Mathematical Methods for Accident Reconstruction: A Forensic Engineering Perspective

by Harold Franck, Darren Franck

Overview

Over the past 25 years Harold and Darren Franck have investigated hundreds of accidents involving vehicle of almost every shape size, and type imaginable. In Mathematical Methods for Accident Reconstruction: A Forensic Engineering Perspective, these seasoned experts demonstrate the application of mathematics to modeling accident reconstructions involving a range of moving vehicles, including automobiles, small and large trucks bicycles motorcycles, all-terrain vehicles, and construction equipment such as hoists and cranes.

The book is anchored on basic principles of physics that may be applied to any of the above named vehicles or equipment Topics covered include:

foundations of measurement

energy methods used in reconstruction

momentum methods

vehicle specifications

failure analysis

geometrical characteristics of highways

visibility, perception, and reaction

crash data recorders

low speed impacts

standards and protocols

Exploring a broad range of accident scenarios and also acknowledging the limits of applicability of the various physical methods employed the breadth and depth of the book s coverage makes it a critical reference for engineers and scientists who perform vehicular accident reconstructions.

Product Details

ISBN-13:	9781032714721
Publisher:	Taylor & Francis
Publication date:	09/25/2025
Edition description:	2nd ed.
Pages:	352
Product dimensions:	6.12(w) x 9.19(h) x (d)

About the Author

Harold Franck and Darren Franck are experts in accident reconstruction and have worked on hundreds of investigations.

Table of Contents

Symbols and Units xiii

Preface xv

1 Introduction 1

2 Basic Principles 7

2.1 Introduction 7

2.2 Coordinate Systems 7

2.3 Basic Principles of Physics 11

2.4 Uniformly Accelerated Linear Motion 12

2.5 Motion in a Plane 13

2.6 Projectile Motion 14

2.7 Uniformly Accelerated Curvilinear Motion 16

2.8 Relation between Angular and Linear Velocity and Acceleration 18

2.9 Newton's First Law 19

2.10 Newton's Second Law 21

2.11 Newton's Third Law 22

2.12 Center of Gravity or Mass 23

2.13 Impulse and Momentum 24

2.14 Conservation of Momentum 28

2.15 Conservation of Energy and Work 29

2.16 Kinetic Energy 32

2.17 Potential Energy due to Gravity 33

2.18 Elastic Potential Energy 34

2.19 Dissipation and Conservation of Forces 36

2.20 Internal Work, Energy, Power, and Velocity 37

2.21 Change in Velocity 39

3 Evidence Gathering 41

3.1 Introduction 41

3.2 Measurements 42

3.2.1 Site Inspection 43

3.2.2 Vehicle Inspection 45

3.2.3 Component Inspection 48

3.3 Photography 51

3.4 Computer Techniques 52

3.5 Photogrammetry 52

4 Energy Methods 59

4.1 Introduction 59

4.2 Friction 61

4.3 Critical Speed-Straight Trajectory 63

4.4 Stopping Distances 67

4.5 Friction and the Speed of a Vehicle 68

4.6 Newton's Second Law Argument 71

4.7 Work-Energy Argument 73

4.8 Critical Speed-Curved Trajectory 75

4.9 Critical Speed to Negotiate a Turn Including Superelevation 80

4.10 Conservation of Energy Analysis 83

4.11 Generalized Critical Speed Analysis 87

4.12 Critical Speed from Yaw and Rollover 91

4.13 Extension on Minimum Speed Calculations when Radius Cannot Be Determinated Uniquely 94

4.14 Crush Analysis 97

4.15 Tree Impacts100

5 Momentum Methods 103

5.1 Introduction 103

5.2 Elastic and Inelastic Collisions 103

5.3 Elastic Collisions 104

5.4 Conservation of Linear Momentum 106

5.5 Conservation of Linear Momentum with Restitution 108

5.6 Conservation of Rotational Momentum 111

5.7 Combined Linear and Rotational Momentum 114

5.8 Rotational Momentum-Alternate Solution 115

5.9 Parametric Analysis for Left of Center Collisions 118

5.10 Plastic-Elastic Analysis 120

6 Vehicle Specifications 125

6.1 Introduction 125

6.2 Calculation of Center of Mass Techniques 125

6.3 Rollover/Tipover 129

6.4 Gouge Spacing 131

7 Vehicle Performance Characteristics 135

7.1 Introduction 135

7.2 Systems and Performance Characteristics 135

7.2.1 Braking 135

7.2.1.1 Vacuum Assisted Brake Booster 136

7.2.1.2 Brake Line Pressure Control Devices 137

7.2.1.3 Step Bore Master Cylinder 137

7.2.1.4 Brake Designs 138

7.2.1.5 Large Vehicles 139

7.2.2 Steering 139

7.2.3 Acceleration 144

7.2.4 Electrical Systems 146

7.3 Failure Analysis 147

7.3.1 Brake Performance Characteristics 147

7.3.2 Seat Failure 150

7.3.2.1 Analysis of the Circular Cross-Section 153

7.3.2.2 Analysis of the Elliptical Cross-Section 155

7.3.3 Seatbelts and Air Bags 159

7.3.4 Light Bulbs and Filaments 160

7.4 Testing of Components 162

8 Bicycles, Motorcycles, and Pedestrians 165

8.1 Introduction 165

8.2 Gyroscopic Action in Two-Wheeled Vehicles 167

8.3 Vaulting of Pedestrians 169

8.4 Determination of Pedestrian Speed from Vehicle Damage 172

8.5 Vehicular Collisions with Pedestrians or Cyclists 173

9 Geometrical Characteristics of Highways 177

9.1 Introduction 177

9.2 Classification and Design Speed 178

9.3 Cross Sectional Elements 180

9.4 Traffic Control Devices (TCDs) 181

9.5 Sight Distance 182

9.6 Horizontal Curves 185

9.7 Vertical Curves--Sag and Crest 187

9.8 Conclusions 189

10 Train Accidents 191

10.1 Introduction 191

10.2 Train Resistance and the Effect of Curvature and Grade 192

10.3 Computation of Train Speed 193

10.4 Train Manifest and Recorders 196

10.5 Train Crossings 196

10.6 Sight Distance Triangle 198

11 Commercial and Off-Road Vehicles 203

11.1 Introduction 203

11.1.1 Federal Motor Vehicle (FMV) Regulations Title 49 Transportation 203

11.2 Tractor/Trailers 204

11.3 Off-Tracking for Coupled Vehicles 209

11.4 Construction Equipment 210

12 Visibility, Perception, and Reaction 219

12.1 Introduction 219

12.2 Limitations of Human Factors Analysis 220

12.3 Design Standards 221

12.4 Capability of Drivers 223

13 Computer Methods and Modern Reconstruction Tools 227

13.1 Introduction 227

13.2 Numerical Methods 228

13.2.1 Systems of Linear Equations 228

13.2.2 Direct Methods 228

13.2.3 Iterative Solutions 230

13.2.4 Polynomial Approximation and Interpolation 233

13.2.5 Numerical Integration and Differentiation 238

13.2.6 Programming 242

13.3 Commercially Available Software 242

13.4 Reconstruction Tools 244

14 Low Speed Impacts and Injury 251

14.1 Introduction 251

14.2 Methods of Analysis 252

14.3 Biomechanics of Injury 254

14.3.1 Soft Tissue Injuries 254

14.3.2 Occupant Kinematics and Vehicle Impacts 256

14.3.3 Bone Fractures 260

14.3.4 Head Injury Criterion (HIC) 263

14.4 Staged Collisions 267

15 Standards and Protocols 271

15.1 Introduction 271

15.2 Protocol for Inspections and Reconstructions 273

15.3 Standard Guide for Vehicular Accident Inspections and Reconstructions 273

15.3.1 Scope 273

15.3.2 Significance and Use 274

15.3.3 Equipment 275

15.3.4 Procedure 275

15.3.5 Reports 277

15.4 American Society of Testing and Materials (ASTM) Standards 277

15.5 Federal Motor Vehicle (FMV) Standards 278

15.6 SAE Standards 279

16 Sensitivity and Uncertainty 281

16.1 Introduction 281

16.2 Sensitivity and Uncertainty 281

Bibliography 285

Index 28