN/rad)   Cα3 = Cα4 = 8100 lb/rad (26.03 kN/rad) Tire Braking (Forward) Coefficients, Cs1 = Cs2 = Cs3 = Cs4 = 10000 lb/rad (44.48 kN/rad) Friction Coefficients, fR = fB = 0.7 Integration Interval, 0.005 s Final Time, 10 s

Solution  The information from Table 13.1 is entered into a simulation software program. Fig. 13.2 shows input data for the simulation and reflects the input data from Table 13.1. For straight-line forward motion, a steer mode for a lane-change maneuver and steer angle amplitude δ = 0 is used. The rear brake wheel slip values s3 and s4 are found by iteration in the following fashion. Start with any forward speed, say 50 ft/s (15.25 m/s), and any common value for s3 and s4 between 0 and 1 (see Chapter 2). Carry out the simulation and examine the tire forces as a percentage of the normal forces (Fig. 13.3). It is found that a value of s3 = s4 = 0.007 causes the braking force to be 14.9% of the normal force, 466.9 lb (2.08 kN), developed at the rear wheels. This means that if the rear brakes are actuated to develop a wheel slip of s3 = s4 = 0.007, a braking force is developed equal to about 15% of the rear wheel normal forces. For example, these values could be used to simulate a condition of 15% powertrain drag at the rear wheels for other simulation conditions. Note that this small amount of slip, s = 0.007, does not mean that the rear brakes are applied. Rather, it accounts for the drag at the rear wheels due to powertrain frictional losses. Consider again the vehicle with properties listed in Table 13.1 with an initial speed of 60 Example 13.2 mph (96.6 km/h) and a frictional drag coefficient of f = 0.7. The driver makes an 8-s lanechange maneuver beginning at time τ = 1.0 s with no braking or acceleration. Determine the necessary (maximum) steer angle that will cause the vehicle to move laterally (to the

327

Chapter 13

Figure 13.2 Input data for Example 13. 1.

driver’s left) 12 ft (3.66 m) in that time. Estimate the maximum tire force that occurs during such a maneuver based on a percentage of the available traction force. Solution  The spreadsheet used for the simulation has a built-in lane-change maneuver feature. It uses a steer angle shape composed of sine and cosine segments, as illustrated in Fig. 13.4, with a specifiable begin time and duration. Here, the begin time is τ = 1 s and the duration is 8 s. For the specified input values, the simulation can be run in an iterative fashion, changing the maximum steer angle, δ, each time and observing the lateral displacement, yc. When this reaches 12 ft (3.66 m), as shown in Fig.13.5, the maximum steer angle is found to be δ = 0.129˚. Examination of the output on Fig. 13.6 shows that the maximum percentage tire force occurs for wheel four (right rear) as 6.1% of 464 lb (2.06 kN), or 28.3 lb (0.126 kN), at 3.5 s.

328

Vehicle Dynamic Simulation

Figure 13.3 Output data for Example 13.1.

The capability to simulate a lane-change maneuver can be useful in the field of accident reconstruction in that it allows the reconstructionist to establish the limit of the capabilities of a particular vehicle to execute such a maneuver. A reconstructionist could also use a simulation’s results to assess or evaluate a witness’ or participant’s claim with respect to timing, level of lateral velocities, or other vehicle motion characteristics. Consider the vehicle described by the parameters listed in Table 13.1, but now it is Example 13.3 pulling a semitrailer. Also, here the frictional drag coefficient of the road surface is f = 1.0. Table 13.2 lists the additional information required to describe and specify the semitrailer. In addition, suppose that while traveling forward at 50 mph (80.5 km/h) the semitrailer is hit from the side by another vehicle and given an initial angular velocity of 40 °/s. Examine the motion of the vehicle when the tow vehicle and semitrailer both have fully locked brakes. Note that the initial conditions of this example may not fully represent the final velocities of an impact of an articulated vehicle (see Chapter 8) because an initial transverse velocity for the semitrailer could also be developed from a side impact as well as initial

329

Chapter 13

Figure 13.4 Steer angle, δ(τ), for simulation of a lanechange maneuver for Example 13.2.

Semitrailer parameters.

Table 13.2 Trailer Weight, W T, 1200 lb (5.34 kN)

Trailer Moment of Inertia, JT, 650 ft-lb-s2 (0.88 m-kN-s2) Lengths, L5 = L6 = 2.0 ft (0.61 m); LCP = 6.0 ft (1.8 m) Widths, W5 = W6 = 3.7 ft (1.13 m); WCP = 2.5 ft (0.8 m) Height of cg, hT = 0.5 ft (0.15 m) Tire Lateral Steer Coefficients, Cα5 = Cα6 = 9200 lb/rad (40.92 kN/rad) Tire Braking (Forward) Coefficients, Cs5 = Cs6 = 10000 lb/rad (44.48 kN/rad)

transverse and rotational velocity components for the tow vehicle. Also note that since this simulation is for a locked-wheel skid, the lateral and longitudinal tire coefficients do not enter into the solution. Solution  For this example, Fig. 13.7 shows the simulation input values. Note that input is specified to simulate a locked-wheel skid for all six wheels, including the semitrailer. Figures 13.8 and 13.9 show output data. The results indicate that the center of gravity of the tow vehicle skids about 82.9 ft (25.3 m) as it rotates approximately 1.2˚ before coming to rest in 2.26 s. The semitrailer rotates approximately 52.3˚. Figure 13.9 shows that the

330

Vehicle Dynamic Simulation

Figure 13.5 Spreadsheet showing positions and velocities as a function of time for Example 13.2.

Figure 13.6 Spreadsheet showing tire forces for Example 13.2.

331

Chapter 13

Figure 13.7 Spreadsheet listing of input data for Example 13.3.

tire forces all are 100% of the frictional limit throughout the motion because wheels are locked. Figure 13.10 shows a plot of the motion of the tow vehicle and semitrailer, including the initial and rest positions. Note that in the solution to Example 13.3, the final velocities were not exactly zero when the simulation ended. This is because integration ends when the total kinetic energy of the system is smaller than when the kinetic energy of the tow vehicle and semitrailer have velocities of 1 ft/s each. Example 13.4 Example 13.3 simulated the motion of a tow vehicle and semitrailer with the properties

listed in Tables 13.1 and 13.2. Consider here the problem of determining the motion of the same vehicle, starting with the same initial conditions and with f = 1.0, but where the brakes on the semitrailer are not applied at any time during the motion. Solution  The conditions in this example are not those of a locked-wheel skid, because the semitrailer brakes are not applied. The solution must be set up in a different way to simulate the desired motion. The lane-change option can be used to simulate this

332

Vehicle Dynamic Simulation

Figure 13.8 Spreadsheet showing positions and velocities as a function of time, Example 13.3.

motion by setting the steer angle amplitude, δ = 0˚, setting the tow vehicle’s brakes to a locked condition using brake slip values of s1 = s2 = s3 = s4 = 1 and leaving the trailer brake slip values set to s5 = s6 = 0. Figures 13.11 through 13.14 show the input and results of a simulation corresponding to this example. Note that the lack of semitrailer braking makes significant and remarkable changes compared to Example 13.3. The total distance traveled by the tow vehicle center of gravity increases to 105 ft (32.1 m) and it takes about 2.9 s to come to rest. With the semitrailer brakes locked, the tow vehicle had an angle at rest of +1.2˚; it now is -37˚. The rotation of the semitrailer in Example 13.3 was +52.3˚ and now is +10.5˚. Removing the locked-wheel braking on the semitrailer wheels (which are at a relatively large distance from the tow vehicle’s center of gravity) has a very large effect on its motion. It should be noted that although the simulation model is based on a single rear axle for the car/cab and a single rear axle for the semitrailer, a simulation can represent tandem axles and dual wheels such as associated with truck tractor and semitrailers

333

Chapter 13

Figure 13.9 Spreadsheet showing tire forces as a function of time, Example 13.3.

Figure 13.10 Plot of the motion of the vehicles, Example 13.3.

with reasonable accuracy [13.7]. This is done by using mid-axle dimensions and using combined tire properties (lateral steering coefficients) of tandem axles and multiple wheel sets (see Section 13.3.2). Example 13.5 This example demonstrates the versatility of dynamic simulations by simulating a

high-speed, sudden-steer maneuver (a maneuver that would likely cause the vehicle’s tires to leave yaw marks on a dry road surface [see Critical Speed, Chapter 4]). In this example, the vehicle (with no trailer) does not slide to a rest position but encounters a rigid pole midway in its trajectory. The goal is to determine the speed of the vehicle when it reaches the pole. The critical speed formula cannot find intermediate positions and velocities since it relates only the initial speed to the initial radius of the trajectory. This example is based on the results of a test [13.25] of a 1991 Honda with partial braking during the maneuver (see Table 13.3). An analysis of the simulation of the full trajectory using different simulation software packages has been presented elsewhere [13.26]. The conditions of the test are used here to simulate the trajectory of the vehicle from an initial speed of 91.13 ft/s (27.8 m/s, 100 km/h) and for the conditions and vehicle properties given in Table 13.3.

334

Vehicle Dynamic Simulation

Figure 13.11 Spreadsheet showing input data for Example 13.4.

Solution  Figure 13.15 shows the trajectory of the vehicle following the sudden steer input by the driver and where it reaches a rigid pole at the time of 2.4 s from the beginning of the steer, at the center-of-gravity position (X, Y) = (179.4 ft, 45.4 ft) = (54.7 m, 13.5 m). When it reaches the pole, the simulation determined that the vehicle has velocity components ( X , Y ) = (49.8 ft/s, 34.7 ft/s) = (15.2 m/s, 10.6 m/s), an overall velocity of 60.7 ft/s (18.5 m/s), and angular velocity of 35.6 ˚/s. At that position, the vehicle has a heading angle of approximately 75˚ relative to the (X, Y) coordinate system (see Fig. 13.15). Figure 13.16 shows a partial output of the simulation giving additional information. For example, the components of the lateral and forward acceleration at time 2.4 s are aL = +25.8 ft/s2 = +7.9 m/s2 and aF = -3.4 ft/s2 = -1.03 m/s2. The resultant acceleration is 26.1 ft/s2 = 7.9 m/s2, or 0.81 g. (An acceleration value of 0.81 g might seem impossible since the frictional drag coefficient is f = 0.75; however, keep in mind that the wheels are not locked and tires can generate forces higher than the locked wheel values for a given f; see Chapter 2.)

335

Chapter 13

Figure 13.12 Spreadsheet showing positions and velocities as a function of time, Example 13.4.

Figure 13.13 Spreadsheet showing tire forces, Example 13.4.

336

Vehicle Dynamic Simulation

Vehicle parameters and initial conditions for Example 13.5.

Table 13.3

Vehicle: 1991 Honda Accord EX Vehicle Test Weight: W = 3186 lb, Distribution 61% ∕ 39% Yaw Radius of Gyration: k = 4.49 ft, 1.37 m Overall Length:  185 in., 4.70 m

Wheelbase: 107 in., 2.72 m

Front Track:  58 in., 1.47 m

Rear Track:  58 in., 1.47 m

Tire Size:  195-60R15

CG Height:  21.2 in, 0.54 m

Tire Side Force Coefficients:  CαF = 13,000 lb/rad, CαR = 10,000 lb/rad Front Wheel Braking Force:  312.3 lb ∕ wheel Rear Wheel Braking Force:  122.9 lb ∕ wheel

   Initial Conditions: X, Y, θ = 0, 0, 0, X , Y , θ \s \* MERGEFORMAT= 91.134, 0, 0 ft/s Front Wheel Steer Angle, δ: linear rise from 0° to 9° in ½ s, then constant at 9° Tire-Road Frictional Drag Coefficient (measured): f = 0.75 Aerodynamic Drag:   Coefficients (forward, lateral/side): CdF = 0.4, CdL = 0.8   Frontal, Lateral/Side Areas: AF = 25 ft2, AL = 60 ft2

Figure 13.14 Plots of vehicle motion, Example 13.4.

Figure 13.15 Trajectory of 1991 Honda for a 9˚ front-wheel sudden steer on an f = 0.75 friction surface. The trajectory path is of the leading front tire (left front).

337

Chapter 13

Figure 13.16 Partial results of the simulation of Example 13.5.

13.5 Appendix: Differential Equations of Planar Vehicular Motion The four differential equations of motion for the simulation of a tow vehicle pulling a semitrailer are listed in this appendix. Equations defining intermediate variables are presented along with a list of symbols. All of the variables are illustrated in Fig. 13.1. These equations already have the pin impulse eliminated and are in the form of four  , Y , θ , and θ . equations and four unknowns. The unknowns are: X C C C T

(mC + mT ) XC = − m ( R θ cos θ T

CP CP

CP

2 + RCPθCP sin θCP + RTPθT cos θTP

6

+ RTPθT2 sin θTP ) + ∑ f iX + FCX + FTX



i =1



(13A.1)



(13A.2)

(mC + mT )YC = − mT (− RCPθC sin θCP + RCPθC2 cos θCP − RTPθT sin θTP 6

+ RTPθT2 cos θTP ) + ∑ fiY + FCY + FTY



i =1

 J + mR ( r cosθ − r sin θ )θ =  C CP PCX CP CPY CP  C 4 4  m 6 m ∑ (riX fiX + riY fiY ) + rCPX  m ∑ fiX − ∑ fiX  + rCPY  m   T i =1  T i =1 i =1 −r m R θ 2 co osθ − R θ sin θ − R θ 2 cosθ CPY

(

CP C

(

CP

TP T

TP

TP T

TP

)

− rCPX m R θ sin θ CP + RTPθT cosθTP + R θ sin θTP

338

2 Cp T

2 TP T

)

6

∑f i =1

iY

4  − ∑ f iY   i =1

(13A.3)

Vehicle Dynamic Simulation

 J + mR ( r sin θ − r cosθ )θ =  T TP TPY TP TPY TP  T 6 4    4 m 6 m + ( r f + r f ) + r f − f r   ∑ iX iX iY iY TPX ∑ iX m ∑ iX  TPY ∑ fiY − m   i =1  i =1 i =5 T i =1 T 2 2 +r m − R θ sin θ + R θ cosθ + R θ cosθ TPY

(

(

CP C

CP

CP C

TP

TP T

TP

) )

+ rTPX m RCPθC cosθ CP + RCPθC2 sin θ CP + RTPθT2 sin θTP

6

∑f i =1

iY

  

(13A.4)

Variables

RCP = [ L2CP + (W3 − WCP ) 2 ]1/2

(13A.5)



RTP = [ L2TP + (WTP − W5 ) 2 ]1/2

(13A.6)



θCP =

π 2

− θC − tan −1[(W3 − WCP ) / LCP ]



(13A.7)



(13A.8)



rTPX = − LTP sin θT − (WTP − W5 ) cos θT

(13A.9)



rTPY = LTP cos θT − (WTP − W5 ) sin θT

(13A.10)



rCPX = LCP sin θC + (W3 − WCP ) cos θC

(13A.11)



rTPX = − LTP sin θT − (WTP − W5 ) cos θT

(13A.12)



rCPY = − LCP cos θC + (W3 − WCP ) cos θC

(13A.13)



θTP =

π 2

− θT − tan −1[(WTP − W5 ) / LTP ]

m=

mC mT mC + mT

(13A.14)

Notation FCX, FCY

External force components on car (tow vehicle)

FTX, FTY

External force components on semitrailer

339

Chapter 13

FZ

Normal force between a wheel and the roadway

fiX, fiY Components of the tangential force between the ith tire and the roadway surface hC , hT Vertical heights of car (tow vehicle) and semitrailer centers of gravity, respectively JC , JT Mass moments of yaw inertia of car (tow vehicle) and semitrailer, respectively, about their centroidal axis LCP, LTP Longitudinal distance from vehicle centroidal axis to the hitch pin in the car (tow vehicle) and semitrailer, respectively Li Distance parallel to heading axis from the lateral centroidal axis to the i’th wheel mC , mT

Mass of car (tow vehicle) and semitrailer, respectively

rCPX, rCPY Moment arms from car (tow vehicle) centroidal axis to force components, fCPX and fCPY, respectively rTPX, rTPY Moment arms from semitrailer centroidal axis to force components, f TPX and f TPY, respectively riX, riY Moment arms from car (tow vehicle) centroidal axis to tire-roadway force components, fiX and fiY, respectively WCP, WTP Transverse distance from wheel 3 and wheel 5 in car (tow vehicle) and semitrailer to the pin, respectively

340

X, Y

Ground fixed inertial coordinates

δ

Front wheel steer angle

C

Subscript, car or cab (tow vehicle)

T

Subscript, semitrailer

Appendix A

Units and Numbers

A.1 Use of SI (Metric Units of Measure in SAE Technical Papers) The long-term goal for SAE is international communication with minimal effort and confusion. Therefore, the use of SI units in all technical publications and presentations is preferred. The Society will strive toward universal usage of SI units and will encourage their use whenever appropriate. However, the Society also recognizes that sectors of the mobility market do not yet use SI units because of tradition, regulatory language, or other reasons. Mandating the use of SI units in these cases will impede rather than facilitate technical communication. Therefore, it is the policy to allow non-SI units and dual dimensioning where communication will be enhanced. This shall not be viewed as an avenue to circumvent the long-term goal of 100 percent SI usage. Instructions on SAE-approved techniques for conversion of units are contained in “SAE Recommended Practices, Rules for SAE Use of SI (METRIC) Units—TSB003.” Copies of TSB003 can be obtained from SAE Headquarters. Although what follows represents a change to the current policy, it is not a change to the SAE Board of Directors’ Policy since it falls within the scope of the words, “where a conflicting industry practice exists.” Dual (metric/U.S. Customary) units for the following vehicle characteristics may be considered where communication will be enhanced.

341

Appendix A

Table A.1

Metric and U.S. customary units. Vehicle characteristic

Metric units

U.S. customary units

Volume, engine displacement

liters, L, or cubic cm, cm3

cubic inches, in3

Liquid volume

liters, L

pints/quarts/gallons

Engine power

kilowatts, kW

brake horse power, bhp

Engine torque

Newton-meters, N-m

foot-pounds, lb-ft

Mass

kilograms, kg

slugs, lb-s2/ft

Pressure, stress

kiloPascals, kPa

pounds per square inch, psi

Temperature

degrees Celsius, ˚C

degrees Fahrenheit, ˚F

Area

square cm, cm

square inches, in2

Linear dimensions

millimeters, mm, meters, m, or kilometers, km

inches, in., feet, ft, miles, mi

Spring rates

Newtons per mm, N/mm

pounds per inch, lb/in

Speed

kilometers per hour, km/h or kph

miles per hour, mph

Fuel economy

kilometers per liter, km/L or kmpL

miles per gallon, mpg

Force

Newtons, N

pounds, lb

Acceleration

kilometers per second per second, km/s2, g

feet per second per second, ft/ s2 , g

2

A.2 Numbers, Significant Figures, and Rounding A.2.1 Significant Figures In all branches of science and technology, numbers are used to express values; i.e., levels or amounts of physical quantities. It is important to state numbers appropriately so that they properly convey the intended information. The number of significant figures contained in a stated number reflects the accuracy to which that quantity is known. For example, suppose the speed of a vehicle is reported as 21 m/s (69 ft/s). Is 21 m/s different from 21.0 m/s? According to the rules of significant figures, yes, but in practice, it may or may not. Could the number 21 m/s imply 20.9 m/s or less or could it imply 21.1 m/s or greater? It could, but such implications or interpretations must be determined from context, not the number 21 itself. Answers to some of these questions are related to the topic of uncertainty (covered in Chapter 1). To properly quantify and communicate a physical measurement or property, it should be stated as a reference value plus and minus an uncertainty. For example, a speed stated as v = 21.0 ± 0.6 m/s clearly is meant to be between 20.4 and 21.6 m/s. This is one of the ways of estimating and revealing the uncertainty of results. But the basic rules of using significant figures and rounding must be understood before uncertainty can be expressed. Some of the rules for handling and interpreting the significance of numbers are covered in this Appendix. Note that the terms significant figures and significant digits are used synonymously.

342

Units and Numbers

The number of significant figures in a number is defined in the following way [A.1, A.2]: 1. The leftmost nonzero digit of a number is the most significant digit. 2. If there is no decimal point, the rightmost nonzero digit is the least significant digit. 3. If there is a decimal point, the rightmost digit is the least significant digit, even if it is a zero. 4. All digits, from the least to the most significant, are counted as significant. So, for example, 2.610 and 2,498 have four significant digits each, whereas 0.125 and 728,000 have three significant digits. The following numbers each has five significant digits: 1000.0, 1206.5, 12,065,000 and 0.00012065. Unless it is stated to be exact, the speed of 21 m/s has two significant figures. If it is exact, then 21 is equivalent to 21.0000 . . . , with an unlimited number of zeros. Each of the speeds 20.4 and 21.6 has three significant figures. When numbers are very large or very small, it is convenient to express them in scientific notation. To use scientific notation, a decimal point is placed immediately after the leftmost significant digit and the number is given a suffix of 10 raised to a power n. The value of n is positive or negative. If the magnitude (disregarding the sign) of the stated number is less than 1, then n < 0. If the stated number is greater than 10, n > 0. If the stated number is between 1 and 10, n = 0. The value of n is the power of 10 that returns the number in scientific notation to its original value. For example, 0.0000687 becomes 6.87 × 10-5 and 12,360,000 becomes 1.236 × 107. Note that the number of significant digits does not change when converting to or from scientific notation.

A.2.2 Rounding of Numbers After completing calculations or when listing the results of measurements, it usually is necessary to round numbers to a lesser number of significant figures by discarding digits. Three possibilities can arise; these are: 1. The leftmost discarded digit is less than 5. When rounding such numbers, the last digit retained should remain unchanged. For example, if 3.46325 is to be rounded to four digits, the digits 2 and 5 would be discarded and 3.463 remains. 2. The leftmost discarded digit is greater than 5 or it is a 5 followed by at least one digit other than 0. In such cases, the last figure retained should be increased by one. For example, if rounded to four digits, 8.37652 would become 8.377; if rounded to three digits, it would be 8.38. 3. The leftmost discarded digit is a 5, followed only by zeros or no other numbers. Here, the last digit retained should be rounded up if it is an odd number, but no adjustment made if it is an even number. For example, 21.165, when rounded to four significant digits, becomes 21.16. The number 21.155 would likewise round to the same value, 21.16.

343

Appendix A

A reason for this last rule [A.2] is to avoid systematic errors that otherwise would be introduced into the average of a group of such numbers. Not all computer software follows this rule, however1, and when rounding for purposes of reporting results of measurements and/or calculations, the even-odd rule is not critical.

A.2.3 Consistency of Significant Figures When Adding and Subtracting When adding and subtracting numbers, proper determination of the number of significant figures is stated as a rule [A.1]. The rule is, the answer shall contain no significant digits farther to the right than occurs in the number with the least significant digits. The simplest way of following this rule is first to add or subtract the numbers using all of the stated significant figures2 followed by rounding of the final answer. For example, consider the addition of the three numbers, 964,532 and 317,880 and 563,000. These have six, five, and three significant figures, respectively. The sum by direct addition is 1,845,412. The answer then is adjusted, or rounded, to conform to the number with the least significant figures (563,000 with three), giving the final result, 1,845,000. This number has no more zero digits to the right of the comma than does 563,000. Now consider the sum of the three numbers, 964,532, -317,880 and -563,000; the direct result is 83,652. As above, this must be made to conform with the significant figures of 563,000 by using the rounding rule and is 84,000. In the last example, the concept being conveyed is that the number 563,000 is “indefinite” to the right of the “3” digit. It is not known if 563,000 could really mean 562,684 or 563,121 or other values, because 563,000, itself, may have been obtained by rounding. If it had been stated as 563,000.0, then everything would be different (since 563,000.0 would have seven significant figures and 317,880 would then have the least significant digits of the three numbers to be added in the above example).

A.2.4 Consistency of Significant Figures When Multiplying and Dividing ASTM SI-10 [A.1] states a rule for multiplying and dividing as the product or quotient shall contain no more significant digits than are contained in the number with the fewest significant digits. For example, consider the product, 125.64 × 829.4 × 1.25, of the three numbers with five, four, and three significant digits, respectively. The answer from straightforward multiplication is 130,257.27. After rounding to three significant figures, the proper end result of the multiplication is 130,000. Note that the answer, 130,000, by itself appears to have only two significant figures. This illustrates that ambiguities sometimes can arise when determining significant figures and that the amount of significant figures of a number may need to be found from context. A way of resolving

1. The reader may wish to try such an example in their favorite software. 2. ASTM SI-10 suggests first rounding each individual number to one significant figure greater than the least before adding or subtracting and then rounding the final answer. Though this may be better, it is not the way most computer software operates. Rounding after summing typically gives the same result.

344

Units and Numbers

such ambiguities is to express results of rounding in scientific notation. In this case the result would be 1.30 × 105.

A.2.5 Other Forms of Number Manipulation Not all calculations are done with addition, subtraction, multiplication, and division. There are the taking of roots, logarithms, trigonometric functions, etc. In addition, sometimes strict adherence of rounding rules can produce paradoxical or impractical results (see the following example). So more general rules are needed. In summary, two very general, but some practical rules are recommended: 1. In rounding of numbers and conversion of units, retain a number of significant digits such that accuracy and precision are neither sacrificed nor exaggerated. 2. When making and reporting calculations, continually carry all of the significant figures of a calculating device without rounding intermediate values, and round only the final answer. 3. Unit conversion should precede rounding. 4. Whenever possible, explicitly state the uncertainty of the results of measurements and calculations. Suppose a vehicle skids to a stop over a distance of d = 33.9 m from an initial speed, v, on a pavement with a uniform frictional drag coefficient of f = 0.7 ± 0.1. Use the minimum Example A and maximum values of f and Eq. 1.1 to calculate bounds on the initial speed. Convert the results to U.S. Customary units of ft/s. Solution  The lower value of speed for f = 0.6:

v = 2 fgd = 2 × 0.6 × 9.806650 × 33.9 = 19.973345... Similarly, the initial speed for f = 0.8 is:

v = 2 × 0.8 × 9.806650 × 33.9 = 23.063233 The frictional drag coefficient and its uncertainty have the fewest number of significant figures of the input values. According to the rules the final results should be rounded to one significant figure. Rounding 19.973345 . . . to a single significant digit gives a speed of v = 20 m/s. Rounding 23.063235 . . . to a single significant digit also gives a speed of v = 20 m/s. Both upper and lower bounds result with the same speed, v = 20 m/s. Clearly the result is an exaggeration of precision. Consider now another approach.

345

Appendix A

The variation of f = ± 0.1 is another way of saying that because of uncertainty, f can take on any value between 0.6 to 0.83. From the above discussion of significant figures and rounding, a point of view can be taken that the lower value, 0.6, for example, could be the result of rounding to one significant figure of any number from 0.55+ to 0.65- (such as 0.551, 0.642, etc.). Similarly, the upper value, 0.8, could be viewed as the result of rounding of any number from 0.75+ to 0.85- (such as 0.751, 0.842, etc.). So the full range of values of the frictional drag coefficient corresponding to the stated uncertainty and from the concepts of significant figures is 0.55 ≤ f ≤ 0.85. At this point the calculations are performed as if all numbers are exact giving a speed range of 19.123022 . . . ≤ v ≤ 23.773036 . . . m/s. Since rounding to one significant figure here produces an exaggeration of precision (as above), rounding is done to an additional significant figure. Consequently, the final result is stated as: 19 ≤ v ≤ 24 m/s, or v = 19.5 ± 2.5 m/s. Precision no longer is exaggerated. An initial ±14 % variation (0.7 ± 0.1) becomes a 12% variation of v (19.5 ± 2.5) through the use of Eq. 1.1. Finally, the speed is to be converted to the U.S. Customary units of ft/s. The proper conversion factor is 1ft = 0.3048 m (this is an exact conversion; see the following unit conversion table). Unit conversions should be done before rounding, so 19.123022 . . . ≤ v ≤ 23.773036 . . . m/s becomes 62.739573 . . . ≤ v ≤ 77.995525 . . . ft/s. Rounding again to one significant figure gives the same result, 70 ft, so another significant figure is acceptable, giving 63 ≤ v ≤ 78 ft/s, or v = 70.5 ± 7.5 ft/s. Another consideration that must be kept in mind when rounding is the use or purpose of the results; for example, if the speed calculated in the last example is to be compared to a speed limit, say 25 m/s. Rounding to a number of significant digits to the right of the decimal point is superfluous. The result 19 ≤ v ≤ 24 m/s is satisfactory to conclude that the calculated speed is less than the speed limit. Instead, suppose that the calculated speed is a measure of vehicle braking performance and is to be compared to a governmental regulation stated to three significant figures. Rounding to an additional significant figure leads to an exaggeration of accuracy. To compare the speed to such a regulation requires a more accurate value of friction, stated at least to two significant figures.

A.3 Unit Conversions for Common Units Factors in boldface are exact. When options exist, units in the first column printed in italics are preferred by the National Institute for Science and Technology. [A.3] To convert from

To

Multiply by

acre (based on U.S. survey foot)

square meter (m 2)

4.046 873

E+03

acre foot (based on U.S. survey foot)

cubic meter (m )

1.233 489

E+03

ampere hour (A • h)

coulomb (C)

3.6

E+03

atmosphere, standard (atm)

pascal (Pa)

1.013 25

E+05

3

3. Note that there is no implication of the likelihood of any of the values within this range.

346

Units and Numbers

atmosphere, standard (atm)

kilopascal (kPa)

1.013 25

E+02

atmosphere, technical (at)

pascal (Pa)

9.806 65

E+04

atmosphere, technical (at)

kilopascal (kPa)

9.806 65

E+01

bar (bar)

pascal (Pa)

1.0

E+05

bar (bar)

kilopascal (kPa)

1.0

E+02

barn (b)

square meter (m )

1.0

E-28

barrel [for petroleum, 42 gallons (U.S.)] (bbl)

cubic meter (m3)

1.589 873

E-01

barrel [for petroleum, 42 gallons (U.S.)] (bbl)

liter (L)

1.589 873

E+02

British thermal unit (mean) (Btu)

joule (J)

1.055 87

E+03

2

bushel (U.S.) (bu)

cubic meter (m )

3.523 907

E-02

bushel (U.S.) (bu)

liter (L)

3.523 907

E+01

calorie (cal) (mean)

joule (J)

4.190 02

E+00

candela per square inch (cd/in2)

candela per square meter (cd/m2)

1.550 003

E+03

carat, metric

kilogram (kg)

2.0

E-04

carat, metric

gram (g)

2.0

E-01

centimeter of mercury (0 °C)

pascal (Pa)

1.333 22

E+03

centimeter of water (4 °C)

pascal (Pa)

9.806 38

E+01

centimeter of water, conventional (cm H2O)

pascal (Pa)

9.806 65

E+01

centipoise (cP)

pascal second (Pa • s)

1.0

E-03

centistokes (cSt)

meter squared per second (m2/s)

1.0

E-06

chain (based on U.S. survey foot) (ch)

meter (m)

2.011 684

E+01

circular mil

square meter (m2)

5.067 075

E-10

cubic meter (m3)

3.624 556

E+00

cubic foot (ft )

cubic meter (m3)

2.831 685

E-02

cubic inch (in )

cubic meter (m )

1.638 706

E-05

cubic mile (mi3)

cubic meter (m3)

4.168 182

E+09

cubic yard (yd )

cubic meter (m )

7.645 549

E-01

cup (U.S.)

cubic meter (m3)

2.365 882

E-04

cup (U.S.)

liter (L)

2.365 882

E-01

day (d)

second (s)

8.64

E+04

day (sidereal)

second (s)

8.616 409

E+04

degree (angle) (°)

radian (rad)

1.745 329

E-02

degree Celsius (temperature) (°C)

kelvin (K)

K = °C + 273.15

degree Celsius (temperature interval) (°C)

kelvin (K)

1.0

degree centigrade (temperature)

degree Celsius (°C)

°C = deg. cent.

degree centigrade (temperature interval)

degree Celsius (°C)

1.0

cord (128 ft3) 3

3

3

3

3

3

E+00 E+00

degree Fahrenheit (temperature) (°F)

degree Celsius (°C)

°C= (°F - 32)/1.8

degree Fahrenheit (temperature) (°F)

kelvin (K)

K = (°F + 459.67)/1.8

degree Fahrenheit (temperature interval) (°F)

degree Celsius (°C)

5.555 556

E-01

347

Appendix A

degree Fahrenheit (temperature interval) (°F)

kelvin (K)

degree Rankine (°R)

kelvin (K)

K = (°R)/1.8

degree Rankine (temperature interval) (°R)

kelvin (K)

5.555 556

E-01

Denier

kilogram per meter (kg/m)

1.111 111

E-07

dyne (dyn)

newton (N)

1.0

E-05

E-01

dyne centimeter (dyn • cm)

newton meter (N • m)

1.0

E-07

dyne per square centimeter (dyn/cm2)

pascal (Pa)

1.0

E-01

erg (erg)

joule (J)

1.0

E-07

erg per second (erg/s)

watt (W)

1.0

E-07

fathom (based on U.S. survey foot)

meter (m)

1.828 804

E+00

fluid ounce (U.S.) (fl oz)

cubic meter (m3)

2.957 353

E-05

fluid ounce (U.S.) (fl oz)

milliliter (mL)

2.957 353

E+01

foot (ft)

meter (m)

3.048

E-01

foot (U.S. survey) (ft)

meter (m)

3.048 006

E-01

Footcandle

lux (lx)

1.076 391

E+01

Footlambert

candela per square meter (cd/m2)

3.426 259

E+00

foot of water, conventional (ftH2O)

pascal (Pa)

2.989 067

E+03

foot of water, conventional (ftH2O)

kilopascal (kPa)

2.989 067

E+00

foot per hour (ft/h)

meter per second (m/s)

8.466 667

E-05

foot per minute (ft/min)

meter per second (m/s)

5.08

E-03

foot per second (ft/s)

meter per second (m/s)

3.048

E-01

foot per second squared (ft/s2)4

meter per second squared (m/s2)2

3.048

E-01

foot poundal

joule (J)

4.214 011

E-02

foot pound-force (ft • lbf)

joule (J)

1.355 818

E+00

foot pound-force per hour (ft • lbf/h)

watt (W)

3.766 161

E-04

foot pound-force per minute (ft • lbf/min)

watt (W)

2.259 697

E-02

foot pound-force per second (ft • lbf/s)

watt (W)

1.355 818

E+00

gal (Gal)

meter per second squared (m/s2)

1.0

E-02

gallon [Canadian and U.K. (Imperial)] (gal)

cubic meter (m3)

4.546 09

E-03

gallon [Canadian and U.K. (Imperial)] (gal)

liter (L)

4.546 09

E+00

gallon (U.S.) (gal)

cubic meter (m3)

3.785 412

E-03

gallon (U.S.) (gal)

liter (L)

3.785 412

E+00

gallon (U.S.) per day (gal/d)

cubic meter per second (m3/s)

4.381 264

E-08

gallon (U.S.) per day (gal/d)

liter per second (L/s)

4.381 264

E-05

gallon (U.S.) per horsepower hour [gal/(hp • h)]

cubic meter per joule (m3/J)

1.410 089

E-09

gallon (U.S.) per horsepower hour [gal/(hp • h)]

liter per joule (L/J)

1.410 089

E-06

4. Standard value of free-fall acceleration is g = 9.80665 m/s2.

348

5.555 556

Units and Numbers

gallon (U.S.) per minute (gpm) (gal/min)

cubic meter per second (m3/s)

6.309 020

E-05

gallon (U.S.) per minute (gpm) (gal/min)

liter per second (L/s)

6.309 020

E-02

grain (gr)

kilogram (kg)

6.479 891

E-05

grain (gr)

milligram (mg)

6.479 891

E+01

grain per gallon (U.S.) (gr/gal)

kilogram per cubic meter (kg/m3)

1.711 806

E-02

milligram per liter (mg/L)

1.711 806

E+01

grain per gallon (U.S.) (gr/gal) gram-force per square centimeter (gf/cm )

pascal (Pa)

9.806 65

E+01

gram per cubic centimeter (g/cm3)

kilogram per cubic meter (kg/m3)

1.0

E+03

hectare (ha)

square meter (m2)

1.0

E+04

horsepower (550 ft • lbf/s) (hp)

watt (W)

7.456 999

E+02

horsepower (boiler)

watt (W)

9.809 50

E+03

horsepower (electric)

watt (W)

7.46

E+02

horsepower (metric)

watt (W)

7.354 988

E+02

horsepower (U.K.)

watt (W)

7.4570

E+02

horsepower (water)

watt (W)

7.460 43

E+02

hour (h)

second (s)

3.6

E+03

hour (sidereal)

second (s)

3.590 170

E+03

hundredweight (long, 112 lb)

kilogram (kg)

5.080 235

E+01

hundredweight (short, 100 lb)

kilogram (kg)

4.535 924

E+01

inch (in)

meter (m)

2.54

E-02

inch (in)

centimeter (cm)

2.54

E+00

inch of mercury, conventional (in. Hg)

pascal (Pa)

3.386 389

E+03

inch of mercury, conventional (in. Hg)

kilopascal (kPa)

3.386 389

E+00

inch of water, conventional (inH2O)

pascal (Pa)

2.490 889

E+02

kelvin (K)

degree Celsius (°C)

t/°C = T/K - 273.15

kilocalorie (mean) (kcal)

joule (J)

4.190 02

E+03

kilogram-force (kgf)

newton (N)

9.806 65

E+00

kilogram-force meter (kgf • m)

newton meter (N • m)

9.806 65

E+00

kilogram-force per square centimeter (kgf/cm2)

kilopascal (kPa)

9.806 65

E+01

kilogram-force per square meter (kgf/m )

pascal (Pa)

9.806 65

E+00

kilometer per hour (km/h)

meter per second (m/s)

2.777 778

E-01

kilopond (kilogram-force) (kp)

newton (N)

9.806 65

E+00

kilowatt hour (kW • h)

joule (J)

3.6

E+06

kilowatt hour (kW • h)

megajoule (MJ)

3.6

E+00

kip (1 kip= 1000 lbf)

newton (N)

4.448 222

E+03

2

2

kip (1 kip= 1000 lbf)

kilonewton (kN)

4.448 222

E+00

kip per square inch (ksi) (kip/in2)

pascal (Pa)

6.894 757

E+06

kip per square inch (ksi) (kip/in2)

kilopascal (kPa)

6.894 757

E+03

349

Appendix A

knot (nautical mile per hour)

meter per second (m/s)

5.144 444

E-01

lambert

candela per square meter (cd/m2)

3.183 099

E+03

light year (l.y.)

meter (m)

9.460 73

E+15

liter (L)

cubic meter (m3)

1.0

E-03

lumen per square foot (lm/ft )

lux (lx)

1.076 391

E+01

microinch

meter (m)

2.54

E-08

microinch

micrometer (μm)

2.54

E-02

micron (μ)

meter (m)

1.0

E-06

micron (μ)

micrometer (μm)

1.0

E+00

mil (0.001 in)

meter (m)

2.54

E-05

mil (0.001 in)

millimeter (mm)

2.54

E-02

mil (angle)

radian (rad)

9.817 477

E-04

mil (angle)

degree (°)

5.625

E-02

mile (mi)

meter (m)

1.609 344

E+03

mile (mi)

kilometer (km)

1.609 344

E+00

mile (based on U.S. survey foot) (mi)

meter (m)

1.609 347

E+03

mile (based on U.S. survey foot) (mi)

kilometer (km)

1.609 347

E+00

mile, nautical

meter (m)

1.852

E+03

mile per gallon (U.S.) (mpg) (mi/gal)

meter per cubic meter m/ m3)

4.251 437

E+05

mile per gallon (U.S.) (mpg) (mi/gal)

kilometer per liter (km/L)

4.251 437

E-01

mile per gallon (U.S.) (mpg) (mi/gal)

liter per 100 kilometer (L/100km)

divide 235.215 by number of miles per gallon

mile per hour (mi/h)

meter per second (m/s)

4.4704

E-01

mile per hour (mi/h)

kilometer per hour (km/h)

1.609 344

E+00

mile per minute (mi/min)

meter per second (m/s)

2.682 24

E+01

mile per second (mi/s)

meter per second (m/s)

1.609 344

E+03

millibar (mbar)

pascal (Pa)

1.0

E+02

millibar (mbar)

kilopascal (kPa)

1.0

E-01

millimeter of mercury, conventional (mmHg)

pascal (Pa)

1.333 224

E+02

millimeter of water, conventional (mm H2O)

pascal (Pa)

9.806 65

E+00

2

350

minute (angle) (N)

radian (rad)

2.908 882

E-04

minute (min)

second (s)

6.0

E+01

minute (sidereal)

second (s)

5.983 617

E+01

ounce (avoirdupois) (oz)

kilogram (kg)

2.834 952

E-02

ounce (avoirdupois) (oz)

gram (g)

2.834 952

E+01

ounce (troy or apothecary) (oz)

kilogram (kg)

3.110 348

E-02

ounce (troy or apothecary) (oz)

gram (g)

3.110 348

E+01

ounce [Canadian and U.K. fluid (Imperial)] (fl oz)

cubic meter (m3)

2.841 306

E-05

Units and Numbers

ounce [Canadian and U.K. fluid (Imperial)] (fl oz)

milliliter (mL)

2.841 306

E+01

ounce (U.S. fluid) (fl oz)

cubic meter (m )

2.957 353

E-05

ounce (U.S. fluid) (fl oz)

milliliter (mL)

2.957 353

E+01

ounce (avoirdupois)-force (ozf)

newton (N)

2.780 139

E-01

ounce (avoirdupois)-force inch (ozf • in)

newton meter (N • m)

7.061 552

E-03

3

ounce (avoirdupois)-force inch (ozf • in)

millinewton meter (mN • m)

7.061 552

E+00

ounce (avoirdupois) per cubic inch (oz/in3)

kilogram per cubic meter (kg/m3)

1.729 994

E+03

peck (U.S.) (pk)

cubic meter (m3)

8.809 768

E-03

peck (U.S.) (pk)

liter (L)

8.809 768

E+00

pennyweight (dwt)

kilogram (kg)

1.555 174

E-03

pennyweight (dwt)

gram (g)

1.555 174

E+00

pica (computer) (1/6 in)

meter (m)

4.233 333

E-03

pica (computer) (1/6 in)

millimeter (mm)

4.233 333

E+00

pica (printer’s)

meter (m)

4.217 518

E-03

pica (printer’s)

millimeter (mm)

4.217 518

E+00

pint (U.S. dry) (dry pt)

cubic meter (m3)

5.506 105

E-04

pint (U.S. dry) (dry pt)

liter (L)

5.506 105

E-01

pint (U.S. liquid) (liq pt)

cubic meter (m3)

4.731 765

E-04

pint (U.S. liquid) (liq pt)

liter (L)

4.731 765

E-01

point (computer) (1/72 in)

meter (m)

3.527 778

E-04

point (computer) (1/72 in)

millimeter (mm)

3.527 778

E-01

point (printer’s)

meter (m)

3.514 598

E-04

point (printer’s)

millimeter (mm)

3.514 598

E-01

poise (P)

pascal second (Pa • s)

1.0

E-01

pound (avoirdupois) (lb)

kilogram (kg)

4.535 924

E-01

pound (troy or apothecary) (lb)

kilogram (kg)

3.732 417

E-01

poundal

newton (N)

1.382 550

E-01

poundal per square foot

pascal (Pa)

1.488 164

E+00

poundal second per square foot

pascal second (Pa • s)

1.488 164

E+00

pound foot squared (lb • ft2)

kilogram meter squared (kg • m2)

4.214 011

E-02

pound-force (lbf)5

newton (N)

4.448 222

E+00

pound-force foot (lbf • ft)

newton meter (N • m)

1.355 818

E+00

pound-force foot per inch (lbf • ft/in)

newton meter per meter (N•m/m)

5.337 866

E+01

pound-force inch (lbf • in)

newton meter (N • m)

1.129 848

E-01

pound-force inch per inch (lbf • in/in)

newton meter per meter (N•m/m)

4.448 222

E+00

5. If the local value of the acceleration of free fall is taken as the standard value g = 9.90665 m/s2, then the exact conversion factor is 4.448 221 615 260 5 E+00.

351

Appendix A

pound-force per foot (lbf/ft)

newton per meter (N/m)

1.459 390

E+01

pound-force per inch (lbf/in)

newton per meter (N/m)

1.751 268

E+02

pound-force per pound (lbf/lb) (thrust to mass ratio)

newton per kilogram (N/kg)

9.806 65

E+00

pound-force per square foot (lbf/ft2)

pascal (Pa)

4.788 026

E+01

pound-force per square inch (psi) (lbf/in )

pascal (Pa)

6.894 757

E+03

pound-force per square inch (psi) (lbf/in2)

kilopascal (kPa)

6.894 757

E+00

2

pound-force second per square foot (lbf • s/ft )

pascal second (Pa • s)

4.788 026

E+01

pound-force second per square inch (lbf• s/in2)

pascal second (Pa • s)

6.894 757

E+03

pound inch squared (lb • in2)

kilogram meter squared (kg •m2)

2.926 397

E-04

pound per cubic foot (lb/ft3)

kilogram per cubic meter (kg/m3)

1.601 846

E+01

pound per cubic inch (lb/in3)

kilogram per cubic meter (kg/m3)

2.767 990

E+04

pound per cubic yard (lb/yd3)

kilogram per cubic meter (kg/m3)

5.932 764

E-01

pound per foot (lb/ft)

kilogram per meter (kg/m)

1.488 164

E+00

pound per foot hour [lb/(ft • h)]

pascal second (Pa • s)

4.133 789

E-04

pound per foot second [lb/(ft • s)]

pascal second (Pa • s)

1.488 164

E+00

pound per gallon [Canadian and U.K. (Imperial)] (lb/gal)

kilogram per cubic meter (kg/m3)

9.977 637

E+01

pound per gallon [Canadian and U.K. (Imperial)] (lb/gal)

kilogram per liter (kg/L)

9.977 637

E-02

pound per gallon (U.S.) (lb/gal)

kilogram per cubic meter (kg/m3)

1.198 264

E+02

pound per gallon (U.S.) (lb/gal)

kilogram per liter (kg/L)

1.198 264

E-01

pound per horsepower hour [lb/(hp • h)]

kilogram per joule (kg/J)

1.689 659

E-07

psi (pound-force per square inch) (lbf/in2)

pascal (Pa)

6.894 757

E+03

2

psi (pound-force per square inch) (lbf/in )

kilopascal (kPa)

6.894 757

E+00

quad (1015 BtuIT)

joule (J)

1.055 056

E+18

quart (U.S. dry) (dry qt)

cubic meter (m3)

1.101 221

E-03

quart (U.S. dry) (dry qt)

liter (L)

1.101 221

E+00

2

352

quart (U.S. liquid) (liq qt)

cubic meter (m )

9.463 529

E-04

quart (U.S. liquid) (liq qt)

liter (L)

9.463 529

E-01

rad (absorbed dose) (rad)

gray (Gy)

1.0

E-02

revolution (r)

radian (rad)

6.283 185

E+00

revolution per minute (rpm) (r/min)

radian per second (rad/s)

1.047 198

E-01

rod (based on U.S. survey foot) (rd)

meter (m)

5.029 210

E+00

3

rpm (revolution per minute) (r/min)

radian per second (rad/s)

1.047 198

E-01

second (angle) (O)

radian (rad)

4.848 137

E-06

second (sidereal)

second (s)

9.972 696

E-01

Units and Numbers

shake

second (s)

1.0

E-08

shake

nanosecond (ns)

1.0

E+01

slug (slug)

kilogram (kg)

1.459 390

E+01

kilogram per cubic meter (kg/m3)

5.153 788

E+02

slug per cubic foot (slug/ft ) 3

slug per foot second [slug/(ft • s)]

pascal second (Pa • s)

4.788 026

E+01

square foot (ft2)

square meter (m2)

9.290 304

E-02

square foot per hour (ft2/h)

square meter per second (m2/s)

2.580 64

E-05

square foot per second (ft2/s)

square meter per second (m2/s)

9.290 304

E-02

square inch (in2)

square meter (m2)

6.4516

E-04

square inch (in )

square centimeter (cm )

6.4516

E+00

square mile (mi2)

square meter (m2)

2.589 988

E+06

2

2

square mile (mi )

square kilometer (km )

2.589 988

E+00

square mile (based on U.S. survey foot) (mi2)

square meter (m2)

2.589 998

E+06

2

2

square mile (based on U.S. survey foot) (mi )

square kilometer (km )

2.589 998

E+00

square yard (yd2)

square meter (m2)

8.361 274

E-01

stokes (St)

meter squared per second (m2/s)

1.0

E-04

tablespoon

cubic meter (m3)

1.478 676

E-05

tablespoon

milliliter (mL)

1.478 676

E+01

teaspoon

cubic meter (m3)

4.928 922

E-06

teaspoon

milliliter (mL)

4.928 922

E+00

therm (EC)

joule (J)

1.055 06

E+08

therm (U.S.)

joule (J)

1.054 804

E+08

ton, assay (AT)

kilogram (kg)

2.916 667

E-02

ton, assay (AT)

gram (g)

2.916 667

E+01

ton-force (2000 lbf)

newton (N)

8.896 443

E+03

ton-force (2000 lbf)

kilonewton (kN)

8.896 443

E+00

ton, long (2240 lb)

kilogram (kg)

1.016 047

E+03

ton, long, per cubic yard

kilogram per cubic meter (kg/m3)

1.328 939

E+03

ton, metric (t)

kilogram (kg)

1.0

E+03

tonne (called “metric ton” in U.S.) (t)

kilogram (kg)

1.0

E+03

ton of refrigeration (12 000 BtuIT/h)

watt (W)

3.516 853

E+03

ton of TNT (energy equivalent)

joule (J)

4.184

E+09

2

2

ton, register

cubic meter (m )

2.831 685

E+00

ton, short (2000 lb)

kilogram (kg)

9.071 847

E+02

ton, short, per cubic yard

kilogram per cubic meter (kg/m3)

1.186 553

E+03

ton, short, per hour

kilogram per second (kg/s)

2.519 958

E-01

3

353

Appendix A

354

torr (Torr)

pascal (Pa)

1.333 224

E+02

watt hour (W• h)

joule (J)

3.6

E+03

yard (yd)

meter (m)

9.144

E-01

year (365 days)

second (s)

3.1536

E+07

year (sidereal)

second (s)

3.155 815

E+07

year (tropical)

second (s)

3.155 693

E+07

Appendix B

Glossary

Common Terms and Acronyms in Accident Reconstruction The terms defined here are for the convenience of the readers, both those learning accident reconstruction as well as those already familiar with the field. Not all terms included in the list are used in this book. 18-wheeler: A tractor, semitrailer with a total of 18 wheels; see tractor, semitrailer. A, B, C, D, pillars and posts: The vertical pillars and posts of a light vehicle forming the major vertical structural members of the body; see Fig. B1. Pillars typically are at window height; posts are below window height. From front to rear, the A post/pillar is the most forward member, the B post/pillar is the second most forward vertical member, etc. Figure B1. Pillars and posts.

355

Appendix B

AACN: Automatic Advanced Crash Notification, General Motors term for ACN (see also ACN). AADT: Average Annual Daily Traffic. AASHTO: American Association of State Highway and Transportation Officials. ABA: American Bus Association. ABS: Antilock Braking System. ACAT: Advanced Collision Avoidance Technology. ACC: Adaptive Cruise Control. acceleration: Change in linear or angular velocity or speed with respect to time. accelerometer: An electromechanical sensing device with an output signal proportional to acceleration. accident investigation: The process of observation, acquisition, and documentation of physical evidence and other information regarding an accident or crash. accident reconstruction: A procedure carried out with the specific purpose of estimating in both a qualitative and quantitative manner how an accident occurred using engineering, scientific, and mathematical principles and based on evidence obtained through accident investigation. accident scene: A place where a traffic accident occurs, both during and immediately following the accident, and before vehicles and participants have departed; see accident site. accident site: A place where a traffic accident occurred, after vehicles and participants have departed the scene; see accident scene. accident, vehicle: An event in which one or more vehicles undergo unexpected action(s), usually involving contact with another vehicle or other object, producing injury, death, and/or property damage; an accident is an unstabilized situation which includes at least one harmful event; see crash. ACM: Airbag Control Module (Chrysler), control module for airbags and related restraint systems, see RCM and SDM. ACN: Automatic Crash Notification System. acoustic levels: see sound levels. ADA: Americans with Disabilities Act. ADR: Accident data recorder; see EDR.

356

Glossary

AE: Algorithm Enable AFV: Alternative Fuel Vehicle. aggressivity: The inertial and structural properties and characteristics of a vehicle that relate to the severity of injuries to occupants in the other vehicle in a crash. agricultural commodity trailer: Trailer designed to transport bulk commodities from harvest sites to process or storage sites. air bag: A device in the interior of a vehicle that inflates and acts between an occupant and an interior vehicle surface to prevent injury in a crash; see supplemental restraint system. angular acceleration: The time rate of change of angular velocity. angular velocity: The time rate of change of rotational displacement. animation: The process by which the movement of objects is illustrated. ANSI: American National Standards Institute. approach speed: Speed of a vehicle just prior to the first significant event such as contact in an accident; see closing speed. aquaplaning: See hydroplaning. area of impact: Area encompassed by the interface between colliding objects projected onto the road; see point of impact. articulated vehicle: A vehicle comprised of two or more distinct, interconnected bodies such as a tractor, semitrailer. asphalt: See bituminous pavement. ATA: American Trucking Association. BAC: Blood Alcohol Concentration. backlite header: The structural body member which connects the upper portions of the rearmost driver and passenger pillars and forms the top edge of the backlite (back window) [B.1]. backlite: The rear or back window which spans from the driver’s to passenger’s side of the vehicle [B.1]. barrier equivalent velocity (BEV): The forward speed and corresponding kinetic energy with which a vehicle contacts a flat fixed rigid barrier at 90° with no rebound; see also equivalent energy speed (EES).

357

Appendix B

BAS: Brake Assist System BCM: Body Control Module. BEV: Barrier Equivalent Velocity, or Battery Electric Vehicle. bicycle model: A two-wheeled vehicle used conceptually in vehicle dynamics studies to represent a four-wheeled vehicle where the side-to-side extent of the vehicle is neglected for simplicity. bituminous pavement: A pavement comprising an upper layer or layers of aggregate with a bituminous binder (asphalt, coal tars, natural tars, etc.) and surface treatments such as chip seals, slurry seals, sand seals, and cape seals. black box: See event data recorder. blacktop: See bituminous pavement. BMP: BitMaP, digital photograph file format. bobtail: A term used to refer to a truck tractor being driven without a semitrailer. brake slip: See wheel slip. braking distance: The distance taken to bring a vehicle to rest during brake application in straight forward motion; see stopping distance. braking force, peak: The largest force that can be developed during brake application as wheel slip is varied over the range of free-rolling slip to locked-wheel slip. braking force: The force over the contact surface between a tire and a road in the direction of heading of the braked wheel that develops as a result of brake application. BTO: Brake Throttle Override. BTS: Bureau of Transportation Statistics. BTSI: Brake Transmission Shift Interlock. Btu: British thermal unit. bus: A vehicle designed to transport more than 15 passengers, including the driver. CAA: Clean Air Act. CAD: Computer Aided Design (often referring to drafting software). CAFE: Corporate Average Fuel Economy. CAN: Controller Area Network, a type of communication bus (see also GMLAN).

358

Glossary

CCD: Charge Coupled Device (such as the sensor in a digital camera). CCM: Cruise Control Module. CDR: Crash Data Retrieval System; a system used to image crash data from certain light vehicles. center of gravity (cg): That point of a body through which the resultant force of gravity (weight) acts irrespective of the orientation of the body. center of impact: See impact center. center of mass: See center of gravity. central impact: An impact in which the contact impulse passes through the center of gravity; see oblique impact. CDL: Commercial Driver’s License CFR: Code of Federal Regulations. change in momentum: The difference of the momentum (product of mass and velocity) of a mass from one time to another; the difference of the momentum of a mass between the beginning and end of contact with another mass; the difference in the momentum of a system of bodies; see also conservation of momentum. change of velocity: The difference between velocity vectors at two points in time; see also ΔV, delta-V. CHB: Crash Imminent Braking. CHOP: a broad shallow gouge in a road surface, beginning with an even, regular, deeper side and terminating in scratches and striations on the opposite shallower side; a depression in pavement made by a strong, sharp metal edge moving under heavy pressure [B.2], more commonly occurring at an impact event as opposed to post-impact trajectory. clearance lamp: Light used on the front or rear of a motor vehicle to indicate overall width or height. closing speed: The magnitude of the relative velocity between two vehicles at a given point in time as they approach each other; the relative velocity between two vehicles as they approach each other at the beginning of a accident; normal component of the closing velocity; see approach speed. closing velocity: The magnitude of the relative velocity between two vehicles at a give point in time as they approach each other; the magnitude of the relative velocity between two vehicles at the beginning of a crash; the vector difference between the velocity of the vehicle and the vehicle/object struck immediately before impact.

359

Appendix B

CMV: Commercial Motor Vehicle. CO: Carbon monoxide. CO2 (CO2): Carbon dioxide. coefficient of friction: A number representing the resistance to sliding of two flat surfaces in contact; defined as the ratio of the resistance force to the normal force between the surfaces; see frictional drag factor. coefficient of restitution: The ratio of the relative normal velocity at the time of separation to the relative normal velocity at the time of initial contact between the point area of contact of two colliding bodies. coefficient of rolling resistance: The ratio of the force of resistance to rolling with zero slip to the vertical load of a wheel or vehicle; see rolling resistance. collision deformation classification (CDC): A classification of the extent of deformation to an automobile, utility vehicle, pickup, and van from a crash [B.3]. collision: Sudden contact of a vehicle with an object or another vehicle, usually resulting in visible damage; see impact, crash. common contact point: See impact center common velocity conditions: Two independent conditions applicable to a collision where at the time of separation the relative normal velocity component is zero (no restitution) and the relative tangential component of velocity is zero (sliding has ended). compatibility, vehicle: Disparities in structural crashworthiness of different sized vehicles due to varying structural geometries, such as different heights of vehicles’ front and side structures; a tendency of some vehicles to inflict more damage on another vehicle in a crash. concrete pavement: A solidified pavement with an upper layer of aggregate (such as sand and stone) mixed with Portland cement paste binder. conservation of momentum: The principle of physics for vehicles in dynamic contact stating that in the absence of external forces, the sum of the preimpact momentum is equal to the sum of the postimpact momentum of the vehicles. contact damage: Deformation sustained in a vehicle from physical engagement with another vehicle or object; see induced damage, residual crush, dynamic crush. contact patch: The area or region of mutual contact between a tire and the surface over which it rests or moves. contact point: The point of intersection of the resultant contact impulse with the intervehicular contact surface of each of two colliding vehicles; see impact center.

360

Glossary

contact surface: See intervehicular contact surface. coordinate system, vehicle: See three-axis vehicle coordinate system. cornering coefficient: See sideslip coefficient. CPR: Crash Pulse Recorder, a device that measures acceleration during a crash; see accelerometer. CRAF: Civil Reserve Air Fleet. crash duration: The period of time defined by the moment when two vehicles come in contact until that time when they separate. crash pulse: The shape of the intervehicular force curve during the crash duration. crash reconstruction: See accident reconstruction. crash: An event in which one or more vehicles make unintended contact with another vehicle or other object producing injury, death, and/or property damage; see accident, impact, collision. CRASH3: An acronym for Calspan Reconstruction of Speeds on the Highway, Version 3; a method of reconstruction that uses the calculation of the crush energy of a collision and an approximate postimpact trajectory spinout simulation. crashworthiness: The characteristics of a motor vehicle which represent occupant protection of that vehicle in a specific collision.

v =

fgR

critical speed formula: A formula, cr , that calculates the speed of a vehicle from its radius of curvature, R, frictional drag coefficient, f, and acceleration of gravity, g. critical speed: The maximum speed at which a vehicle can traverse a path with a specific radius of curvature without loss of directional control; the speed of a vehicle undergoing a sudden turn maneuver at which the tires leave visible sideslip marks. crumple zone: That portion of the front or rear of a vehicle designed to absorb energy of a collision for the protection of the occupants. crush area: Area defined by the original vehicle exterior and a crush profile. crush equivalent speed: See energy equivalent speed, EES. crush profile: The geometric shape in a specified plane (e.g., vertical, horizontal) which describes the vehicle damage resulting from an impact. crush stiffness coefficient: An empirical quantity used in the calculation of the energy dissipated in a collision and associated with each vehicle’s velocity change, ΔV; see CRASH3.

361

Appendix B

crush stiffness: See crush stiffness coefficient. CTE: Coefficient of Thermal Expansion. curb weight: The weight of a motor vehicle with standard equipment and maximum fuel capacity. CVT: Continuously Variable Transmission. delta-t (Δt or Δτ): A time interval associated with an event such as vehicle-to-vehicle contact; the time duration of impulse. delta-v (ΔV): The difference or change of a velocity vector over a time interval; the difference in the velocity vector of the center of gravity of a vehicle between separation and first contact in a crash. departure velocity: See separation velocity. deployment (event): Actuation of a supplementary restraint, based on an enabling algorithm, following which acceleration and other data are recorded and made available to an event data recorder. deployment level (event): An acceleration level sufficient to cause the GM SDM’s crashsensing algorithm to “enable” and anticipate a collision severity which otherwise warrants a deployment for that vehicle but a deployment had been previously commanded. decibel: A logarithmic measure of the level, L, of a time-varying signal, s(τ), relative to a reference value sref.

L = 10 log

s2 2 sref

where s2 is the mean square value of the signal. DGPS: Differential Global Positioning System. direction of principal force (DOPF): See principal direction of force, PDOF. divot: A piece of turf or sod torn up by dynamic contact. DLC: Diagnostic Link Connector, may also be seen as Data Link Connector. DOE: U.S. Department of Energy. DOI: U.S. Department of the Interior. DoT, DOT: United States Department of Transportation; see NHTSA.

362

Glossary

drag factor: An equivalent acceleration expressed as a fraction of the acceleration of gravity, g; also see frictional drag coefficient. drag sled: A weighted device (whose bottom surface is covered with a portion of tire tread) which is pulled along a roadway surface and provides a sliding friction coefficient of that device and roadway surface by computing the ratio of the pull force to its weight. dwt: Deadweight tons. drop axle: An unpowered auxiliary axle on a truck that can be raised or lowered to change the vertical load distribution of permanent axles (also called tag axle). DTC: Diagnostic Trouble Code. dual wheels: The use of two closely spaced wheels on one side of an axle, typically used on trucks and semitrailers. DXF: Drawing Exchange Format, graphical format for drawings made with CAD programs. dynamic crush: The deformation formed by the external surface of a vehicle at any time during an impact, usually measured relative to the corresponding as-manufactured undeformed surface; see crush area, crush profile, static crush, residual crush. eccentric impact: See oblique impact. ECE: Economic Commission for Europe (United Nations). ECI: Electronic Conductive Immunity. EDS: Explosive Detection Systems. EDR: Event Data Recorder; a function within a vehicle module (ACM, PCM . . .) which has the capability to save certain crash data parameters after primary functions are completed. elastic deformation: Deformation which is fully recovered after an applied force is removed. elastic impact: An idealized impact where the kinetic energy at separation equals the kinetic energy at the initiation of contact; a fully elastic impact is an impact where the coefficient of restitution is equal to one. electronic control module (ECM): See Electronic Control Unit (ECU). electronic control unit (ECU): The computer in a vehicle that controls vehicle system operation, including functions such as engine operation, On Board Diagnostics (OBD), Stability Control, Safety System Operation, etc.

363

Appendix B

electronic data recorder (EDR): See event data recorder. EMC: Electromagnetic Compatibility. EMI: Electromagnetic Interference. energy equivalent speed (EES): The speed and corresponding kinetic energy with which a vehicle must contact a fixed rigid object with no rebound for equivalence to conditions of another collision; for example, the energy may be equal to a specified level of residual crush; EES is a preferred term, broader than barrier equivalent velocity (BEV), equivalent barrier speed (EBS), and equivalent test speed (ETS). energy equivalent velocity: See energy equivalent speed and equivalent barrier speed. engine control module (ECM): An electronic device in a vehicle (especially heavy trucks) that controls engine operation. EPA: U.S. Environmental Protection Agency. EPS: Electronic Power Steering. ERI: Electronic Radiated Immunity. ESD: Electrostatic Discharge. ESC: Electronic stability control. ESP: An acronym for Chrysler’s Electronic Stability Program (see ESC). equivalent barrier speed (EBS): The forward speed and corresponding kinetic energy with which a vehicle must contact a flat, fixed, rigid barrier at 90° with no rebound for equivalence to conditions of another collision; for example, the energy may be equal to a specified level of residual crush; see also equivalent energy speed (EES). equivalent test deformation: See EES. equivalent test speed (ETS): ISO term and is a non-preferred term, see EBS and EES. ETC: Electronic Throttle Control (can also mean: electronic toll collection). ETMS: Enhanced Traffic Management System. EU: European Union. EVC: Electronic Vehicle Control. event data recorder (EDR): An onboard electronic module or device capable of monitoring, recording, and displaying precrash, crash, and postcrash data and information from a vehicle, event, and driver.

364

Glossary

EXIF: Exchangeable Image File Format (photographs). FAA: Federal Aviation Administration. FAF: Freight Analysis Framework. farm tractor: A powered farm vehicle designed to pull farm implements (such as a plow, farm trailer, manure spreader, etc.) FARS: Fatality Analysis Reporting System. FARs: Federal Aviation Regulations. FHWA: U.S. Federal Highway Administration. first contact position: The position, or location, at an accident scene (measured relative to a coordinate system fixed to the earth) of a vehicle, pedestrian, or other object at the time it first has contact with another body in a collision. first contact velocity: The velocity of the center of gravity of a vehicle, pedestrian, or other object at its first contact position. fixed object: A stationary object such as a guardrail, bridge railing or abutment, construction barricade, impact attenuator, tree, embedded rock, utility pole, ditch side, steep earth or rock slope, culvert, fence, or building [B.4]. flip: Movement of a vehicle from a place where the forward velocity of a part of the vehicle suddenly is stopped by an object below its center of gravity such as a curb, rail, or furrow with the result that the ensuing rotation lifts the vehicle from the ground. FMCSA: Federal Motor Carrier Safety Administration. FMCSR: Federal Motor Carrier Safety Regulations. FMEA: Failure Modes and Effects Analysis. FMVSS: Federal Motor Vehicle Safety Standard; see NHTSA. forward projection pedestrian collision: A frontal collision of vehicle and pedestrian or cyclist where the initial contact area is at or above the height of the center of gravity of the pedestrian or cyclist and where a single impact with the frontal geometry of the vehicle causes the pedestrian or cyclist to be projected straight forward relative to the vehicle. four point transformation: A photogrammetric technique whereby points positioned on a surface reasonably approximated by a plane with unknown locations can be located through the use of four additional points whose locations are known; see photogrammetry.

365

Appendix B

FRA: Federal Railroad Administration. friction: Resistance to sliding over a contact surface between two materials. friction coefficient: See coefficient of friction. frictional drag coefficient: An average, uniform (constant) value of a sliding friction coefficient applied to a specific sliding event such as when an object slides from an initial speed to a stop over a distance, d, or during a speed change, ΔV. frictional drag factor: See frictional drag coefficient. frontal impact: An impact or collision involving the front of a vehicle. FTA: Federal Transit Administration, also Fault Tree Analysis. full trailer: A towed vehicle with a fixed rear axle and a front axle that pivots and is made to be pulled by a powered tow vehicle (an example is a farm trailer). furrow: A channel in a loose or soft material, such as snow or soil, made by a vehicle tire or some other part of a moving vehicle. GA: General Aviation. GAW: Gross Axle Weight is the total weight carried by an individual axle (front and rear) including the vehicle weight and cargo. GAWR: Gross Axle Weight Rating is the maximum allowable weight that can be carried by a single axle (front or rear). GCW: Gross Combined Weight is the weight of a loaded vehicle plus the weight of a fully loaded semitrailer. GCWR: Gross Combined Weight Rating is the maximum allowable weight of a vehicle and loaded semitrailer. GHG: Greenhouse Gas. GIF: Graphics Interchange Format, digital photograph file format. GIS: Geographic Information Systems. glare: Interference to a driver’s vision due to natural or artificial, direct or reflected light. gouge or gouge mark: Pavement or ground scar deep enough to be easily felt with the fingers; see Fig. B6 and also chop and groove. GMLAN: A General Motors implementation of the Controller Area Network (CAN) type serial communication protocol.

366

Glossary

GPS: Global Positioning System. groove: A long, narrow, pavement gouge or a channel in a pavement. gross vehicle weight rating: The upper limit of combined weight and cargo for a vehicle established by design, regulation, or both gross vehicle weight: The combined weight of a vehicle and its cargo. GVWR: See Gross Vehicle Weight Rating. HAPs: Hazardous Air Pollutants. heading angle, ψ: The angle between a reference axis fixed in the vehicle and a reference axis fixed in the roadway, giving a measure of vehicle yaw rotation or directional orientation relative to the roadway; see Fig. B2 [B.5]. head-on impact: Frontal impact where the PDOF is at or near zero degrees. heavy truck classifications: See truck classifications. HEV: Hybrid Electric Vehicle. Figure B2. SAE coordinate system showing the vehicle heading angle, ψ, and vehicle sideslip angle, β.

HELP: Heavy Vehicle Electronic License Plate. HMIS: Hazardous Materials Information System. HPMS: Highway Performance Monitoring System.

367

Appendix B

HSI: Human-System Integration. HSR: High-Speed Rail. HTF: Highway Trust Fund. HV: Heavy Vehicle. hydroplaning: A phenomenon where a layer of fluid (usually water) on a roadway separates the load-bearing surface of one or more tires of a moving vehicle from the road surface and causes a full loss of traction (longitudinal) and steering (transverse) force components. IBET: Intermodal Bottleneck Evaluation Tool. illumination: Placement or existence of natural or artificial light on an area presented to a driver. impact center: The point of intersection of the contact impulse and the intervehicular contact surface for an impact; see contact point. impact force (lever arm) moment arm: See impulse moment arm. impact velocity: The velocity of an object’s center of gravity relative to a coordinate system fixed in the earth during an impact; see preimpact velocity, postimpact velocity. impact: The striking of one body against another; short-duration, high-force contact of two objects; a collision of a vehicle with another vehicle, a pedestrian, or some other object; see collision, crash. impulse moment arm: The perpendicular distance from an object’s center of gravity to the line of action of an impulse; see Fig. B3; see also impact force moment arm.

Figure B3. Illustration of the moment arm, d, of an impulse and the PDOF, Principal Direction of Force.

368

Glossary

impulse ratio: The ratio of the tangential and normal impulse components in planar impact mechanics; see impulse. impulse: A combination of force, F, and time, τ, defined as a mathematical integral, ∫Fdτ, of the force over a specific time duration. induced damage: Residual deformation caused without direct contact by virtue of being adjacent to deformation caused by direct contact; see residual crush. initial contact: The point in time and space when two objects begin to touch or interact with no significant force. The beginning of an impact. INS: Immigration and Naturalization Service. intervehicular contact surface: A single, planar surface that represents the average (over time and space) deformed contact surface between two vehicles or a vehicle and barrier. intervehicular crush plane: See intervehicular contact surface. intrusion: Reduction of the pre-crash space within the passenger space compartment [B.6]. IPCC: Intergovernmental Panel on Climate Change. ISO: International Organization for Standardization, Geneva, Switzerland. ISTEA: Intermodal Surface Transportation Efficiency Act. ITS: Intelligent Transportation System. JPEG: Joint Photographic Experts Group, digital photograph file format. KPH: Kilometers Per Hour (also km/h). leading edge: The foremost part of a vehicle with respect to the vehicle’s motion and attitude. light vehicle: An automobile, passenger van, pickup truck, or sport utility vehicle. LNG: Liquefied Natural Gas. LPG: Liquefied Petroleum Gas. LTV: Light Trucks and Vans. LV: See light vehicle. MAIT: Multidisciplinary Accident Investigation Team (NHTSA). maximum crush depth: Deepest part of a crush profile; see dynamic crush or residual crush.

369

Appendix B

maximum engagement: The point in time when the maximum dynamic crush occurs. MDB: Moving Deformable Barrier. MMUCC: Model Minimum Uniform Crash Criteria. moment of inertia: A physical property of a body that represents its resistance to rotational acceleration. MPG: Miles Per Gallon. MPH: Miles Per Hour. MTBE: Methyl-tertiary-butyl-ether. MTC: Mechanical Throttle Control. MUTCD: Manual on Uniform Traffic Control Devices. NASS: National Automotive Sampling System. NCAP: New Car Assessment Program (DoT, NHTSA). NDR: National Driver Register. neutral steer: When a vehicle, traveling on a circular path at constant speed and a constant front wheel steer angle, is accelerated it will remain on a path with the same radius, tend to increase its radius, or tend to decrease its radius; these are defined as neutral steer, understeer, and oversteer, respectively; see oversteer and understeer. NHTSA: National Highway Traffic Safety Administration; see DoT. NO2 (NO2): Nitrogen dioxide. NOX: Nitrogen oxides. NPTS: Nationwide Personal Transportation Survey. NTL: National Transportation Library. NTSB: National Transportation Safety Board. OBD: On Board Diagnostics. oblique impact: An impact in which the contact impulse does not pass through the center of gravity; see central impact. occupant compartment: That portion of a vehicle’s interior designed for the use of passengers during operation of the vehicle. ODI: Office of Defects Investigation (NHTSA).

370

Glossary

offset: The distance between the longitudinal heading axes of two vehicles in frontal contact; see Fig. B4; see overlap. Figure B4. Illustration of offset and overlap.

OPEC: Organization of Petroleum Exporting Countries. ORC: Occupant Restraint Controller, see event data recorder (EDR). OTR: Over The Road. overhang, front rear: The longitudinal dimension of a vehicle from the center of the front/rear wheels to the foremost/rearmost point on the vehicle including bumper, bump guards, tow hooks, and/or rub strips if standard equipment. overlap: The length of mutual contact damage; see Fig. B4; see offset. override: A condition in a collision where the main structural members such as a bumper of the striking vehicle are above the main structural members such as frame rails of the struck vehicle; see Fig. B5; see underride. Figure B5. Illustration of override in a collision.

oversteer: When a vehicle, traveling on a circular path at constant speed and a constant front wheel steer angle, is accelerated it will remain on a path with the same radius, tend to increase its radius, or tend to decrease its radius; these are defined as neutral steer, understeer, and oversteer, respectively; see neutral steer and understeer. PCM: Powertrain Control Module; also Pulse Code Modulation (a way of digitally transmitting analog data).

371

Appendix B

PAR: Police Accident Report. PCR: Police Crash Report. PDOF: See principal direction of force. PDR time: See perception-decision-reaction time. perception-decision-reaction time: The time required by a person to complete a response to an event or stimulus; see reaction time. PFD: Personal Flotation Device. PHEV: Plug-In Hybrid Electric Vehicle. photogrammetry: The process of determining the quantitative dimensional information of objects in two or three dimensions through the process of recording, interpreting, and transforming measurements from a flat photographic image. pitch, roll, yaw: Terms that distinguish rotations of a vehicle about three perpendicular axes with origin at the vehicle’s center of gravity; pitch is rotation about the horizontal, side-to-side axis; roll is rotation about a horizontal front-to-rear axis; and yaw is rotation about the vertical axis; see Fig. B7; see yaw angle. planar impact: An impact in which all forces, moments, and motion takes place in a plane. plastic impact: An impact with little or no rebound at the end of impact; a perfectly plastic impact is where the coefficient of restitution is equal to zero. PM-10: Particulate matter of 10 microns in diameter or smaller. PM-2.5: Particulate matter of 2.5 microns in diameter or smaller. PMT: Passenger-Miles of Travel. point mass: An idealized concept from mechanics where an object is considered to have mass but no extent, no finite dimensions, and as a consequence, its rotation is irrelevant; see rigid body. point of contact: The point of intersection of the contact impulse and the intervehicular contact surface during an impact; see also impact center, first contact position, PDOF, DOPF . postcollision trajectory, postimpact trajectory: The path of a vehicle from the time of separation to its rest position. postcrash damage: Damage existing to a vehicle after it came to rest, including damage that may result during rescue, towing, and salvage operations.

372

Glossary

postimpact speed: The magnitude of the velocity of an object in a collision at the time of separation, or end of contact; see postimpact velocity and separation speed. postimpact velocity: The velocity of an object in a collision at the time of separation, or end of contact; see postimpact speed and separation speed. PNG: Portable Network Graphics, digital photograph file format. preimpact velocity: The velocity of a vehicle in a collision at the instant of its initial contact. principal direction of force (PDOF ): The direction of the line of action of the contact impulse in a planar collision expressed in degrees, measured clockwise from the longitudinal axis of a vehicle; see Fig. B3. PRNDL: Park-Reverse-Neutral-Drive-Low (shift mechanism sequence). PTC: Positive Train Control. PUV: Personal-Use Vehicle. radius of gyration: The square root of the quotient of the moment of inertia and the mass of a rigid body; see moment of inertia. RCM: Restraint Control Module (Ford). reaction time: See perception-decision-reaction time. residual crush: The permanent deformation formed by the nominal external surface of a vehicle caused by an impact, usually measured relative to the corresponding asmanufactured undeformed surface; see crush area, crush profile. rest position: The location of the center of gravity of a vehicle following an accident measured relative to a coordinate system fixed in the earth. reverse projection photogrammetry: The photogrammetric procedure of inserting a transparency that contains outlines of transient and fixed objects into a camera for the purpose of determining the position and orientation of the camera at the time the original photograph was taken to facilitate the re-location of the transient information. RFG: Reformulated gasoline. RFI: Radio Frequency Interference. rigid body: A concept from mechanics where an object is considered to have mass and dimensions (such as length, width, radius, etc.) that remain constant and which provide resistance to rotation; see point mass, moment of inertia. roll out: Part or all of a postimpact trajectory in which little or no sideslip of a vehicle’s wheels occur; see spinout.

373

Appendix B

roll: See pitch, roll, yaw, and yaw angle. rollbar: A structural member placed over the occupant compartment of a vehicle to protect the occupants against the effects of roof crush during vehicle rollover: also used in some busses and construction machinery. rolling resistance: The retarding force of a freely rolling wheel due to interaction with a contact surface, parallel to the heading axis of a wheel of a moving vehicle; also: • a force opposite to the direction of travel resulting from deformation of a rolling tire [B.7, B.8] • several resistances to motion that may be classified as due to friction in the wheel bearings, friction in the tire walls and tread as they flex when rolling along the road surface, deformation of road surface, impact resistance due to irregularities of road surface, and churning of air by wheels [B.9]. rollover: Vehicle motion where its wheels leave the road surface and at least one side or top of the vehicle contacts the ground; see flip and vault. ROPS: Rollover Protection System. ROR: Run-Off-the-Road. SAE coordinate system: See three-axis vehicle coordinate system. SAI (SA): Sudden Acceleration Incident (Sudden Acceleration). scrape: A mark on a surface that is wider than it is deep that can usually be felt with fingers. scratch: A light and usually irregular scar made on a hard surface, such as paving, by a sliding metal part without great pressure [B.2]. Scratches are visible but not normally distinguishable to the touch. Figure B6. Examples of scuff marks (broad, dark, sweeping marks on surface) and gouge marks (lightcolored marks into surface).

374

SCTG: Standard Classification of Transported Goods. scuff marks: Relatively short marks made by a moving tire on a road or other surface in an erratic fashion with no specific, consistent features; for example, acceleration scuff, impact scuff, flat tire mark; see Fig. B6; see yaw marks, skid marks.

Glossary

SDD: Sudden Deceleration Data (Cummins Engines). SDM: Sensing and Diagnostic Module; an electronic device in a vehicle that captures and stores information in the event of a crash in which air bags may or may not deploy; see EDR. second impact: An impact between an occupant and an interior surface of a vehicle caused by and following an impact between the vehicle and another object. secondary impact: A second or subsequent impact between the same two vehicles during the crash. semi: See tractor, semitrailer. semitrailer: A semitrailer is a towed vehicle equipped with one or more axles to the rear of its laden center of gravity and whose front end forms part of a pivot joint attached to a truck tractor or other powered tow vehicle (examples are truck, cargo, recreational, boat, and livestock trailers). separation speed: The speed at the time of loss of contact of two vehicles in a collision, can refer to the speed of the centers of gravity or of the contact point. separation velocity: The vector velocity at the time of loss of contact of two vehicles in a collision can refer to the speed of the centers of gravity or at the contact point. service brake system: The primary brake system used for slowing and stopping a vehicle. SI System of Units: Metric system, (Système International d’Unitès). side rail: The outermost edge on the side of a vehicle’s roof connecting the upper ends of the A, B, C, and D pillars. sideslip angle, tire: See tire slip angle. sideslip angle, vehicle: The angle between the vehicle’s heading and its velocity vector (β in Fig. B2); see sideslip. sideslip coefficient: The slope of the initial linear portion of the lateral force-slip angle curve of a tire. sideslip stiffness: See sideslip coefficient sideslip: Lateral/transverse translation of a vehicle perpendicular to its heading; see Fig. B2; see tire slip angle. sideswipe collision: A collision of a vehicle where sliding (relative tangential motion) over the intervehicular contact surface does not end at or before separation; see common velocity conditions.

375

Appendix B

simulation: The use of mathematics and mechanics, usually done using a computer, to represent, reproduce or model a physical process. SIR: Supplemental Inflatable Restraint. skid: Motion over a surface of a vehicle with its wheels locked from rotation. skid number: A number representing tire-pavement frictional drag determined by measurements made according to standard equipment, conditions, and procedures and usually stated as 100 times a friction coefficient. skidmark: A friction mark on a pavement made by a tire that is sliding without rotation and, if along the heading axis of the tire, displays a tread pattern. sliding friction coefficient: See coefficient of friction. slip angle, tire: See wheel sideslip angle. slip speed: The speed of a single wheel in the direction of its heading at a given value of longitudinal wheel slip. slip stiffness: See wheel slip coefficient SLIP: See wheel slip, sideslip. SO2 (SO2): Sulfur dioxide. SOC: State of Charge (electric vehicles). sound level:

LP = 10 log

Sound Pressure Level, and pref = 2×10–5 N/m2

p2 2 pref , where s2 is the mean square acoustic pressure

LW = 10 log10 Sound Power Level, 1×10 –6 W

W Wref

, where W is the acoustic power and Wref =

speed: The rate of change of vehicle displacement with respect to time; the magnitude of velocity. spinout: A descriptive term for postimpact vehicle motion including significant yaw rotation; see postimpact trajectory. static crush: See residual crush. stiffness coefficient: See crush stiffness coefficient.

376

Glossary

stopping distance: The distance taken by a driver to bring a vehicle to rest in straight forward motion by braking, including the distance traveled during perception-decisionreaction time prior to brake application; see braking distance. STRAHNET: Strategic Highway Network. striations: Periodic stripes that appear transverse to the tire marks from a yawing vehicle. superelevation: A side-to-side slope of a road measured in degrees or percent. supplemental restraint system: An interior vehicle device that inflates when actuated by accelerometers and/or crash sensors and acts between an occupant and an interior vehicle surface to prevent injury due to sudden contact; see air bag. SUV: Sport Utility Vehicle. tag axle: See drop axle. tandem axles: The use of two closely spaced axles, front-to-rear, usually for buses, heavy trucks, and trailers. TAU: Throttle Actuation Unit. TEA-21: Transportation Equity Act for the 21st Century. three-axis vehicle coordinate system: Fig. B7 shows the standard, three-dimensional vehicle coordinate system. Figure B7. Diagram of a three-axis SAE coordinate system.

throw distance: The distance a pedestrian is propelled (in the direction of vehicle motion at impact) between the location of the pedestrian at first contact and pedestrian’s rest position.

377

Appendix B

TIFF (TIF): Tagged Image File Format, digital photograph file format. tire marks: General term for marks on a surface generated by tires; can be scuffs, skids, yaw marks, prints, etc. tire slip angle (also tire sideslip angle): See wheel slip angle. total station: An electro-optical device, usually mounted on a tripod, used to make position measurements such as in-site surveys. tractor, semitrailer: A truck tractor (cab) with two or more axles pulling a semitrailer; see Fig. B8. Figure B8. Illustration of an automobile riding under the rear of a semitrailer.

tractor trailer: A truck tractor (cab) with two or more axles pulling a trailer. tractor: See truck tractor or farm tractor. trailer: A trailer is a towed vehicle equipped with two axles; the front axle is attached to the tow vehicle and pivoted for turning, whereas the rear axle is fixed. trailing edge: The term used to describe that portion of a vehicle component (such as door, window, fender, quarter, etc.) which is closest to the rear of the vehicle. The rearmost part of a vehicle with respect to a vehicle’s motion and attitude. trajectory: The path of the center of gravity of a body as it moves through space; usually associated with coordinates of the center of gravity as a function of time; see Fig. B2. TRB: Transportation Research Board. trip point: That location along a ground surface at which the motion of a vehicle component is suddenly halted followed by a flip, rollover, or vault. truck classifications: Trucks are classified according to their GVWR

378

Class

GVWR range, lb

Class

GVWR range, lb

1 3 5 7

0 - 6,000 10,001 - 14,000 16.001 - 19,500 26,001 - 33,000

2 4 6 8

6,001 - 10,000 14,001 - 16,000 19,501 - 26,000 33,001 and above

Glossary

truck deformation classification (TDC): A classification system used to appropriately describe a collision-damaged truck. It consists of seven alphanumeric characters arranged in specific order to form a descriptive composite of the vehicle damage [B.6]. truck tractor: A motor vehicle designed for pulling semitrailers. Basic types are cabover-engine and conventional. TSA: Transportation Security Administration. TSB: Technical Service Bulletin. TSC: Transportation Systems Center (NHTSA). UA: Unintended Acceleration. underride: A condition in a collision where the main structural components of one vehicle are below the main structural components of the other vehicle; see Fig. B8; see override. understeer: When a vehicle, traveling along a circular path on a flat, level surface with a constant speed and a constant front-wheel steer angle is accelerated, it will remain on a path with the same radius, tend to increase its radius, or tend to decrease its radius; these are defined as neutral steer, understeer, and oversteer, respectively; see neutral steer and oversteer. USGS: U.S. Geological Survey. UST: Underground Storage Tank. V2I: Vehicle To Infrastructure (electronic communication). V2V: Vehicle To Vehicle (electronic communication). VCE: Vehicle Control Electronics. VIIC: Vehicle Information Integration Consortium. V-MAC: Vehicle Management and Control Unit (Mack Trucks). vault: A roll or pitch motion of a vehicle made following loss of ground contact. vehicle coordinate system: See Figs. B2 and B7; see three-axis vehicle coordinate system. vehicle length: The maximum dimension measured longitudinally between the foremost point and the rearmost point in the vehicle, including bumper, bumper guards, tow hooks, and/or rub strips, if standard equipment [B.1]. Also known as overall length (OAL). vehicle width: The maximum dimension measured between the widest point on the vehicle, excluding exterior mirrors, flexible mud flaps, and marker lamps, but including

379

Appendix B

bumpers, moldings, sheet metal protrusions, or dual wheels if standard equipment [B.1]. Also known as overall width (OAW). velocity: The rate of change of displacement with both a magnitude and direction; the magnitude of velocity is referred to as speed. velocity-time curve (v-τ or v-t curve): A graphical depiction of velocity of the center of gravity of a vehicle as it changes over time. vmt: Vehicle-miles of travel. VOC: Volatile Organic Compounds. VOQ: Vehicle Owner Questionnaire (NHTSA). VRTC: Vehicle Research and Test Center (NHTSA). wheel base: The perpendicular distance between axes through front and rear wheel centerlines of a vehicle. In case of dual axles, the distance is to the midpoint of the centerlines of the dual axles. wheel slip angle (also wheel sideslip angle): The angle between a wheel’s heading axis (x axis) and direction of the velocity vector of the center of the wheel. wheel slip coefficient: The slope of the initial linear portion of the longitudinal forcewheel slip curve of a tire. wheel slip: The ratio of the forward velocity of a tire at the road contact patch to the forward velocity at the center of the wheel (for braking) or the ratio of the forward velocity of a tire at the center of the wheel to the forward velocity at the road contact patch (for traction or acceleration). windshield header: The structural body member which connects the upper portions of the left and right A-pillars and is above the top edge of the windshield. WOT: Wide Open Throttle. wrap pedestrian collision: A frontal collision of vehicle and pedestrian or cyclist where initial contact occurs at a point below the center of gravity of the pedestrian or cyclist and where the frontal geometry of the vehicle allows the pedestrian to move rearward relative to the vehicle and strike another portion of the vehicle, such as a windshield. The latter impact causes the pedestrian or cyclist to develop an airborne trajectory followed by an impact with the ground. WWC: Windshield Wiper Control. yaw angle: The angle between the heading of a vehicle and a fixed reference; see Fig. B2.

380

Glossary

yaw mark: A tire mark caused by a sideslipping tire, often showing a striped pattern called striations. yaw moment of inertia: The moment of inertia about a vertical axis of a vehicle; see moment of inertia, radius of gyration. yaw rate: Angular velocity about the z-axis; see Fig. B2. yaw: See pitch, roll, yaw, and yaw angle.

381

References

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Goudie, D.W., J.J. Bowler, C.A. Brown, B.E. Heinrichs, C.A. Brown, and G.P. Siegmund, “Tire Friction During Locked Wheel Braking,” Paper No 2000-011314, SAE International, Warrendale, PA, 2000.

1.3.

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1.4.

Beers, Y., Introduction to the Theory of Error, Addison-Wesley, Reading, MA, 1957.

1.5.

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1.6.

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1.7.

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1.9.

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1.10. Brach, R.M. and P.F. Dunn, Uncertainty Analysis for Forensic Science, 2nd Ed. Lawyers and Judges Publishing Co., Tucson, AZ, 2009. 1.11. Ball, J., D. Danaher, R. Ziernicki, “Considerations for Applying and Interpreting Monte Carlo Simulation Analyses in Accident Reconstruction,” Paper 2007-010741, SAE International, Warrendale, PA, 2007. 1.12. Andreas, M., A. Spek, H. Steffan, W. Makkinga, “Application of the Monte Carlo Methods for Stability Analysis Within the Accident Reconstruction Software PC- Crash 03/03/2003,” Paper 2003-01-0488, SAE International, Warrendale, PA, 2003. 1.13. Wood, D., S. O’Riordain, “Monte Carlo Simulation Methods Applied to Accident Reconstruction and Avoidance Analysis,” Paper 940720, SAE International, Warrendale, PA, 1994. 1.14. Kost, G. and S. Werner, “Use of Monte Carlo Simulation Techniques in Accident Reconstruction,” Paper 940719, SAE International, Warrendale, PA, 1994. 1.15. Daily, J., “Monte Carlo Techniques for Correlated Variables in Crash Reconstruction,” Paper 2009-01-0104, SAE International, Warrendale, PA, 2009. 1.16. Wach, W. and J. Unarski, “Determination of Vehicle Velocities and Collision Location by Means of Monte Carlo Simulation Method,” Paper 2006-01-0907, SAE International, Warrendale, PA, 2006. 1.17. Bartlett, W., “Conducting Monte Carlo Analysis With Spreadsheet Programs,” Paper 2003-01-0487, SAE International, Warrendale, PA, 2003. 1.18. Rose, N. , S. Fenton, C. Hughes, “Integrating Monte Carlo Simulation, MomentumBased Impact Modeling, and Restitution Data to Analyze Crash Severity,” Paper 2001-01-3347, SAE International, Warrendale, PA, 2001. 1.19. Wach, W. and J. Unarski, “Uncertainty Analysis of the Preimpact Phase of a Pedestrian Collision,” Paper 2007-01-0715, SAE International, Warrendale, PA, 2007. 1.20. Kimbrough, S., “Determining the Relative Likelihoods of Competing Scenarios of Events Leading to An Accident,” Paper 2004-01-1222, SAE International, Warrendale, PA, 2004.

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1.21. Bartlett, W. and A. Fonda, “Evaluating Uncertainty in Accident Reconstruction With Finite Differences,” Paper 2003-01-0489, SAE International, Warrendale, PA, 2003. 1.22. Hermitte, T., C. Thomas, Y. Page, T. Perron, “Real-world car accident reconstruction methods for crash avoidance system research,” Paper 2000-050221, SAE International, Warrendale, PA, 2000. 1.23. Brach, R., “Design of experiments and parametric sensitivity of planar impact mechanics,” XVI Europäischen Vereinigung für Unfallforschung und Unfallanalyse (EVU)—Conference Uncertainty in Reconstruction of Road Accidents, Krakow, Poland, 2007. 1.24. Ferson, S., V. Kreinovich, J. Hajagos, W. Oberkampf, and L. Ginzburg, “Experimental Uncertainty Estimation and Statistics for Data Having Interval Uncertainty,” SAND2007-0939, Sandia National Laboratories, Albuquerque, 2007. 1.25. Salicone, S., Measurement Uncertainty: An Approach via the Mathematical Theory of Evidence, Springer, New York, 2007. 1.26. Kolmogorov, A., “Confidence limits for an unknown distribution function,” Annals of Mathematical Statistics. 12: 461-463, 1941. 1.27. Smirnov, N., “On the estimation of the discrepancy between empirical curves of distribution for two independent samples,” Bulletin de l’Université de Moscou, Série internationale (Mathématiques), 2: (fasc. 2), 1939.

Chapter 2 2.1. Vehicle Dynamics Terminology, SAE Recommended Practice, SAE J670e, SAE International, Warrendale, PA, 1976. 2.2.

Milliken, W. F. and D. L. Milliken, Race Car Vehicle Dynamics, SAE International, 1995.

2.3

Gim, G. and P.E. Nikravesh, “An Analytical Model of Pneumatic Tyres for Vehicle Dynamic Simulations. Part 1: Pure Slips,” Int. J. of Vehicle Design, Vol. 11, No. 6., 1990.

2.4

Gim, G. and P.E. Nikravesh, “An Analytical Model of Pneumatic Tyres for Vehicle Dynamic Simulations. Part 2: Comprehensive Slips,” Int. J. of Vehicle Design, Vol. 12, No. 1, 1991.

2.5.

Gim, G. and P.E. Nikravesh, “An Analytical Model of Pneumatic Tyres for Vehicle Dynamic Simulations. Part 3: Validation Against Experimental Data,” Int. J. of Vehicle Design, Vol. 12, No. 2, 1991.

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2.6.

Allen, R.W., T. Rosenthal, and J. Chrstos, “A Vehicle Dynamics Tire Model for Both Pavement and Off-Road Conditions,” Paper 970559, SAE International, Warrendale, PA, 1997.

2.7

Clark, S.K. [ed.], Mechanics of Pneumatic Tires, DOT HS 805 952, U.S. Department of Transportation, National Highway Traffic Safety Administration, 1981.

2.8.

Wong, J.Y., Theory of Ground Vehicles, John Wiley and Sons, Inc, 1993.

2.9.

Nielsen, L. and Sandberg, T., “A New Model For Rolling Resistance of Pneumatic Tires,” Paper 2002-01-1200, SAE International, Warrendale, PA, 2002.

2.10. Adler, Ulrich [ed.], Automotive Handbook, 3rd Edition, Robert Bosch, 1993. 2.11. “Tires and Passenger Vehicle Fuel Economy,” TRB Special Report 286, Transportation Research Board, Washington, D.C., 2006. 2.12. Salaani, K., “Analytical Tire Forces and Moments Model with Validated Data,” Paper 2007-01-0816, SAE International, Warrendale, PA, 2007. 2.13. Warner, C.Y., G.C. Smith, M.B. James, and G.J. Germane, “Friction Applications in Accident Reconstruction,” Paper 830612, SAE International, Warrendale, PA, 1983. 2.14. Statement of Work, Truck Tire Characterization Phase 1 Part 2, SAE Cooperative Research, Funding by NHTSA Contract No. DTNH22-92-C-17189, Warrendale, PA, 1995. 2.15. Nguyen, P. K. and E.R. Case, “Tire Friction Models and Their Effect on Simulated Vehicle Dynamics,” Proceedings of a Symposium on Commercial Vehicle Braking and Handling, UM-HSRI-PF-75-6, 1975. 2.16. Schuring, D.J., W. Pelz, M.G. Pottinger, “A Model for Combined Tire Cornering and Braking Forces,” Paper 960180, SAE International, Warrendale, PA, 1996. 2.17. Brach, R. Matthew and Brach, Raymond M., “Modeling Combined Braking and Steering Forces,” Paper 2000-01-0357, SAE International, Warrendale, PA, 2000. 2.18. Bakker, E., L. Nyborg, and H.B. Pacejka, “Tyre Modeling for Use in Vehicle Dynamic Studies,” Paper 870421, SAE International, Warrendale, PA, 1987. 2.19. Schuring, D.J., W. Pelz, M.G. Pottinger, “The BNPS Model—An Automated Implementation of the ‘Magic Formula’ Concept,” Paper 931909, SAE International, Warrendale, PA, 1993. 2.20. d’Entremont, K.L., “The Behavior of Tire-Force Model Parameters Under Extreme Operating Conditions,” Paper 970558, SAE International, Warrendale, PA, 1997.

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2.19. Nicolas, V.T. and T.R. Comstock, “Predicting Directional Behavior of Tractor Semitrailers When Wheel Anti-Skid Brake Systems Are Used,” Paper No. 72— WA/Aut-16, ASME Winter Annual Meeting, November 26-30, 1972. 2.20. Brach, Raymond and Matthew Brach, “Tire Models for Vehicle Dynamic Simulation and Accident Reconstruction,” Paper 2009-01-0102, SAE International, Warrendale, PA, 2009. 2.21. Brach, Raymond and Matthew Brach, “Tire Models used in Accident Reconstruction Vehicle Motion Simulation,” XVII Europäischen Vereinigung für Unfallforschung und Unfallanalyse (EVU)—Conference, Nice, France, 2008. 2.22. Rill, G, Vehicle Dynamics Lecture Notes, University of Applied Sciences, Hochschule für Technik Wirtschaft Soziales, Germany, 2007. 2.23. Gäfvert, M and J. Svedenius, Construction of Novel Semi-Empirical Tire Models for Combined Braking and Cornering, ISSN 0280-5316, Lund Inst of Tech., 2003. 2.24. Bartlett, W. and W. Wright, “The Effect of ABS on Braking Deceleration on Pavement and Gravel,” Paper 2010-01-0066, SAE International, Warrendale, PA, 2010. 2.25. Fricke, L.B., “Traffic Accident Reconstruction,” Vol 2, Traffic Accident Investigation Manual, Northwestern University Traffic Institute, 1990. 2.26. Garrott, W., D. Guenther, R. Houk, J. Lin, and M. Martin, “Improvement of Methods for Determining Pre-Crash Parameters from Skid Marks,” DOT, NHTSA, NTIS Report PB82-183716, May1981. 2.27. American Jurisprudence, Proof of Facts, Vol 10 Bancroft-Whitney, San Francisco, 1961. 2.28. Machine Design magazine, “Sand-Versus-Skid Research,” Jan 25, 1979. 2.29. Ebert, N.E., “SAE Tire Braking Traction Survey: A Comparison of Public Highways and Test Surfaces,” Paper 890638, SAE International, Warrendale, PA, 1989. 2.30. Varat, M.S., J.F. Kerkhoff, S.E. Husher, C.D. Armstrong, and K.F. Shuman, “The Analysis and Determination of Tire-Roadway Frictional Drag,” Paper 2003-010887, SAE International, Warrendale, PA, 2003. 2.31. Goudie, D.W., J. J. Bowler, C.A. Brown, B.E. Heinrichs, and G.P. Sigmund, “Tire Friction During Locked Wheel Braking,” Paper 2000-01-1314, SAE International, Warrendale, PA, 2000.

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2.32. “A Policy On Geometric Design of Highways and Streets,” American Association of State Highway and Transportation Officials (AASHTO), 5th Ed., Washington, DC, 2004. 2.33. Moyer, R., “Skidding Characteristics of Automobile Tires on Roadway Surfaces and Their Relation to Highway Safety,” Bulletin No 120, Iowa Engineering Experiment Station, Ames, IA, 1934. 2.34. Stonex, K. and C. Noble, “Curve Design and Tests on the Pennsylvania Turnpike,” Proceedings, Vol 20, Highway Research Board, 1940. 2.35. Moyer, R. and D. Berry, “Marking Highway Curves with Safe Speed Indications,” Proceedings, Vol 20, Highway Research Board, 1940. 2.36. Adams, M. and R. Knight, “Drag Factors—Applications and Sensitivity,” Accident Reconstruction Journal, Jan/Feb 1990. 2.37. Dunn, A.L. and R.L. Hoover, Class 8 Truck Tractor Braking Performance Improvement Study, Report 1, “Straight Line Stopping Performance on a High Coefficient of Friction Surface,” DOT HS 809 700, National Highway Traffic Safety Administration, Washington, DC, May 2004. 2.38. Federal Motor Carrier Safety Regulations, USDOT, Part 393, Section 52, Brake Performance, August, 2001. 2.39. Garrott, W., D. Guenther, R. Houk, J. Lin, and M. Martin, “Improvement of Methods for Determining Pre-Crash Parameters from Skid Marks,” Accident Reconstruction Journal, Sep/Oct 1990. 2.40. Thompson, T. C., “Heavy Truck Skid Tests,” Accident Reconstruction Journal, Jan/Feb, 1991. 2.41. Fancher, P.S., R.D. Ervin, C.B. Winkler, and T.D. Gillespie, A Factbook of the Mechanical Properties of the Components for Single-Unit and Articulated Heavy Trucks, UMTRI -86-12. 2.42. Wang, J., L. Alexander and R. Rajamani, “GPS Based Real-Time Tire-Road Friction Coefficient Identification,” MN/RC 2005-04, Minnesota Department of Transportation, September 2004. 2.43. Bartlett, W., “Calculation of Heavy Truck Deceleration Based on Air Pressure Rise-Time and Brake Adjustment,” Paper 2004-01-2632, SAE International, Warrendale, PA, 2004. 2.44. Dunn, A., J. Wiechel, D. Guenther, C. Tanner, E. Sauer, B. Boggess, F. Bayan, A. Cornetto, A. Pearlman, D. Morr, and S. Noll, “The Influence of Complete Disablement of Various Brakes on the Dry Stopping Performance of a TractorSemitrailer,” Paper 2009-01-0099, SAE International, Warrendale, PA, 2009.

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2.45. “Heavy Duty Truck Wet Braking Tests,” Accident Reconstruction Journal, Nov/ Dec, 2009. 2.46. Blythe, W. and T.D. Day, “Single Vehicle Wet Road Loss of Control; Effects of Tire Tread Depth and Placement,” Paper 2002-01-0553, SAE International, Warrendale, PA, 2002. 2.47. Browne, A.L. “Mathematical Analysis for Pneumatic Tire Hydroplaning,” ASTM STP 583, American Society for Testing Materials, pp. 75-94, 1975.

Chapter 3 3.1. Gillespie, T., Fundamentals of Vehicle Dynamics, SAE International, Warrendale, PA, 1992. 3.2.

Wong, J.Y., Theory of Ground Vehicles, John Wiley, New York, 2001.

3.3.

Tires and Passenger Vehicle Fuel Economy, TRB Special Report 286, Transportation Research Board, Washington, D.C., 2006

3.4.

Farber, E., P. Olson, and R. Dewar , Forensic Aspects of Driver Perception and Response, Third Edition, Lawyers and Judges Publishing, Tucson, AZ, 2010.

3.5.

Dewar, R. and P. Olson, Human Factors in Traffic Safety, Second Edition, Lawyers and Judges Publishing, Tucson, AZ, 2007.

3.6.

Green, M., with M.J. Allen, B.S. Abrams, and L. Weintraub, Forensic Vision with Application to Highway Safety, 3rd Edition, 2010.

3.7.

Muttart, J.W., “Development and Evaluation of Driver Response Time Predictors Based on Meta Analysis,” Paper 2003-01-0885, SAE International, Warrendale, PA.

3.8.

Tustin, B.H., H. Richards, H. McGee, and R. Peterson, Railroad-Highway Grade Crossing Handbook, 2nd Ed., Federal Highway Administration, FHWA TS-86215, McLean, VA, 1986.

3.9.

Reed, W.S. and A.T. Keskin, “A Comparison of Emergency Braking Characteristics of Passenger Cars,” Paper 880231, SAE International, Warrendale, PA.

3.10. Baker, J. Standard and Lynn B. Fricke, The Traffic Accident Investigation Manual, Northwestern University Traffic Institute, Ninth Edition, Evanston, IL, 1986. 3.11. Daily, John, N. Shigemura, Jeremy Daily, Fundamentals of Traffic Crash Reconstruction, Volume 2, 2009.

389

References

Chapter 4 4.1. Brach, R.M., “An Analytical Assessment of the Critical Speed Formula,” Paper 970957, SAE International, Warrendale, PA, 1997. 4.2.

Fittano, D.A. and J. Puig-Suari, “Using a Genetic Algorithm to Optimize Vehicle Simulation Trajectories: Determining Initial Velocity of a Vehicle in Yaw,” Paper 2000-01-1616, SAE International, Warrendale, PA, 2000.

4.3.

Sledge, N.H. and K.M. Marshek, “Formulas for Estimating Vehicle Critical Speed from Yaw Marks—A Review,” Paper 971147, SAE International, Warrendale, PA, 1997.

4.4.

Brach, Raymond and M. Brach, “Tire Models for Vehicle Dynamic Simulation and Accident Reconstruction,” Paper 2009-01-0102, SAE International, Warrendale, PA, 2009.

4.5.

Brach, Raymond and M. Brach, “Tire Models used in Accident Reconstruction Vehicle Motion Simulation,” (with R. Matthew Brach), XVII Europäischen Vereinigung für Unfallforschung und Unfallanalyse (EVU)—Conference, Nice, France, 2008.

4.6. Semon, M., “Determination of Speed from Yaw Marks,” Chapter 4, Forensic Accident Investigation: Motor Vehicles, Bohan and Damask, Ed., Michie Butterworth., 1995. 4.7.

Dickerson, C.P., M.W. Arndt, S.M. Arndt, and G.A. Mowry, “Evaluation of Vehicle Velocity Predictions Using the Critical Speed Formula,” Paper 950137, SAE International, Warrendale, PA, 1995.

4.8.

Lambourn, R.F., “The calculation of motor car speeds from curved tire marks,” Journal of the Forensic Science Society (UK), 29, 371-386, 1989.

4.9.

Beauchamp, G., D. Hessel, N.A. Rose, and S.J. Fenton, “Determining Vehicle Steering and Braking from Yaw Mark Striations,” Paper 2009-01-0092, SAE International, Warrendale, PA, 2009.

4.10. Fricke, L.B., Traffic Accident Reconstruction, Traffic Accident Investigation Manual, Vol 2, Northwestern University Traffic Institute, Evanston, IL, 1990. 4.11. Garber, N.J. and L.A. Hoel, Traffic and Highway Engineering, PWS Publishing Co., Boston, 1997. 4.12. Baxter, A.T. and J.R. Mentzer, “Critical Speed Field Testing of a Passenger Vehicle,” Annual Meeting of the National Association of Traffic Accident Reconstructionists and Investigators, Atlantic City, NJ, 1991. 4.13. Milliken, W.F and D.L. Milliken, Race Car Vehicle Dynamics, SAE International, Warrendale, PA, 1995.

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References

4.14. Wong, J.Y., Theory of Ground Vehicles, John Wiley & Sons, Inc., New York, 1993. 4.15. Sledge, N.H. and K.M. Marshek, “Vehicle Critical Speed Formula—Values for the Coefficient of Friction—A Review,” Paper 971148, SAE International, Warrendale, PA, 1997. 4.16. Shelton, Sgt. Thomas, “Validation of the Estimation of Speed from Critical Speed Scuffmarks,” Accident Reconstruction Journal, January/February 1995. 4.17. Amirault, G and S.A. Macinnis, “Variability Analysis of Yaw Calculations from Field Testing,” Paper 2009-01-0103, SAE International, Warrendale, PA, 2009. 4.18. Asay, A.F. and R.L. Woolley, “Rollover Testing on an Actual Highway,” Paper 2009-01-1544, SAE International, Warrendale, PA, 2009. 4.19. Heusser, R., J. Hunter, and D. Martin, “Critical Speed Evaluation with Acceleration/Deceleration,” Newsletter of WATAI (Washington Association of Traffic Accident Investigators), 1994. 4.20. Martinez, L, “Estimating Speed from Yawmarks—An Empirical Study,” Accident Reconstruction Journal, May/June1993.

Chapter 5 5.1. Winkler, C.B., D. Blower, R.D. Ervin, and R.M. Chalasani, “Rollover of Heavy Commercial Vehicles,” SAE Research Report, SAE International, Warrendale, PA, 2000. 5.2.

Orlowski, K.F., R.T. Bundorf, and E.A. Moffatt, “Rollover Crash Tests— The Influence of Roof Strength on Injury Mechanics,” Paper 851734, SAE International, Warrendale, PA, 1985.

5.3.

Bahling, G.S., R.T. Bundorf, G.S. Kaspzyk, E.A. Moffatt, K.F. Orlowski, and J.E. Stocke, “Rollover and Drop Tests--The Influence of Roof Strength on Injury Mechanics Using Belted Dummies,” Paper 902314, SAE International, Warrendale, PA, 1990.

5.4.

Moffatt, E. A. and J. Padmanaban, “The Relationship Between Vehicle Roof Strength and Occupant Injury in Rollover Crash Data,” Association for the Advancement of Automotive Medicine, Chicago, October 1995.

5.5.

Rains, G.C. and J.N. Kanianthra, “Determination of the Significance of Roof Crush on Head and Neck Injury to Passenger Vehicle Occupants in Rollover Crashes,” Paper 950655, SAE International, Warrendale, PA, 1995.

5.6.

Anonymous, Code of Federal Regulations, Title 49, Chapter 5, Part 571.208, Occupant Crash Protection, 10-01-2002 Edition.

391

References

5.7.

Larson, R.E., J.W. Smith, S.M. Werner, and G.F. Fowler, “Vehicle Rollover Testing in Recreating Rollover Collisions,” Paper 2000-01-1641, SAE International, Warrendale, PA, 2000.

5.8.

Boyd, Patrick L., “NHTSA’s NCAP Rollover Resistance Rating System,” Paper Number 05-0450, National Highway Traffic Safety Administration, 2005.

5.9.

SAE J2114, “Dolly Rollover Recommended Test Procedure,” SAE International, Warrendale, PA, 1999.

5.10. Cooperrider, N.K., T.M.Thomas, and S.A. Hammoud, “Testing and Analysis of Vehicle Rollover Behavior,” Paper 900366, SAE International, Warrendale, PA, 1990. 5.11. Cooperrider, N.K., S.A. Hammoud, and J. Colwell, “Characteristics of SoilTripped Rollovers,” Paper 980022, SAE International, Warrendale, PA, 1998. 5.12. D’Entremont, K.L., “The Effects of Light-Vehicle Design Parameters in TrippedRollover Maneuvers—A Statistical Approach Using and Experimentally Validated Computer Model,” Paper 950315, SAE International, Warrendale, PA, 1995. 5.13. Asay, A.F. and R.L. Woolley, “Rollover Testing on an Actual Highway,” Paper 2009-01-1544, SAE International, Warrendale, PA, 2009. 5.14

Dickerson, C. P., S.M. Arndt, G.A. Mowry, and M.W. Arndt, “Effects of Outrigger Design on Vehicle Dynamics,” Paper 940226, SAE International, Warrendale, PA, 1994.

5.15. Carter, J.W., J.L. Habberstad, and J. Croteau, “A Comparison of the Controlled Rollover Impact System (CRIS) with the J2114 Rollover Dolly,” Paper 2002-010694, SAE International, Warrendale, PA, 2002. 5.16. Orlowski, K.R., E.A. Moffatt, R.T. Bundorf, and M.P. Holcomb, “Reconstruction of Rollover Collisions,” Paper 890857, SAE International, Warrendale, PA, 1989. 5.17. Martinez, J.E. and R.J. Schlueter, “A Primer on the Reconstruction and Presentation of Rollover Accidents,” Paper 960647, SAE International, Warrendale, PA, 1996. 5.18. Baker, J.S. and L.B. Fricke, The Traffic Accident Investigation Manual, Northwestern University Traffic Institute, Evanston, IL, 1986. 5.19. Beauchamp, G., D. Hessel, N. Rose, and S. Fenton, “Determining Vehicle Steering and Braking from Yaw Mark Striations,” Paper 2009-01-0092, SAE International, Warrendale, PA, 2009.

392

References

5.20. Fay, R.J. and J.D. Scott, “New Dimensions in Rollover Analysis,” Paper 1999-010448, SAE International, Warrendale, PA, 1999. 5.21. Jones, I.S. and L.A. Wilson, “Techniques for the Reconstruction of Rollover Accidents Involving Sport Utility Vehicles, Light Trucks and Minivans,” Paper 2000-01-0851, SAE International, Warrendale, PA, 2000. 5.22. Cliff, W.E., J.M. Lawrence, B.E. Heinrichs, and T.R. Fricker, “Yaw Testing of an Instrumented Vehicle with and without Braking,” Paper 2004-01-1187, SAE International, Warrendale, PA, 2004. 5.23. Clarke, S.K. [ed.], Mechanics of Pneumatic Tires, DOT HS 805 952, August 1981. 5.24. Pacejka, H.B., Tire and Vehicle Dynamics, Butterworth-Heinemann, Woburn, MA, 2002. 5.25. Bernard, J., J. Shannon, and M. Vanderploeg, “Vehicle Rollover on Smooth Surfaces,” Paper 891991, SAE International, Warrendale, PA, 1989. 5.26. Gillespie, T., Fundamentals of Vehicle Dynamics, SAE International, Warrendale, PA, 1992. 5.27. Hac, A., “Rollover Stability Index Including Effects of Suspension Design,” Paper 2002-01-0965, SAE International, Warrendale, PA, 2002. 5.28. Erdogan, L., D. Guenther, and G. Heydinger, “Suspension Parameter Measurement Using Side-Pull Test to Enhance Modeling of Vehicle Roll,” Paper 1999-01-1323, SAE International, Warrendale, PA, 1999. 5.29. Heydinger, G., N.J. Durisek, D.A. Coovert, D.A. Guenther, and S.J. Novak, “The Design of a Vehicle Inertia Measurement Facility,” Paper 950309, SAE International, Warrendale, PA, 1995. 5.30. Lund, Y.I. and J.E. Bernard, “Analysis of Simple Rollover Metrics,” Paper 950306, SAE International, Warrendale, PA, 1995. 5.31. Marine, M.C., J.L. Wirth, and T.M. Thomas, “Characteristics of On-Road Rollovers,” Paper 1999-01-0122, SAE International, Warrendale, PA, 1999. 5.32. Chou, C.C., and F. Wu, “Analysis of Vehicle Kinematics in Laboratory-based Rollover Test Modes,” Paper 2006-01-0724, SAE International, Warrendale, PA, 2006. 5.33 Rose, N.A., G. Beauchamp, and S.J. Fenton, “Factors Influencing Roof-to-ground Impact Severity: Video Analysis and Analytical Modeling,” Paper 2007-01-0726, SAE International, Warrendale, PA, 2007.

393

References

5.34. Rosenthal, T.J., H.T. Szostak, and R.W. Allen, “User’s Guide and Program Description For Tripped Roll Over Vehicle Simulation,” DOT HS 807 140 Final Report, 1987. 5.35. Day, T.D. and J.T. Garvey, “Applications and Limitations of 3-Dimensional Vehicle Rollover Simulation,” Paper 2000-01-0852, SAE International, Warrendale, PA, 2000. 5.36. Tamny, S., “Friction Induced Rollover from Lift-Off to Launch,” Paper 2000-011649, SAE International, Warrendale, PA, 2000. 5.37 Anonymous, ADAMS User Documentation, MSC Software Corporation, Santa Ana, CA, 92707, www.mscsoftware.com. 5.38. Chace, M.A. and T.J. Wielenga, “A Test and Simulation Process to Improve Rollover Resistance,” Paper 1999-01-0125, SAE International, Warrendale, PA, 1999. 5.39. Rose, N.A. and G. Beauchamp, “Analysis of a Dolly Rollover with PC-Crash,” Paper 2009-01-0822, SAE International, Warrendale, PA, 2009. 5.40. Rose, N.A., S.J. Fenton, G. Beauchamp, and R.W. McCoy, “Analysis of Vehicle-toground Impacts During a Rollover with an Impulse-Momentum Impact Model,” Paper 2008-01-0178, SAE International, Warrendale, PA, 2008. 5.41. Meyer, S.E., M. Davis, S. Forrest, D. Chng, and B. Herbst, “Accident Reconstruction of Rollovers—A Methodology,” Paper 2000-01-0853, SAE International, Warrendale, PA, 2000. 5.42. Kaplan, M., D. Bilek, S. Kaplan, D. Vellos, and M.G. Gilbert, “An Examination of Rim Gouging and Its Relation to On-Road Vehicle Rollover,” SOARce, Society Of Accident Reconstructionists, Spring, 2004. 5.43. Carter, J.W., P. Luepke, K.C. Henry, G.J. Germane, and J.W. Smith, “Rollover Dynamics: An Exploration of the Fundamentals,” Paper 2008-01-0172, SAE International, Warrendale, PA, 2008. 5.44. Rose, N. and Beauchamp, G., “Development of a Variable Deceleration Rate Approach to Rollover Crash Reconstruction,” Paper 2009-01-0093, SAE International, Warrendale, PA, 2009. 5.45. Bready, J.E., May, A.A., and Allsop, D., “Physical Evidence Analysis and Roll Velocity Effects in Rollover Accident Reconstruction,” Paper 2001-01-1284, SAE International, Warrendale, PA, 2001. 5.46. Perl, T.R., Bready, J.E., Nordhagen, R.P., and Warner, M.H., “Glass Debris in Rollover Accidents,” Paper 2008-01-0167, SAE International, Warrendale, PA, 2008.

394

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5.47. Stevens, D.C., “Passenger Vehicle Rollover Reconstruction: Investigation, Analysis and Presentation of Results,” Passenger Vehicle Rollover TOPTEC: Causes, Prevention and Injury Prevalence, April 22-23, 2002, Scottsdale, AZ, SAE International, Warrendale, PA, 2002. 5.48. Chen, H.F. and D. Guenther, “Modeling of Rollover Sequences,” Paper 931976, SAE International, Warrendale, PA, 1993. 5.49. Luepke, P., J.W. Carter, K.C. Henry, G.J. Germane, and J.W. Smith, “Rollover Crash Tests on Dirt: An Examination of Roll Dynamics,” Paper 2008-01-0156, SAE International, Warrendale, PA, 2008.

Chapter 6 6.1. Brach, R.M., Mechanical Impact Dynamics, John Wiley & Sons, New York, 1991 (also, Revised Edition, Brach Engineering, 2005, Granger, IN). 6.2.

Varat, M.S. and S.E. Husher, “Vehicle Impact Response Analysis through the Use of Accelerometer Data,” Paper 2000-01-0850, SAE International, Warrendale, PA, 2000.

6.3.

Prasad, A.K., “Coefficient of Restitution of Vehicle Structures and its Use in Estimating the Total ΔV in Automobile Collisions,” AMD Vol 126/BED Vol 19, Crashworthiness and Occupant Protection in Transportation Systems, ASME, 1991.

6.4.

McHenry, B.G. and R.R. McHenry, “SMAC-97—Refinement of the Collision Algorithm,” Paper 970947, SAE International, Warrendale, PA, 1997.

6.5.

McHenry, B.G. and R.R. McHenry, “CRASH-97—Refinement of the Trajectory Solution Procedure,” Paper 970949, SAE International, Warrendale, PA, 1997.

6.6.

McHenry, B.G. and R.R. McHenry, “Effects of Restitution in the Application of Crush Coefficients,” Paper 970960, SAE International, Warrendale, PA, 1997.

6.7.

Brach, R.M., “Restitution in Point Collisions,” Chapter in Computational Aspects of Impact and Penetration, edited by R. F. Kulak & L. Schwer, Elmepress International, Lausanne Switzerland, 1991.

6.8.

Stronge, W.J., Impact Mechanics, Cambridge University Press, Cambridge, U.K., 2000.

6.9.

Brogliato, B., Nonsmooth Impact Mechanics: Models, Dynamics and Control, Springer, London, NY, 1996.

6.10. Monson, K.L. and G.J. Germane, “Determination and Mechanisms of Motor Vehicle Structural Restitution from Crash Test Data,” Paper 1999-01-0097, SAE International, Warrendale, PA, 1999.

395

References

6.11. Ishikawa, H., “Computer Simulation of Automobile Collision—Reconstruction of Accidents,” Paper 851729, SAE International, Warrendale, PA, 1985. 6.12. Ishikawa, H., “Impact Model for Accident Reconstruction—Normal and Tangential Restitution Coefficients,” Paper 930654, SAE International, Warrendale, PA, 1993. 6.13. Ishikawa, H., “Impact Center and Restitution Coefficients for Accident Reconstruction,” Paper 940564, SAE International, Warrendale, PA, 1994. 6.14. Brach, Raymond and Matthew Brach, “Analysis of Collisions, Conservation of Linear Momentum: Can we do Better?”, Collision Magazine, Vol 2, No 1, June , 2007. 6.15. Brach, R.M., “Formulation of Rigid Body Impact Problems using Generalized Coefficients”, Int J of Engrg Sci, Vol 36, 1, p 61-72, 1998. 6.16. Lord Kelvin and P. Tait, Treatise on Natural Philosophy, Cambridge University Press, Cambridge, UK, 1903. 6.17. Brach, R., M. Brach, and K. Welsh, “Residual Crush Energy Partitioning, Normal and Tangential Energy Losses,” SAE 2007 International Congress, Paper 200701-0737, SAE International, 2007. 6.18. Anonymous, “CRASH3 User’s Guide and Technical Manual,” DOT Report DOT HS 805 732, Feb. 1981. 6.19. Jones, I.S. and A.S. Baum, “Research Input for Computer Simulation of Automobile Collisions (RICSAC),” Vol IV: “Staged Collision Reconstructions,” DOT HS-805 040, Dec. 1978. 6.20. Brach, R.M., “Impact Analysis of Two-Vehicle Collisions,” Paper 830468, SAE International, Warrendale, PA, 1987. 6.21. Brach, R.M., “Energy Loss in Vehicle Collisions,” Paper 871993, SAE International, Warrendale, PA, 1987. 6.22. Brach, R.M. and R.A. Smith, “Re-Analysis of the RICSAC Car Crash Accelerometer Data,” Paper 2002-01-1305, SAE International, Warrendale, PA, 2002. 6.23. McHenry, B.G. and R.R. McHenry, “RICSAC-97—A Revaluation of the Reference Set of Full Scale Crash Tests,” Paper 970961, SAE International, Warrendale, PA, 1997. 6.24. Woolley, R.L., “The ‘IMPAC’ Program for Collision Analysis,” Paper 870046, SAE International, Warrendale, PA, 1987. 6.25. Campbell, K.L., “Energy Basis for Collision Severity,” Paper 740565, SAE International, Warrendale, PA, 1974.

396

References

6.26. Germane, G. and R. Gee, “Introduction of Pulse Shapes and Durations into Impulse-Momentum Collision Models,” Paper 2005-01-1183, SAE International, Warrendale, PA, 2005.

Chapter 7 7.1. Jones, I.S. and A.S. Baum, “Research Input for Computer Simulation of Automobile Collisions (RICSAC),” Volume IV: “Staged Collision Reconstructions,” DOT HS805 040, 1978. 7.2.

Brach, R.M. and R.A. Smith, “Re-Analysis of the RICSAC Car Crash Accelerometer Data,” Paper 2002-01-1305, SAE International, Warrendale, PA, 2002.

7.3

Brach, R.M., “Energy Loss in Vehicle Collisions,” Paper 871993, SAE International, Warrendale, PA, 1987.

7.4.

Brach, Raymond M. and Brach, R. Matthew, “Crush Energy and Planar Impact Mechanics for Accident Reconstruction,” Paper 980025, SAE International, Warrendale, PA, 1998.

7.5.

Brach, R.M., Mechanical Impact Dynamics, John Wiley & Sons, New York, 1991 (also, Revised Edition, Brach Engineering, 2005, Granger, IN).

7.6.

Brach, R.M., “Modeling of Low-Speed, Front-to-Rear Vehicle Impacts,” Paper 2003-01-0491, SAE International Warrendale, PA, 2003.

7.7.

Heinrichs, B.E. and J-F Goulet, “Predicting Low-Speed Collisions Descriptors using Dissimilar Collision Data,” Paper 2008-01-0169, SAE International, Warrebdale, PA, 2008.

7.8.

Happer, A., M. Hughes, M. Peck, and S. Boehme, “Practical Analysis Methodology for Low-Speed Vehicle Collisions involving Vehicles with Modern Bumper Systems,” Paper 2003-01-0492, SAE International, Warrendale, PA, 2003.

7.9.

Rosenbluth, W., Investigation and Interpretation of Black Box Data in Automobiles, SAE International, Warrendale, PA, 2001.

7.10. Chan, Ching-Yao, Fundamentals of Crash Sensing in Automotive Air Bag Systems, SAE International, Warrendale, PA, 2000. 7.11. Gabler, H.C., J.A. Hinch, J. Steiner, Event Data Recorders, PT-139, SAE International, 2008. 7.12. Wilkinson, C.C., Lawrence, J.M., Heinrichs, B.E. and King, D.J., “The Accuracy and Sensitivity of 2003 and 2004 General Motors Event Data Recorders and LowSpeed Barrier and Vehicle Collisions,” Paper 2005-01-1190, SAE International, Warrendale, PA, 2005.

397

References

7.13. Lawrence, J.M. and Wilkinson, C.C., “The Accuracy of Crash Data from Ford Restraint Control Modules Interpreted with Revised Vetronix Software,” Paper 2005-01-1206, SAE International, Warrendale, PA, 2005. 7.14. Gabler, H.C., C.E. Hampton, and J.A. Hinch, “Crash Severity: A Comparison of Event Data Recorder Measurements with Accident Reconstruction Estimates,” Paper 2004-01-1194, SAE International, Warrendale, PA, 2004. 7.15. Lawrence, J.M., C.C. Wilkinson, B.E. Heinrichs, and G.P. Siegmund, “The Accuracy of Pre-Crash Speed Captured by Event Data Recorders,” Paper Number 2003-01-0889, SAE International, Warrendale, PA, 2003.

Chapter 8 8.1. Brach, Raymond M., “Impact of Articulated Vehicles,” Paper 860015, SAE International, Warrendale, PA, 1986. 8.2.

Brach, Raymond M., “An Impact Moment Coefficient for Vehicle Collision Analysis,” Paper 770014, SAE International, Warrendale, PA, 1977.

8.3.

Brach, R.M., “Formulation of Rigid Body Impact Problems using Generalized Coefficients,” Int J of Engrg Sci, Vol 36, 1, p 61-72, 1998.

8.4.

Leonard, M., et al., “HVE EDSMAC4 Trailer Model Simulation Comparison with Crash Test Data,” Paper 2000-01-0468, SAE International, Warrendale, PA, 2000.

8.5.

Day, T.D. “An Overview of the EDSMAC4 Collision Simulation Model,” Paper 1999-01-0102, SAE International, Warrendale, PA, 1999.

8.6.

McHenry, B.G. and R.R. McHenry, “SMAC-87,” Paper 880227, SAE International, Warrendale, PA, 1987.

8.7.

Steffan, H. and A. Moser, “The Trailer Simulation Model of PC-Crash,” Paper 980372, SAE International, Warrendale, PA, 1998.

8.8.

Brach, R.M., Mechanical Impact Dynamics, John Wiley & Sons, New York, 1991 (also, Revised Edition, Brach Engineering, 2005, Granger, IN).

8.9.

Test performed in Wildhaus (Switzerland) by DEKRA (Germany) and Winterthur Insurance (Switzerland), 1995.

8.10. Machinery’s Handbook, 22nd Edition, Industrial Press, New York, NY, 1984.

Chapter 9 9.1. Campbell, K., “Energy Basis for Collision Severity,” Paper 740565, SAE International, Warrendale, PA, 1974.

398

References

9.2.

Cheng, P.H., M.J. Sens, J.F. Weichel, and D.A. Guenther, “An Overview of the Evolution of Computer Assisted Motor Vehicle Accident Reconstruction,” Paper 871991, SAE International, Warrendale, PA, 1987.

9.3.

Prasad, A.K., “CRASH3 Damage Algorithm Reformulation for Front and Rear Collisions,” Paper 900098, SAE International, Warrendale, PA, 1990.

9.4.

Warner, Mark H. and Ronald P. Nordhagen, “Development of Pole Impact Testing at Multiple Vehicle Side Locations as Applied to the Ford Taurus Structural Platform,” Paper 2006-01-0062, SAE International, Warrendale, PA, 2006.

9.5

Anonymous, “CRASH3 User’s Guide and Technical Manual,” NHTSA, DOT Report HS 805 732, 1981.

9.6.

Tumbas, N.S. and R.A. Smith, “Measurement Protocol for Quantifying Vehicle Damage From an Energy Basis Point of View,” Paper 880072, SAE International, Warrendale, PA, 1988.

9.7.

Willke, D.T. and M.W. Monk, “CRASH III Model Improvements: Derivation of New Side Stiffness Parameters from Crash Tests,” Accident Reconstruction Journal, Spring 1995.

9.8

Marine, Micky, Jeffrey Wirth, and Terry Thomas, “Crush Energy Considerations in Underride/Override Impacts,” Paper 2002-01-0556, SAE International, Warrendale, PA, 2002.

9.9

Marine, Micky, Jeffrey Wirth, Brian Peters, and Terry Thomas, “Override/ Underride Crush Energy Results from Vertically Offset Barrier Impacts,” Paper 2005-01-1202, SAE International, Warrendale, PA, 2005.

9.10

Struble, Donald, Kevin Welsh, and John Struble, “Crush Energy Assessment in Frontal Underride/Override Crashes,” Paper 2009-01-0105, SAE International, Warrendale, PA, 2009.

9.11. Woolley, R.L., “Nonlinear Damage Analysis in Accident Reconstruction,” Paper 2001-01-0504, SAE International, Warrendale, PA, 2001. 9.12. Asay, A.F., D.B. Jewkes, and R.L. Woolley, “Narrow Object Impact Analysis and Comparison with Flat Barrier Impacts,” Paper 2002-01-0552, SAE International, Warrendale, PA, 2002. 9.13. Welsh, K.J. and D.E. Struble, “Crush Energy & Structural Characterization,” Paper 1999-01-0099, SAE International, Warrendale, PA, 1999. 9.14. Varat, M.S. and S.E. Husher, “Vehicle Crash Severity Assessment in Lateral Pole Impacts,” Paper 1999-01-0100, SAE International, Warrendale, PA, 1999.

399

References

9.15. Prasad, A.K., “Energy Absorbed by Vehicle Structures in Side Impacts,” Paper 910599, SAE International, Warrendale, PA, 1991. 9.16. Prasad, A.K., “Energy absorbing properties of vehicle structures and their use in estimating impact severity in automobile collisions,” Paper 925209, SAE International, Warrendale, PA, 1992. 9.17. Accident Reconstruction Journal, Sample Problems, Waldorf, MD, November/ December, 1998. 9.18 Brach, Raymond M., Kevin J. Welsh, and R. Matthew Brach, “Residual Crush Energy Partitioning, Normal and Tangential Energy Losses,” Paper 2007-010737, SAE International, Warrendale, PA, 2007. 9.19 Prasad, A.K., “Missing Vehicle Algorithm (OLDMISS) Reformulation,” Paper 910121, SAE International, Warrendale, PA, 1992. 9.20 Chen, H. Fred, Brian Tanner, Phillip Cheng, and Dennis Guenther, “Application of Force Balance Method in Accident Reconstruction,” Paper 2005-01-1188, SAE International, Warrendale, PA, 2005. 9.21 Neptune, J.A. and J.E. Flynn, “A Method for Determining Accident Specific Crush Stiffness Coefficients,” Paper 940913, SAE International, Warrendale, PA, 1994. 9.22

Wood, D.P., A. Ydenius, and D. Adamson, “Velocity changes, mean accelerations and displacements of some car types in frontal collisions”, I J Crash, Vol 8, No 6., pp 591-603, 2003.

9.23 Singh, Jai, “The Effect of Residual Damage Interpolation Mesh Fineness on Calculated Side Impact Stiffness Coefficients,” Paper 2005-01-1205, SAE International, Warrendale, PA, 2005.

Chapter 10 10.1. Eubanks, J.J., Pedestrian Accident Reconstruction, Lawyers & Judges Publishing Company, Tucson, AZ, 1994. 10.2. van Wijk, J., J. Wismans, J. Maltha, and L. Wittebrood, “MADYMO pedestrian simulations,” Paper 830060, SAE International, Warrendale, PA, 1983. 10.3. Moser, A., H. Steffan, H. Hoschopf, G. and Kasanicky, “Validation of the PCCrash Pedestrian Model,” Paper 2000-01-0847, SAE International, Warrendale, PA, 2000. 10.4. Happer, A., M. Araszewski, A. Toor, R. Overgaard, and R. Johal, “Comprehensive Analysis Method for Vehicle/Pedestrian Collision”, Paper 2000-01-0846, SAE International, Warrendale, PA, 2000.

400

References

10.5. Han, I. and R.M. Brach, “Throw Model for Frontal Pedestrian Collisions,” Paper 2001-01-0898, SAE International, Warrendale, PA, 2001. 10.6. Toor, A., “Theoretical Versus Empirical Solutions for Vehicle/Pedestrian Collisions,” Paper 2003-01-0883, SAE International, Warrendale, PA, 2003. 10.7. Russell, C.G., Pedestrian Formulas for Excel, Lawyers & Judges Publishing Company, CD-ROM, 2005. 10.8. Wood, D. and C.K. Simms, “A Hybrid Model for Pedestrian Impact and Projection,” I J Crashworthiness, Vol. 5, No. 4, pp 393-403, UK, 2000. 10.9. Wood, D.P. and D.G. Walsh, “Pedestrian forward projection impact,” I J Crashworthiness, Vol. 7, No. 3, 2002. 10.10. Wood, D.P., “Impact and Movement of Pedestrians in Frontal Collisions with Vehicles,” Proc. of the Institution of Mechanical Engineers, Vol. 202, No. D2, pp. 101-110, UK, 1988. 10.11. Backaitis, Stanley H., Accident Reconstruction Technologies — Pedestrians and Motorcycles in Automotive Collisions, PT-35, 1990, SAE International, Warrendale, PA, 1990. 10.12. Fugger, T.F., B.C. Randles, J.L. Wobrock, A.C. Stein, and W.C. Whiting, “Pedestrian Behavior at Signal-Controlled Crosswalks,” Paper 2001-01-0896, SAE International, Warrendale, PA, 2001. 10.13. Vaughan R. and J. Bain, “Acceleration and Speeds of Young Pedestrians Phase II,” Paper 2000-01-0845, SAE International, Warrendale, PA, 2000. 10.14. Vaughan R. and J. Bain, “Acceleration and Speeds of Young Pedestrians,” Paper 1999-01-0440, SAE International, Warrendale, PA, 1999. 10.15. Evans, A.K. and R. Smith, “Vehicle speed calculation from pedestrian throw distance,” IMechE, Proc Instn Mech Engrs, Vol 213, Prt D, 1999. 10.16. Wood, D. and C.K. Simms, “Coefficient of friction in pedestrian throw,” Impact, Vol. 9, No. 1, pp. 12-15, UK, 2000. 10.17. Hill, G.S., “Calculations of vehicle speed from pedestrian throw,” Impact, J. Inst. Traffic Accident Investigation, pp. 18-20, 1994. 10.18. Funk, J.R., G. Beauchamp, N. Rose, S. Fenton, and J. Pierce, “Occupant Ejection Trajectories in Rollover Crashes: Full-Scale Testing and Real World Cases,” Paper 2008-01-0166, SAE International, Warrendale, PA, 2008. 10.19. @RISK, Palisade Corporation, Newfield, NY, 1998.

401

References

Chapter 11 11.1. Slama, C.C., Manual of Photogrammetry, Fourth Edition, American Society of Photogrammetry, Falls Church, VA, 1980. 11.2. Hallert, B. Photogrammetry, Basic Principles and General Survey, McGraw-Hill Book Company, 1960. 11.3. Moffitt, F.H. and E.M. Mikhail, Photogrammetry, Harper and Row Publishers, New York, Third Edition, 1980. 11.4. Baker, J.S. and L.B. Fricke, The Traffic Accident Investigation Manual, Northwestern University Traffic Institute, Ninth Edition, Evanston, IL, 1986. 11.5. Husher, S.E., M.S. Varat, and J.F. Kerkhoff, “Survey of Photogrammetric Methodologies for Accident Reconstruction,” Proceedings of the Canadian Multidisciplinary Road Safety Conference VII, Vancouver, British Columbia, June 1991. 11.6. Bleyl, R.L., “Using Photographs to Map Traffic Accident Scenes: A Mathematical Approach,” J. of Safety Research, v 8, n 2, June 1976. 11.7. Woolley, R., et al., “Determination of Vehicle Crush from Two Photographs and the Use of 3D Displacement Vectors in Accident Reconstruction,” Paper 910118, SAE International, Warrendale, PA, 1991. 11.8. Williamson, J.R., “Hazards of Reverse Projection from Hand-held Cameras,” SPIE vol 1943, State of the Art Mapping, pp 137-147, 1993. 11.9. Massa, D.J., “Using Computer Reverse Projection Photogrammetry to Analyze Animations,” Paper 1999-01-0093, SAE International, Warrendale, PA, 1999. 11.10. Fenton, S., et al., “Determining Crash Data Using Camera Matching Photogrammetric Technique,” Paper 2001-01-3313, SAE International, Warrendale, PA, 2001. 11.11. Autodesk, http://usa.autodesk.com/. 11.12. Sunbear Associates, http://home.pacbell.net/vondaa. 11.13. Pappa, R.S., et al., “Photogrammetry Methodology for Gossamer Spacecraft Structures,” Sound and Vibration, August 2002. 11.14. Randles, B., et al., “The Accuracy of Photogrammetry vs. Hands-on Measurement Techniques used in Accident Reconstruction,” Paper 2010-01-0065, SAE International, Warrendale, PA, 2010. 11.15. Townes, H., Photogrammetry in Accident Reconstruction, SAE Professional Development Seminar, July 1998.

402

References

11.16. Townes, H.W. and J.R. Williamson, Close Range Photogrammetry in Accident Reconstruction, Chapter 9, Forensic Accident Investigation: Motor Vehicles—2, Thomas L. Bohan Editor, Lexis Law Publishing, Charlottesville, VA, 1997. 11.17. D’Souza, A.F. and V.K. Garg, Advanced Dynamics, Modeling and Analysis, Prentice-Hall, 1984. 11.18. Photomodeler Pro 4.0, Eos Systems, Inc., Vancouver, BC, Canada, http://www. photomodeler.com/. 11.19. iWitness Close Range Photogrammetry, DeChant Consulting Services—DCS, Inc., Bellevue, WA, http://www.iwitnessphoto.com/.

Chapter 12 12.1. Railroad-Highway Grade Crossing Handbook, Second Edition, U.S Department of Transportation, Federal Highway Administration, Washington, D.C., 1986. 12.2. Railroad-Highway Grade Crossing Handbook, Revised Second Edition, U.S Department of Transportation, Federal Highway Administration, Washington, D.C., 2007. 12.3. “Use of Locomotive Horns at Highway-Rail Grade Crossings,” Final Rule, 49 CFR Parts 222 and 229, Federal Register, Part IV, 2006. 12.4. Rapoza, A.S., T.G. Raslear, and E.J. Rickley, Railroad Horn Systems Research, DOT-VNTSC-FRA-98-2, 1999. 12.5. Beranek, L.L. [ed.], Noise and Vibration Control, Institute of Noise Control Engineering, Washington, D.C., 1988. 12.6. ANSI Standard S1.13-2005, Measurement of Sound Pressure Levels in Air, Acoustical Society of America, Melville, NY, 2010. 12.7. Seshagiri, B. and B. Stewart, “Investigation of the Audibility of Locomotive Horns,” Am. Ind. Hyg. Assoc. J. 9530, Nov 1992. 12.8. Rapoza, A.S. and G.G. Fleming, The Effect of Installation Location on Railroad Horn Sound Levels, DOT Letter Report DTS-34-RR297-LR, September 2000. 12.9. 49 CFR, Part 229 Amended, Use of Locomotive Horns at Highway-Rail Grade Crossings, Final Rule, Vol 71, No 159, August 17, 2006. 12.10. Brach, Raymond and Matthew Brach, “Insertion Loss: Train & Light-Vehicle Horns and Railroad-Crossing Sound Levels,” 158th Meeting, Acoustical Society of America, San Antonio, TX, Proceedings of Meetings on Acoustics, http:// scitation.aip.org/POMA, 2009.

403

References

12.11. Dolan, T.G. and J.E. Rainey, “Audibility of Train Horns in Passenger Vehicles,” Human Factors, Vol 47, No 3, pp 613-29, 2005. 12.12. Rapoza, A.S., T.G. Raslear, and E.J. Rickley, Railroad Horn Systems Research, DOT-VNTSC-FRA-98-2, 1999. 12.13. NTSB, “Safety at Passive Grade Crossings,” Vol 1, Analysis, NTSB/ss-9802, 1998.

Chapter 13 13.1. McHenry, B.G. and R.R. McHenry, “HVOSM-87,” Paper 880228, SAE International, Warrendale, PA, 1988. 13.2. Leatherwood, M.D. and D.D. Gunter, “Vehicle Dynamics of a Heavy Truck/ Trailer Combination using Simulation,” Paper 1999-01-0119, SAE International, Warrendale, PA, 1999. 13.3. Day, T.D., “Validation of the EDVSM 3-dimensional Vehicle Simulator,” Paper 970958, SAE International, Warrendale, PA, 1997. 13.4. McHenry, B.G. and R.R. McHenry, “SMAC-97: Refinement of the Collision Algorithm,” Paper 970947, SAE International, Warrendale, PA, 1997. 13.5. Day, T.D., “Further Validation of EDSMAC using the RICSAC Staged Collisions,” Paper 900102, SAE International, Warrendale, PA, 1990. 13.6. Chrstos, J.P. and G.J. Heydinger, “Evaluation of VDANL and VDM RoAD for Predicting the Vehicle Dynamics of a 1994 Ford Taurus,” Paper 970566, SAE International, Warrendale, PA, 1997. 13.7. Brach, R.M., “Vehicle Dynamics Model for Simulation on a Microcomputer,” Int J of Veh Des, Vol. 12, No. 4, 1991. 13.8. Day, T.D. and D.E. Siddall, “Validation of Several Reconstruction and Simulation Models in the HVE Scientific Visualization Environment,” Paper 960891, SAE International, Warrendale, PA, 1996. 13.9. Salaani, M.K., D.A. Guenther, and G.J. Heydinger, “Vehicle Dynamics Modeling for the National Advanced Driving Simulator of a 1997 Jeep Cherokee,” Paper 1999-01-0121, SAE International, Warrendale, PA, 1999. 13.10. Day, T.D., S.G. Roberts, and A.R. York, “SIMON: A New Vehicle Simulation Model for Vehicle Design and Safety Research,” Paper 2001-01-0503, SAE International, Warrendale, PA, 2001. 13.11. PC-Crash, www.PC-Crash.com. 13.12. VCRware, www.brachengineering.com.

404

References

13.13. Fiala, E., “Seitenkrafte am rollenden Luftreifen,” VDI-Zeitschrift 96:973, 1964. 13.14. Allen, R.W., T.J. Rosenthal, and J.P. Chrstos, “A Vehicle Dynamics Tire Model for both Pavement and Off-Road Conditions,” Paper 970559, SAE International, Warrendale, PA, 1999. 13.15. Bakker, E., L. Nyborg, and H.B. Pacejka, “Tyre Modeling for Use in Vehicle Dynamic Studies,” Paper 870421, SAE International, Warrendale, PA, 1987. 13.16. Brach, R. Matthew and Raymond M. Brach, “Tire Forces: Modeling the Combined Braking and Steering Forces,” Paper 2000-01-0357, SAE International, Warrendale, PA, 2000. 13.17. Todd, J., Editor, Survey of Numerical Analysis, McGraw-Hill, New York, 1962. 13.18. Brach, R.M., “Impact of Articulated Vehicles,” Paper 860015, SAE International, Warrendale, PA, 1986. 13.19. Gent, A.N. & J.D. Walter, The Pneumatic Tire, NHTSA, 2005, (http://www. tiresociety.org/mainpages/nhtsa.html). 13.20. Pacejka, H.B., Tire and Vehicle Dynamics, SAE International, Warrendale, PA, 2002. 13.21. Bosch, R., Bosch Automotive Handbook, 7th Ed, SAE International, 2007. 13.22. Jazar, R.N., Vehicle Dynamics, Theory and Application, Springer, NY, 2009. 13.23. Salaani, K., “Analytical Tire Forces and Moments Model with Validated Data,” Paper 2007-01-0816, SAE International, Warrendale, PA, 2007. 13.24. Fancher, P., Generic Data for Representing Truck Tire Characteristics in Simulations of a Braking and Braking-in-a-Turn Maneuvers, UMTRI Report UMTRI-95-34, September 1995. 13.25. Cliff, W.E., J.M. Lawrence, B.E. Heinrichs, and T.R. Fricker, “Yaw Testing of an Instrumented Vehicle with and Without Braking,” Paper 2004-01-1187, SAE International, Warrendale, PA, 2004. 13.26. Brach, R. and M. Brach, “Tire Models for Vehicle Dynamic Simulation and Accident Reconstruction,” Paper 2009-01-0102, SAE International, Warrendale, PA, 2009.

Appendix A 1. IEEE/ASTM-SI-10 Standard for Use of the International System of Units (SI):The Modern Metric System, ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA, 19428-2959 USA.

405

References

2.

Bevington, P.R. and D.K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd Edition, McGraw-Hill, New York, 1992.

3.

National Institute for Science and Technology (NIST), “Guide for the Use of International System of Units (SI),” Special Publication 811, abridged, Gaithersburg, MD, 1995.

Appendix B 1. “Motor Vehicle Dimensions,” SAE Standard J1100, SAE International, Warrendale, PA, July 2002.

406

2.

Baker, J. Stannard, from the Traffic Accident Investigation Manual, Northwestern University Traffic Institute, Evanston, IL, pp 313—321, 1975.

3.

“Collision Deformation Classification,” SAE Standard J224, SAE International, Warrendale, PA, March 1980.

4.

“Manual on Classification of Motor Vehicle Traffic Accidents,” 6th ed., ANSI D16.1, National Safety Council, Chicago, IL, 1996.

5.

Handbook of Motor Vehicle Safety and Environmental Terminology, SAE HS-215, SAE International, Warrendale, PA, December 1995.

6.

“Truck Deformation Classification,” SAE Standard J1301, SAE International, Warrendale, PA, August 2003.

7.

“Vehicle Dynamics Terminology,” SAE Standard J670, SAE International, Warrendale, PA, July 1976.

8.

“A Dictionary of Terms for the Dynamics and Handling of Single Track Vehicles (Motorcycles, Mopeds and Bicycles),” SAE Standard J1451, SAE International, Warrendale, PA, January 2000.

9.

“Commercial Truck and Bus SAE Recommended Procedure for Vehicle Performance Prediction and Charting,” SAE Standard J2188, SAE International, Warrendale, PA, October 2003.

Bibliography of Vehicle Dynamics Books

1.

Dukkipati, R., J. Pang, M. Qatu, G. Sheng, and Z. Shuguang, Road Vehicle Dynamics, SAE International, Warrendale, PA, 2008.

2.

Ellis, J.R., Vehicle Dynamics, Business Books Limited, London, 1969.

3.

Genta, G., Motor Vehicle Dynamics, Modeling and Simulation, World Scientific, Singapore, 1997.

4.

Gillespie, T.D., Fundamentals of Vehicle Dynamics, SAE International, Warrendale, PA, 1992.

5.

Jazar, R.N., Vehicle Dynamics, Theory and Applications, Springer, New York, 2008.

6.

Mastinu, G. and M. Plochl [ed.], Road Vehicle Dynamics, CRC Press, 2010.

7.

Milliken, W.F. and D.L. Milliken, Race Car Vehicle Dynamics, SAE International, Warrendale, PA, 1995.

8.

Rajamani, R., Vehicle Dynamics and Control, Springer, NY, 2006.

9.

Rill, G., Road Vehicle Dynamics, Fundamentals and Modeling, CRC Press, 2011.

10.

Sauvage, G., The Dynamics of Vehicles on Roads and Tracks, CRC Press, 1992.

11.

Wong, J.Y., Theory of Ground Vehicles, John Wiley, New York, 2001.

407

Index

Acceleration equations, 51 constant acceleration, 51–53 rotational acceleration, 55 Accident reconstruction, braking distance vs. stopping distance, 59 Accident site documentation of, 88 rollover accident, information about, 110–111 Airbags, 189–191 Antilock braking systems (ABS), 39–40 Articulated vehicles collisions, 193–224 examples, 197–198, 209–224 impact equations, 198–205 validation of, 206–222 results of analysis, 221t spreadsheet appendix, 222–224 Bakker-Nyborg-Pacejka model, 31–33 Calculations terminology, 3 uncertainty in, 1–15

Close-range photogrammetry, see Photo­ grammetry for accident reconstruction Coefficient of friction, 22–23, 43n dry pavement, 45, 46t wet pavement, 45, 47t Coefficient of restitution, 134, 136, 141–143, 142t, 147, 153, 189, 237 mass equivalent collision, 143–144 stiffness equivalent collision, 142 Collisions, articulated vehicles, 193–224; see also Impulse-momentum examples, 197–198, 209–224 impact center, 199, 202 impact equations, 198–205 validation of, 206–222 planar impact mechanics, assumptions about, 195–198 results of analysis, 221t spreadsheet appendix, 222–224 Collisions, frontal vehicle-pedestrian, 244–258 analysis model, 249 pedestrian motion, 249 vehicle motion, 252 forward projection model, 248 hybrid wrap model, 248

409

Index

Collisions, frontal vehicle-pedestrian (continued) reconstruction model, 254–258 speed reconstruction approaches, 247 throw distance, 247, 252, 254, 255f, 256f examples, 253–254, 257 types of contact, 245–246 values of physical variables, 252–254 pedestrian-ground drag coefficient, 253–254 variables for vehicle-pedestrian throw model, 251t Collisions, light vehicles, analysis of, 129– 162, see also Impulse-momentum controlled collisions, 139–144 barrier collisions, 139–141 coefficients of restitution 141–144, 142t example, 140–141 low-speed in-line, 186–189 example, 189 quantitative concepts, 131 example, 131–132 RICSAC collisions, 158–162 sideswipe collisions, 137–139 example, 137–139 using a solution of the planar impact equations, 171–172 using a spreadsheet solution of the planar impact equations, 172–185 vehicle collisions with rotation, 164–168 Collisions, railroad grades, 299–319 insertion loss, sound pressure level, 316–319, 317t, 318t locomotive horn sound levels, 313–319 calculation of, 313–316 example, 316 Railroad Highway Grade Crossing Handbook equations, 309–312 clearing sight distance, 311–312, 313f stopping distance, 309–311 stopping sight distance, 311 sight distance for stopping before a crossing, 304–307 example, 307–309

410

sight triangle for clearing a crossing, 300–304 example, 304–305 sound pressure level, calculation example, 318–319 train-vehicle converging example, 303–304, 307–309 Collisions, road intersections, 299–319, see also Collisions, railroad grades sight distance for stopping before an intersection, 304–307 sight triangle for clearing an intersection, 300–304 Common-velocity conditions, 230 Conservation of energy, 98–99 Conservation of momentum, 129–130 Controlled collisions, 139–144 barrier collisions, 139–141 coefficients of restitution 141–144, 142t example, 140–141 Conversion nomograms, 41f, 42f Coordinate system, 15–17, 16f Coulomb, Charles Augustin de, 22 Coulomb’s laws, 22–23 CRASH program, xv, 141 CRASH3, 150, 157, 176, 195, 227–235 assumptions of model, 230 Critical speed, 69–83 calculated vs. measured, 80 defined, 69 drawbacks to sophisticated formulas, 70–71 equation, 70, 73 use of in example, 119 examples, 74–75, 82, 119 Crush energy, 225–243 arbitrary number of crush measurements, 243 average crush, rigid barrier tests, 236–241 CRASH3, 227–235, 237–241 assumptions of model, 230 crush stiffness coefficient, 225, 236– 241 example, 236 side crush, 243 vehicle-to-vehicle collisions, 241–243

Index

equation, 225 examples, 226–227, 232–236, 237–142 nonlinear models, 243 planar impact mechanics, 155–158 for calculating DV, 232–235 principal direction of force, 228–229 Crush stiffness coefficients, 225, 236–241 side crush, 243 vehicle-to-vehicle collisions, 241– 243 Crush surface, 155 angle, 151–152, 166, 171–172 Curved motion, 63 Driver control modes, 77 Edge dropoff, 79 EDSVS, 322 EDVSM (Engineering Dynamics Vehicle Simulation Model), 322 EDVTS, 322 Effective drag factor equation, 90 Equations acceleration, 51–56 constant acceleration, 51–53 rotational acceleration, 55 articulated vehicle impact equations, 198–205 validation of, 206–222 constant mass, 131 crush energy, 225 effective drag factor, 90 forward-motion performance, 54–59 notation for, 56t frictional drag coefficient, 1 of motion, 51, 99–101 photogrammetry, collinearity equations, 287 coplanarity equations, 287 projection equations, 285–287 planar vehicle motion, differential equations, 338–340 Railroad Highway Grade Crossing Handbook equations, 309–312 clearing sight distance, 311–312, 313f

stopping distance, 309–311 stopping sight distance, 311 solution equations, 134–135 sound pressure level, 313 system equations, 134 vehicle fall, 64–68 velocity of a vehicle, estimating, 1 Event data recorders, 189–192 crash data, 191–192 Examples articulated vehicle collision, 197–198, 209–224 controlled collision, 140–141 critical speed, 74–75, 82, 119 crush energy, 157–158, 236, 237–142 crush stiffness coefficient, 236 low-speed in-line collision, 189 mean and standard deviation, 11 pedestrian throw distance, 253–254, 257 perception-reaction time, 9 planar impact mechanics, 151–153, , 159–162, 171–172, 172–186 preimpact speed, 165–168 quantitative concepts, 131–132 railroad grade and road intersection collisions, 304–305, 307–309, 316, 318–319 rollover accidents pretrip phase, 91–95 trip phase, 105 roll phase, 108 sideswipe collisions, 137–139 straight-line motion braking distance, 57–58 pitch angle of vehicle, 68–69 speed when distance is known, 60–62 stopping time and distance, 62– 63 uniform grade, 53 vertical force and frictional drag coefficient, 58–59 uncertainty, 5, 7, 82 vehicle dynamic simulation, 326–338 vehicle-pedestrian wrap collision, 253–254

411

Index

Federal Motor Carrier Safety Regulations (FMCSR), 44 Federal Motor Vehicle Safety Standards (FMVSS), 43–44, 86–87 Force diagram, 96f Forward-motion performance equations, 54–59 notation for, 56t Forward projection collisions, see Collisions, frontal vehicle-pedestrian Friction, coefficient of, 22–23, 43n dry pavement, 45, 46t wet pavement, 45, 47t Friction circle, 27–31, 76–77 actual, 30 idealized, 28–30 Friction ellipse, 27–31 idealized, 28–30 Frictional drag coefficient, 22–23, 30, 37, 39–46, 76 equation for, 1 for heavy vehicles, 41–46 for light vehicles, 40–41 for combined trip and roll phase, 108t Grade and equivalent drag coefficients, 54 HVOSM (Highway Vehicle Obstacle Simulation Model), 322 Hydroplaning, 46–49 Impact center (point C), 155, 199, 202 Impulse-momentum, 95–96, 98–99, 102, 131 controlled collisions, 139–144 barrier collisions, 139–141 coefficients of restitution 141–144, 142t planar impact mechanics, 144–158 applications of, 164–168 and articulated vehicles, assumptions about, 195–198 coefficient of restitution, 147, 153–154 crush energy, 155–158 crush energy example, 157–158 crush surface, 155

412

examples, 151–153, 159–162, 171– 172, 172–186 impact center (point C), 155 impulse ratio, 153–154 overview of, 150–153 sideswipe collision, 154 solution equations, 148, 171–172 spreadsheet solution, 172–185 point-mass theory, 132–138 coefficient of restitution, 134, 136 common velocity conditions, 137–139 critical impulse ratio, 136–137 frictionless point-mass collisions, 136 preimpact speed examples, 165–168 sideswipe collisions, 137–139 solution equations, 134–135 system equations, 134 theory, 129–224 Impulse ratio, 134, 147, 153 critical, 136–137, 149 Insertion loss, sound pressure level, 316–319, 317t, 318t Lateral tire forces, 25–27, 26f modeling of, 31–37 Locomotive horn sound levels, 313–319 calculation of, 313–316 example, 316 Longitudinal tire forces, 23–24 modeling of, 31–37 peak, 47–48 Low-speed in-line (central) collisions, 186–189 Measurements, uncertainty in, 1–15 terminology, 3 Modified Nicolas-Comstock model, 33, 78 Monte Carlo method, 257 Motion, equation of, 51 National Highway Traffic Safety Administration (NHTSA), 87, 98 Neutral stability position, 98

Index

New Car Assessment Program (NHTSA), 87, 139 Newton’s laws of motion, 51, 54–56, 64, 146 second law, 74, 131, 132, 147, 323 third law, 200, 242 Nicolas-Comstock model, modified, 33, 78 PC-Crash, 322 Peak friction, 39 Pedestrian collisions, see Collisions, frontal vehicle-pedestrian Perception-decision-reaction time, 59–60, 299, 301 examples, 9, 303, 308–309, 316 Perception-reaction time, see Perceptiondecision-reaction time Peterbilt vehicles, frictional drag coefficient, 44 Photogrammetry for accident reconstruction, 259–298 close-range, 259 defined, 259 examples, 269–271, 272–274, 276–277, 278–281, 291–296 planar photogrammetry, 260, 274–281 assumptions about, 275 example, 276–277, 278–281 field example, 279–281 image and object planes, 298f obtaining coordinates, 275–276 projective relation, 297–298 quality of photograph used, 276 reverse projection photogrammetry, 260–274 analog approach, 263 defined, 261 full-frame photographs, importance of, 262–263 landmarks, importance of, 262 overlays, 263–265 practical difficulties, 271–272 procedure, 263–268 requirements, 261–263 seven degrees of freedom, 267t size of image, choosing, 264 size of viewable area, determining, 266–267

summary of main steps, 268–269 three-dimensional photogrammetry, 281–297 accuracy of, 282 collinearity equations, 287 coordinate system, 284 coplanarity equations, 287 example, residual crush, 291–296 fundamentals, 283 mathematical basis, 284 multiple image considerations, 287 practical use of, 288–290 projection equations, 285–287 Planar impact mechanics, 144–158 applications of, 164–168 and articulated vehicles, assumptions about, 195–198 coefficient of restitution, 147, 153–154 crush energy, 155–158 crush energy example, 157–158 crush surface, 155 examples, 151–153, 159–162, 171–172, 172–186 impact center (point C), 155 impulse ratio, 153–154 overview of, 150–153 sideswipe collision, 154 solution equations, 148, 171–172 spreadsheet solution, 172–185 Planar photogrammetry, 260, 274–281 assumptions about, 275 example, 276–277, 278–281 field example, 279–281 image and object planes, 298f obtaining coordinates, 275–276 projective relation, 297–298 quality of photograph used, 276 Point-mass theory, 132–138 coefficient of restitution, 134, 136 common velocity conditions, 137–139 critical impulse ratio, 136–137 frictionless point-mass collisions, 136 preimpact speed examples, 165–168 sideswipe collisions, 137–139 solution equations, 134–135 system equations, 134 Postimpact speed, calculating, 164

413

Index

Principal direction of force, 176–177, 181, 191, 228–229 Railroad grade collisions, see Collisions, railroad grades Railroad Highway Grade Crossing Handbook equations, 309–312 clearing sight distance, 311–312, 313f stopping distance, 309–311 stopping sight distance, 311 Reaction time, see Perception-reaction time Reconstruction applications, 163–192; see also Impulse-momentum airbags, 189–191 event data recorders, 189–191 low-speed in-line (central) collisions, 186–189 point-mass collision applications, 164–168 planar impact mechanics applications, 164–168 using a solution of the planar impact equations, 171 using a spreadsheet solution of the planar impact equations, 172–185 vehicle collisions with rotation, 164–168 Reconstruction techniques CRASH program, xv, 141, 150, 157 SMAC, 141, 195 Reverse projection photogrammetry, 260– 274 analog approach, 263 defined, 261 full-frame photographs, importance of, 262–263 landmarks, importance of, 262 overlays, 263–265 practical difficulties, 271–272 procedure, 263–268 requirements, 261–263 seven degrees of freedom, 267t size of image, choosing, 264 size of viewable area, determining, 266–267 summary of main steps, 268–269

414

RICSAC collisions, 158–162 RICSAC 7, 180 RICSAC 9, 172–179 test conditions, 176t Rigid barrier tests, 139, 142t, 236–241 Rigid-body impact theory, 144 application of, 168–171 Road intersection collisions, see Collisions, road intersections Roadway, superelevation, 74–75 Rollover accidents, reconstruction of, 85–128 conservation of energy, 98–99 documentation of accident site, 88 example, 117–126 detailed analysis, 120–125 speed analysis, 117–120 phases of, 85 pretrip phase, 89–85, 118–120 example, 91–95 roll phase, 106–117, 108f accident scene, gathering information about, 110–111 analysis of rolling vehicle, 109–110 example, 108 frictional drag coefficients, 108t, 109t inspection of accident vehicle, 111–115 police report and photographs, importance of, 110–111 speed analysis, 107–109 rollover reconstruction tools, 115–116 validation of reconstruction method, 106 SAE recommended practice, 87 scratch patterns, 113 segmented method, 94 static stability factor, 87, 98 test methods, 86–88 tire marks, 90–94 trip phase, 95–105 analysis of vehicle trip, 96–101 complex vehicle trip models, 101–102 constant force trip model, 99 difficulties applying trip models, 101–102

Index

drag factors, 104t example, 105 frictional drag coefficients, 108t reconstruction of, 102–105 rim contact, 105–106 vehicle drag factor, 89–90 vehicle dynamics simulation, 92–93 vehicle roll rate, 126–127, 128t vehicle sideslip angle, 89–90, 90f Rollover reconstruction tools, 115–116 Rollover Resistance Rating system (NHTSA), 98 Rollover tests, 81, 86–88 Rounding of numbers, 343–344 Runge-Kutta-Gill numerical integration, 323 Scratch patterns, evaluating, 113–114 SI units metric vs. customary units, 342t use of, 341 Sideswipe collisions, 137–139 example, 137–139 Sight distance for stopping before a crossing, 304–307 example, 307–309 Sight triangle for clearing a crossing, 300– 304 example, 304–305 Significant figures, 342 consistency of, 344–345 SMAC, 141, 195, 322 Sound pressure levels, 313–319 example of calculation, 318–319 insertion loss, 316–319, 317t, 318t Speed, critical, 69–83 calculated vs. measured, 80 defined, 69 drawbacks to sophisticated formulas, 70–71 equation, 70, 73 use of in example, 119 examples, 74–75, 82, 119 Static stability factor, 87, 98 Steering forces, combined with tire forces, modeling of, 31–37 Bakker-Nyborg-Pacejka model, 31–33

Modified Nicolas-Comstock model, 33–37 Stopping distance, 59–63 calculating distance from speed, 60 calculating speed from distance, 60–61 Straight-line motion, 51–68 equation of motion, 51 examples braking distance, 57–58 pitch angle of vehicle, 68–69 speed when distance is known, 60–62 stopping time and distance, 62–63 uniform grade, 53 vertical force and frictional drag coefficient, 58–59 stopping distance, 59–63 calculating distance from speed, 60 calculating speed from distance, 60–61 uniform acceleration and braking motion 51 equations of acceleration, 51–56 grade and equivalent drag coefficients, 54 vehicle fall equations, 64–68 vehicle forward-motion performance equations, 54 notation for, 56t Three-dimensional photogrammetry, 281–297 accuracy of, 282 collinearity equations, 287 coordinate system, 284 coplanarity equations, 287 example, residual crush, 291–296 fundamentals, 283 mathematical basis, 284 multiple image considerations, 287 practical use of, 288–290 projection equations, 285–287 Tire forces, 15–49 acceleration slip vs. braking slip, 20 contact patch, 17 coordinate system, 15–17 friction, 22

415

Index

Tire forces (continued) lateral force, 25–27, 26f modeling of, 31–37 longitudinal force, 23–24 modeling of, 31–37 peak, 47–48 rolling resistance, 17, 18f slip force, 17–27, 18f longitudinal, 19–20 Tire forces, combined with steering forces, modeling of, 31–37 Bakker-Nyborg-Pacejka model, 31–33 Modified Nicholas-Comstock model, 33–37 Tire mark striations, 90 Tire model, important aspects, 322 Tire rims contact with roadway, 105–106 evaluating deformation, 113 Tire sideslip, 71–72 Tire treads depths, 48t Tire yaw marks, 69–83 curved, 76 estimation of speed, 69 defined, 71 guidelines for measuring, 76 radius from, 73, 77 severe, 77–78 vs. skid marks, 72 Tires, evaluating deformation, 113 Treads, tire, see Tire treads Uncertainty of articulated vehicle reconstructions, 217 of critical speed calculations, 79–83 design of experiments, 12–13 differential variations, 5–7, 79–80 distribution-free methods, 13 examples, 5, 7, 82 norm, defined, 6n propagation of error, 5 sensitivity coefficients, 6 statistical distributions, 8–9 arbitrary functions, 9–11

416

linear functions, 8–9 statistical variations, 81–82 upper and lower bounds, 4–5 Uncertainty, epistemic, 13 Uncertainty, parametric, 1 Uncertainty, modeling, 1 Unit conversion, 346–354 VCRware, 322 VDANL (Vehicle Dynamics Analysis Nonlinear), 322 VDM RoAD (Vehicle Dynamics Models for Roadway Analysis and Design), 322 VdynVB, 322 Vehicle drag factor, 89–90 Vehicle dynamic simulation, 92–93, 321–340 conditions applied, 323–325 defined, 321 examples, 326–338 planar vehicle dynamics simulation, 322 planar vehicle motion, differential equations, 338–340 Runge-Kutta-Gill numerical integration, 323 tire model, important aspects, 322 tire side-force stiffness coefficients, 325 for heavy vehicle, 326 for light vehicle, 325 vehicle dynamic simulation vs. vehicle dynamics, 321 Vehicle fall equations, 64–68 Vehicle roll rate, 126–127, 128t Vehicle sideslip angle, 89–90, 90f Velocity of a vehicle calculating DV with planar impact mechanics, 232–235 DV, 225–243 estimating, 2 predicted, 95t Volvo vehicles, heavy, frictional drag coefficient, 44 Yaw marks, see Tire yaw marks

About the Authors

Raymond M. Brach Raymond M. Brach, PhD, PE, is a consultant in the field of accident reconstruction and a professor emeritus of the Department of Aerospace and Mechanical Engineering, University of Notre Dame. He is a fellow member of SAE International. Other professional memberships include the American Society of Mechanical Engineers (ASME), The Acoustical Society of America (ASA), The Institution of Noise Control Engineers (INCE), and the National Association of Professional Accident Reconstruction Specialists (NAPARS). He was granted a PhD in Engineering Mechanics from the University of Wisconsin, Madison, and a BS and MS in Mechanical Engineering from Illinois Institute of Technology, Chicago. His specialized areas of teaching and research include mechanical design, mechanics, vibrations, acoustics, applications of statistics and quality control, vehicle dynamics, accident reconstruction, and microparticle dynamics. He is a licensed professional engineer in the state of Indiana. In addition to more than 100 research papers and numerous invited lectures, he has authored Mechanical Impact Dynamics published by Wiley Interscience in 1991 and is a coauthor of Uncertainty Analysis for Forensic Science, Lawyers and Judges Publishing Company, 2004.

R. Matthew Brach R. Matthew Brach, PhD, PE, is an engineering consultant with the firm Brach Engineering, LLC. His principal areas of professional activities include vehicle impact analysis, vehicle dynamics, and automotive accident reconstruction. He has a PhD in Mechanical Engineering from Michigan State University, East Lansing (1995), an MS in Mechanical Engineering from the University of Illinois at

417

About the Authors

Chicago (1986), and a BS in Electrical Engineering from the University of Notre Dame (1982). He served as an adjunct professor in the Mechanical Engineering Department at Lawrence Technological University, Southfield, Michigan from 1994 to 2000. He is a member of SAE International, the American Society of Mechanical Engineers (ASME), the Institute of Electrical and Electronics Engineers (IEEE), and the National Association of Professional Accident Reconstruction Specialists (NAPARS). He is a licensed professional engineer in the states of Indiana and Michigan. He is the author of technical papers covering a range of topics that includes nonlinear vibrations, automotive engine mount design, vehicular accident analysis methodologies, and tire forces.

418

Accident Analysisan Reconstruction Methods,

Raymond M. Brach and R. Matthew Brach

Second Edition

This book is an essential tool for experienced and novice engineers and accident reconstructionists involved in accident analysis, reconstruction, and vehicle design. The methods presented in the book enable analysis that provides insight into how, when, and why accidents occur. This second edition of Vehicle Accident Analysis and Reconstruction Methods is enhanced with new material that covers articulated vehicle accidents, event data recorders (EDRs), frictional drag coefficients for sliding tires, railroad gradecrossing collisions, and new practical applications of mathematical methods. It includes an expanded glossary of automotive terms and improved chapters on tire forces, rollover accidents, crush energy, pedestrian collisions, vehicle dynamic simulations, and many more topics. Accident reconstruction is a field that increasingly relies on science, mathematics, and engineering. The theories, experimental data, and examples presented are grounded in physics, mechanics, and mathematics. A bibliography at the end of the book provides further reading suggestions. Vehicle Accident Analysis and Reconstruction Methods incorporates real-world examples to illustrate the use of the methods, clarify important concepts, and provide practical applications of the methods to those working in the field. Police officers, attorneys, and insurance professionals will also find the book to be a definitive resource in reconstructing accident scenes.

About the Authors Raymond M. Brach, PhD, PE, is a consultant in the field of accident reconstruction and a professor emeritus of the Department of Aerospace and Mechanical Engineering, University of Notre Dame. He is a fellow member of SAE International. He received a PhD in Engineering Mechanics from the University of Wisconsin, Madison, and a BS and MS in Mechanical Engineering from Illinois Institute of Technology, Chicago. His specialized areas of teaching and research include mechanical design, mechanics, vibrations, acoustics, applications of statistics and quality control, vehicle dynamics, accident reconstruction, and microparticle dynamics. R. Matthew Brach, PhD, PE, is an engineering consultant with the firm Brach Engineering, LLC. His principal areas of professional activities include vehicle impact analysis, vehicle dynamics, and automotive accident reconstruction. He has a PhD in Mechanical Engineering from Michigan State University, East Lansing (1995), an MS in Mechanical Engineering from the University of Illinois at Chicago (1986), and a BS in Electrical Engineering from the University of Notre Dame (1982). He served as an adjunct professor in the Mechanical Engineering Department at Lawrence Technological University, Southfield, Michigan from 1994 to 2000. R-397 ISBN 978-0-7680-3437-0

9780768 034370