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Vehicle Crash Mechanics
Vehicle Crash Mechanics
Matthew Huang
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Vehicle Crash Mechanics clarifies the complexities of this multifaceted area of study. It sets forth the principles of engineering mechanics and applies them to the issue of crashworthiness. It explores the three primary elements of crashworthiness, which are vehicle, occupant, and restraints, and illustrates their dynamic interactions through analytical models, experimental methods, and test data from actual crash tests. Parallel development of the analysis of actual test results and the interpretation of mathematical models related to the test provide additional insight, and case studies present realworld crash tests, accidents, and the effectiveness of air bag and crash sensing systems.
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Year:
2002
Edition:
1
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CRC Press
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english
Pages:
488
ISBN 10:
0849301041
ISBN 13:
9780849301049
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crush^{544}
pulse^{524}
mph^{523}
restraint^{499}
crc press^{492}
crc press llc^{490}
press llc^{490}
dynamic^{467}
deceleration^{451}
shown in fig^{445}
displacement^{410}
truck^{391}
acceleration^{332}
stiffness^{314}
frequency^{312}
dynamic crush^{307}
crash pulse^{272}
responses^{237}
coefficients^{223}
loading^{221}
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curve^{213}
damping^{212}
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initial^{206}
zero^{205}
analysis^{189}
computed^{188}
rigid barrier^{186}
air bag^{180}
ratio^{176}
collision^{169}
chest^{165}
fir^{160}
normalized^{157}
vehicles^{156}
transient^{155}
deflection^{155}
sled^{155}
magnitude^{154}
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peak^{153}
input^{151}
esw^{150}
maximum^{149}
output^{145}
coefficient^{145}
pulses^{143}
excitation^{142}
sensor^{140}
respectively^{140}
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VEHICLE CRASH MECHANICS MATTHEW HUANG CRC PR E S S Boca Raton London New York Washington, D.C. Library of Congress CataloginginPublication Data Catalog record is available from the Library of Congress This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the authors and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. Visit the CRC Press Web site at www.crcpress.com © 2002 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 0849301041 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acidfree paper PREFACE This textbook, Vehicle Crash Mechanics, has grown out of a series of my lectures on vehicle crashworthiness at the University of Michigan, Dearborn. Since 1991, these lectures have been presented to automotive engineers from the Ford Motor Company, full service suppliers to the Ford Motor Company, and engineers from various consulting firms. The primary goals of this book are to provide the funda; mentals of engineering mechanics and to apply these fundamentals to the study of vehicle crashworthiness. Also the book was written to present a number of interesting and informative ancillary topics related to vehicle crashes but extending beyond purely fundamental theory. In the automotiverelated industry, the goal of engineering effort in the field of crashworthiness is to satisfy, or, to the extent possible, exceed the safety requirements mandated by the Federal Motor Vehicle Safety Standards (FMVSS) and administered by the National Highway Traffic Safety Administration (NHTSA). Governed as it is by strict adherence to regulations and the balancing of complex interactions among the variables, the application of mechanics to crashworthiness is not a simple task. The importance of understanding the fundamentals of mechanics cannot be overemphasized. In this book, I have strived to present the fundamentals as clearly as possible, and with an aim toward applications to problems. This field can be subdivided into four groups: (1) Vehicle crash dynamics, (2) Computer aided engineering, (3) Occupant impact dynamics, and (4) Design analysis and accident reconstruction. In each of these groups, knowledge of the fundamental principles of mechanics is essential. Also, the ability to apply such knowledge to hardware, to developmental work, or analytical modeling is required. First, the fundamentals, which range from particle dynamics to rigid body kinetics, are presented. Then, Newton's Second Law, the principle of impulse and momentum, and the principle of work and energy are applied to engineering problems. It is assumed that the reader has had courses in mathematics through calculus and engineering statics. Formulas are presented as needed; as each one is presented for the first time, a short derivation of the formula is provided. Throughout the book, when analyzing vehicle tests, both the analysis of actual test results and the interpretation of mathematical models related to the tests will be developed in parallel. This approach is done in an orderly fashion in order to provide an insight into the parameters and interactions that influence the results. In the study of crashworthiness, three main elements can be defined: vehicle, occupant, and restraints (VOR). In this book, the dynamic interactions among these three elements will be illustrated by the use of analytical models, experimental methods, and test data from actual vehicle crash tests. As an example, the occupantvehicle kinematics in the restraint coupling phase and the use of the ridedown concept are presented in both analytical and experimental terms. The book contains seven chapters, each having an introduction which describes the objectives of the chapter and the material to be covered. Chapter 1 presents an overview of the crash pulse and kinematics, the kinematic principles used in VOR analysis and digital filtering techniques which satisfy the frequency response requirements of SAE J211. The filtering process is used to analyze vehicle, occupant, and crash sensor test data for crash severity. Also covered are analyses of crash sensor and air bag performance for an accident using onboard recorder data as they are related to the crash pulse analysis. © 2002 by CRC Press LLC Chapter 2 presents ways of characterizing the crash pulse. An approximation method is developed which describes the crash pulse with a few parameters. Eight approximation methods are presented, ranging from the Tipped Equivalent Square Wave (TESW) with two boundary conditions to the Fourier Equivalent Wave (FEW) with or without boundary conditions. Physical significance of crash pulse centroid and residual deformation are discussed. Use of crash pulse approximation to the testings of an air bag crash sensor and vehicle interiorheadform impact is illustrated. Chapter 3 deals with the use of digital convolution methods for the prediction of responses of an object in a system test such as vehicle/Hyge sled test, and in a component test such as body mount. The basic operation of convolution theory, the derivation of the transfer function, and an algorithm using a snowball effect to increase the computation efficiency are discussed. A dynamic system is characterized by a set of FIR coefficients, i.e., a transfer function. Applications of FIR in vehicle, occupant, and component test forward prediction (predicting the low frequency output from a high frequency input) are presented. Applications of FIR transfer functions and inverse filtering method yield the RIF (response inverse filtering) method, which is utilized to make backward prediction (predicting the high frequency output from a low frequency input). Case studies on the use of transfer functions include: (1) effect of the fullpowered and depowered air bags, (2) effect of the front and rearloaded crash pulses, (3) effect of the different body mounts and restraint systems, and (4) effect of the approximated crash pulse (such as a halfsine sled test pulse) and test crash pulse. Chapter 4 covers the basic modeling techniques using Newton's Second Law. Transient and major model responses are formulated starting with simple models using Kelvin elements to hybrid models using both Kelvin and Maxwell elements. Since any crash event involves impact and excitation, the formulas derived are applicable to the analysis of model with a slack. Factors affecting the system output, such as natural frequency, damping factor, and coefficient of restitution are described. Applications of the closed form formulas to the VOR analysis are illustrated. Chapter 5 covers the numerical methods applied to the response prediction. The solution to an impact or excitation model with more than two masses and/or nonlinear energy absorbers becomes too complex to solve in closedform. In such cases, numerical evaluation and integration techniques are necessary to solve for the dynamic responses. In a multimass model, the unloading characteristics of a spring element are as equally important as the loading characteristics. The unloading of one mass in a model may become a loading to the neighboring masses, therefore affecting the total system model responses. Power curve loading and unloading simulation with hysteresis energy loss and permanent deformation are covered. To help solve some dynamic models quickly, a lumpedparameter model, CRUSH II, is utilized. The forcedeflection formulas of some simple structures are listed for ease of determining the spring stiffness for the modeling. Some lumpedparameter models for the full frontal, side, and frontal offset impacts are described. The basic concepts of splitting a simple spring mass model for the frontal offset impact and the model validation are also presented. In Chapter 6, the principle of impulse and momentum and the principle of work and energy, derived from the Newton’s Second Law, are utilized to solve the impulsive loading problems. The CG (center of gravity) motion theorem in the multiple vehicle collision analysis and the circle of constant acceleration (COCA) on a rigid body subjected to an eccentric loading are analyzed. Specific design analysis is presented. The formulation of critical sliding velocity, rollover dynamics, and detection of an incipient rollover are introduced. Methods of determining the vehicle inertia properties, such as the CG height and the moment of inertia of a vehicle, are covered. The formulation of the critical sliding velocity (CSV), the rollover dynamics, and detection of an incipient rollover using a simple vehicle model are introduced. Chapter 7 discusses vehicle and occupant impact severity and accident reconstruction methodology. Vehicle components, such as body mounts, and engine size and location are evaluated for their roles in the absorption of vehicle energy, deceleration, and dynamic crush. Restraint devices, such as a pretensioner in a belt restraint system, are also evaluated for their values in reducing the © 2002 by CRC Press LLC severity of occupant impact. The test results, principles, and functions of the pretensioner are analyzed. The use of the damage boundary curve (DBC) in assessing the vehicle, occupant, and sensor impact is covered. In the section on accident reconstruction, the derivation of a formula used to compute the vehicle stiffness coefficients is presented and discrepancies between the builtin stiffness coefficients in the data base and those obtained from crash data are analyzed. The consequences of using improper coefficients are illustrated by drawing upon realworld accident cases. There are many aspects to vehicle crashworthiness, and it is hoped that this book will provide the fundamentals of engineering mechanics, which can be revealing in applications and will also serve as a helpful reference on uptodate techniques used in this field of study. In preparing this book I am greatly appreciative of the considerable help from my former colleague, Mr. Calvin C. Matle, a retired Senior Research Engineer at the Ford Motor Company. His critiques on the details of the subjects on mechanics were enlightening and the review of the entire manuscript was not a simple task. I am grateful that Mr. Matle created several artworks for use in the text including the car and dummy drawing on the book cover. Also, I would like to express my sincere appreciation to Dr. Clifford C. Chou, a Senior Staff Technical Specialist at the Ford Motor Company for his technical input and review of the manuscript. I would also like to gratefully acknowledge Mr. Jianming Li, an outstanding senior student at the University of Michigan, Dearborn, for his artistic skills in creating a major portion of the artwork used in this book. As inspired by those before us, and enhanced by those among us, I wish to extend many thanks to all of my students, my colleagues for their contributions. Although the list is too long to mention individually, it is hoped that those who have shared in the discussions and who have helped will accept this recognition, CRC Press staff has been very helpful and its contributions to this endeavor are gratefully recognized. Lastly, I am deeply indebted to my wife, Becky, for her everlasting patience and care over the years while I worked on this book. I am also grateful to my daughters Dr. Caroline B. Huang and Ms. Kelly M. Huang for their concerns and understandings; and to my nephew David C. Huang for his help making the contents consistent. In closing, I would like to dedicate this book to my parents who were so helpful in my life. Matthew Huang Dearborn, Michigan, USA May, 2002 © 2002 by CRC Press LLC TABLE OF CONTENTS CHAPTER 1 CRASH PULSE AND KINEMATICS 1.1 INTRODUCTION 1.2 VEHICLE IMPACT MODES AND CRASH DATA RECORDING 1.2.1 Accelerometer Mounting and Coordinate Systems 1.3 DIGITAL FILTERING PRACTICE PER SAE J211 AND ISO 6487 1.3.1 Relationship Between Two Points in a Frequency Response Plot 1.3.2 Chebyshev and Butterworth Digital Filters 1.3.3 Filter Type, Deceleration Magnitude, and Phase Delay 1.3.4 Moving Window Averaging and Equivalent Cutoff Frequency 1.3.4.1 Moving Window Averaging 1.3.4.2 Equivalent Cutoff Frequency 1.4 BASIC KINEMATIC RELATIONSHIPS 1.4.1 Computing Acceleration from a VelocityDisplacement Curve 1.4.2 Particle Kinematics in a Gravitational Field 1.4.2.1 Car Jumping and Landing 1.4.3 Slipping on an Incline & Down Push and Side Push 1.4.4 Calculation of Safe Distance for Following Vehicle 1.5 IMPACT AND EXCITATION: VEHICLE AND SLED TEST KINEMATICS 1.5.1 Vehicle Kinematics in a Fixed Barrier Impact 1.5.2 Unbelted Occupant Kinematics 1.5.2.1 Kinematics Based on Accelerometer Data 1.5.2.2 Kinematics Based on Crash Film Records 1.5.2.3 Vehicle Crush, Sled Displacement, and Crash Pulse Centroid 1.6 VEHICLE AND OCCUPANT KINEMATICS IN FIXED OBJECT IMPACT 1.6.1 Vehicle Kinematics in Different Test Modes 1.6.2 Vehicle Energy Density 1.6.3 Occupant Kinematics in Different Test Modes 1.7 KINEMATIC VARIABLES 1.7.1 Use of Residual Energy Density in Air Bag Sensor Activation 1.7.2 Time Requirement for Air Bag Sensor Activation 1.7.3 VehicleOccupantRestraint (VOR) Interaction 1.8 CASE STUDY: SINGLE VEHICLETREE IMPACT ACCIDENT 1.8.1 Analysis of the Recorder Crash Data 1.8.2 Frequency Spectrum Analysis for Electronic Crash Sensing 1.8.3 Application of a Residual Energy Density Algorithm 1.9 RESTRAINT COUPLING 1.9.1 Restraint Specific Stiffness and Onset Rate of Occupant Deceleration 1.9.2 Occupant Response in the Restraint Coupling Phase 1.9.3 Maximum Occupant Response, Timing, and Onset Rate 1.9.4 Vehicle, Occupant, and Restraint (VOR) Analysis Charts 1.9.4.1 3D Contour Plots of the Occupant Response and Timing 1.9.4.2 Vehicle, Occupant, and Restraint (VOR) Analysis Charts 1.9.5 VOR Trend Analysis Based on Car and Truck Test Results © 2002 by CRC Press LLC 1.10 OCCUPANT RIDEDOWN ANALYSIS AND ENERGY MANAGEMENT 1.10.1 Energy Density Model 1.10.1.1 Equations of Motion and Energy Density of a Crash Model 1.10.1.2 Ridedown, Restraint Energy Densities, and Timings 1.10.2 Validation of Energy Density Model in High Speed Crash 1.10.2.1 Test Energy Densities 1.10.2.2 Model Energy Densities 1.10.3 Contour Plots of Ridedown Efficiency and Occupant Response 1.10.4 Restraint Design with Constant Occupant Deceleration 1.10.5 Design Constraint and TradeOff 1.11 REFERENCES CHAPTER 2 CRASH PULSE CHARACTERIZATION 2.1 INTRODUCTION 2.2 MOMENTAREA METHOD 2.2.1 Displacement Computation Without Integration 2.2.2 Centroid Time and Characteristics Length 2.2.3 Construction of Centroid Time and Residual Deformation 2.2.3.1 Centroid of a QuarterSine Pulse 2.2.3.2 Residual Deformation of a QuarterSine 2.3 PULSE APPROXIMATIONS WITH NONZERO INITIAL DECELERATION 2.3.1 ASW (Average Square Wave) 2.3.2 ESW (Equivalent Square Wave) 2.3.2.1 ESW Transient Analysis 2.3.3 Tipped Equivalent Square Wave (TESW) – Background 2.3.4 Derivation of TESW Parameters 2.3.4.1 Deformation and Rebound Phase 2.3.5 Construction of TESW Parameters 2.3.5.1 Relationships Between TESW and ASW 2.3.6 Kinematic Comparisons of Test Pulse and Approximated Pulses 2.3.6.1 RearLoaded 2.3.6.2 FrontLoaded 2.4 PULSE APPROXIMATIONS WITH ZERO INITIAL DECELERATION 2.4.1 Fourier Equivalent Wave (FEW) 2.4.2 FEW Sensitivity Analysis with Boundary Conditions 2.4.3 Kinematics and Energy Comparison 2.4.4 Use of FEW and Power Rate Density in Crash Severity Detection 2.4.4.1 Discrimination of Pole Impact Crash Severity 2.4.4.2 Use of All Negative FEW Coefficients in Pole Tests 2.4.5 Use of Pulse Curve Length in Crash Severity Detection 2.4.6 FEW Analysis on Body Mount Attenuation 2.4.6.1 Frame Impulse Attenuation by Body Mount 2.4.7 FEW Analysis on Resonance 2.4.7.1 Air Bag Sensor Bracket Design Analysis 2.4.7.2 Resynthesis of a Crash Pulse Without Resonance 2.4.8 Trapezoidal Wave Approximation (TWA) 2.4.8.1 Deriving the Closedform Solutions for TWA Parameters 2.4.9 Bislope Approximation (BSA) 2.4.9.1 Comparison of Test Pulse, BSA, and TWA © 2002 by CRC Press LLC 2.4.10 Harmonic Pulses – Background 2.4.11 Halfsine Approximation 2.4.12 Haversine Approximation 2.4.13 Comparison of Halfsine and Haversine Pulses 2.4.14 Response of Air Bag Sensor to Harmonic Pulses 2.4.14.1 Sensor Dynamic Equations 2.4.14.2 GasDamped Sensor Mathematical Relationship 2.4.15 Head Injury Criteria 2.4.15.1 HIC Topographs 2.4.16 Application of HIC Formula in Head Interior Impact 2.5 REFERENCES CHAPTER 3 CRASH PULSE PREDICTION BY CONVOLUTION METHOD 3.1 INTRODUCTION 3.2 TRANSFER FUNCTION VIA CONVOLUTION INTEGRAL 3.2.1 Convolution Method and Applications 3.2.2 Solution by the Least Square Error Method 3.2.3 Matrix Properties and SnowBall Effect 3.2.4 Case Studies: Computing Transfer Functions 3.3 TRANSFER FUNCTION AND A SPRINGDAMPER MODEL 3.3.1 FIR Coefficients and KC Parameters of a SpringDamper Model 3.3.2 Transfer Functions of Special Pulses 3.4 BELTED AND UNBELTED OCCUPANT PERFORMANCE WITH AIR BAG 3.4.1 Test Vehicle and Occupant Responses 3.4.2 Truck #1: Unbelted Occupant with FullPowered Air Bag 3.4.2.1 Restraint FIR Model Validation Using Test Results 3.4.2.2 Filtered Signals of FIR Coefficients 3.4.2.3 Response Prediction using TWA 3.4.3 Truck #2: Belted Occupant with Depowered Air Bag 3.4.3.1 Restraint Transfer Function Validation 3.4.3.2 Response Prediction Using TWA 3.4.3.3 Response Prediction Using Fourier Equivalent Wave (FEW) 3.5 BODY MOUNT AND TORSO RESTRAINT TRANSFER FUNCTIONS 3.5.1 Body Mount Characteristics and Transient Transmissibility 3.5.2 Types F and T Body Mount Transfer Functions 3.5.3 Body Response Prediction of Truck T with Type F Body Mount 3.5.3.1 Frame Impulse Duration and Transient Transmissibility 3.5.3.2 Testing Frame Rail for a Desired Impulse Duration 3.5.4 Torso Restraint Transfer Functions 3.5.4.1 Vehicle and Belted Occupant Performances in Trucks F and T 3.5.4.2 Truck T Response Prediction with Truck F Restraints 3.6 EFFECT OF SLED AND BARRIER PULSES ON OCCUPANT RESPONSE 3.7 OTHER APPLICATIONS 3.8 RESPONSE INVERSE FILTERING (RIF) 3.8.1 Forward Prediction by Finite Impulse Response (FIR) 3.8.2 Inverse Filtering (IF) 3.8.3 Crash Pulse Prediction using FIR and RIF 3.8.3.1 Transferring [X] to [Y] with [H] 3.8.3.2 Transfer [Y] to [X] with [H]N © 2002 by CRC Press LLC 3.8.3.3 Transferring [Y] to [X] using [IF] 3.8.4 RIF Application in Frame Pulse Prediction 3.9 REFERENCES CHAPTER 4 BASICS OF IMPACT AND EXCITATION MODELING 4.1 INTRODUCTION 4.2 IMPACT AND EXCITATION – RIGID BARRIER AND HYGE SLED TESTS 4.2.1 Vehicle and Sled/Unbelted Occupant Impact Kinematics 4.2.1.1 A VehicletoBarrier Displacement Model 4.2.1.2 Unbelted Occupant Kinematics 4.3 RIDEDOWN EXISTENCE CRITERIA AND EFFICIENCY 4.3.1 Vehicle and Occupant Transient Kinematics 4.3.1.1 EOM for Vehicle 4.3.1.2 EOM for Occupant 4.3.2 Derivation of Ridedown Existence Criteria 4.3.2.1 Method 1 4.3.2.2 Method II 4.3.3 Application of Ridedown Existence Criteria 4.3.3.1 Case Study – High Speed Crash 4.3.3.2 Case Study – Low Speed Crash 4.3.4 Occupant Response Surface and Sensitivity 4.3.4.1 Restraint Design Optimization by Response Contour Plots 4.3.4.2 Sensitivity of Occupant Response to ESW 4.3.4.3 Sensitivity of Occupant Response to Dynamic Crush 4.3.4.4 Statistical Regression of Test Data and Model Responses 4.3.4.5 Response Prediction and Ridedown Efficiency 4.4 BASICS OF SPRING AND DAMPER DYNAMIC MODELING 4.4.1 Spring and Damper Elements 4.4.2 Properties of Viscoelastic Materials and Damping 4.4.2.1 Equivalent Viscous Damping 4.4.3 2Mass (VehicletoVehicle) Impact Model 4.4.4 Dynamic Equivalency Between TwoMass and Effective Mass Systems 4.5 VEHICLE TO BARRIER (VTB) IMPACT: SPRINGMASS MODEL 4.5.1 Model Formulation 4.5.2 Design and Trend Analysis 4.5.2.1 Acceleration Function 4.5.2.2 Dynamic Crush Function 4.5.2.3 Estimating Time of Dynamic Crush, Tm 4.5.2.4 Response Properties as a Function of V and C 4.5.2.5 Mass and Stiffness Ratios in VehicletoVehicle Impact 4.5.3 Effect of Test Weight Change on Dynamic Responses 4.6 SPRINGMASS OCCUPANT MODEL SUBJECTED TO EXCITATION 4.6.1 Response Solutions due to TESW and Sinusoidal Excitation 4.6.1.1 Model with TESW Excitation, (E + j t) 4.6.1.2 Sine Excitation (E sin Tt) 4.6.2 Model Response due to Sinusoidal Displacement Excitation 4.7 VEHICLETOVEHICLE (VTV) IMPACT: SPRINGMASS MODEL 4.7.1 Crash Pulse Approximation by TESW and Sinusoidal Waves 4.7.1.1 Relative Motion Analysis (An Effective Mass System) © 2002 by CRC Press LLC 4.7.1.2 Individual Vehicle Response Analysis 4.7.2 Comparison of Sinusoidal Wave with Test Crash Pulse 4.7.3 Truck and Car Occupant Responses due to Halfsine Excitation 4.7.4 Elastoplastic Modeling 4.8 A MAXWELL MODEL 4.8.1 A DamperMass System (without Oscillatory Motion) 4.8.2 The Maxwell SpringDamper Model 4.8.3 Alternate Method: Zero Mass Between Maxwell Spring and Damper 4.8.4 Transition and Infinite Damping Coefficients 4.8.4.1 Transition Damping Coefficient, c* 4.8.4.2 Infinite Damping Coefficient, c=4 4.8.5 Model Response Characteristics with Transition Damping Coefficient 4.9 IMPACT ON KELVIN MODEL!VEHICLE OR COMPONENT 4.9.1 Transient and Major Responses of Kelvin Model 4.9.1.1 Underdamped System (. < 1) 4.9.1.2 Critically Damped System (. = 1) 4.9.1.3 Overdamped System (. > 1) 4.9.1.4 Normalized Response Comparisons of Three Damping Systems 4.9.2 Factors Affecting the Pulse Shape of System with Various Damping 4.9.3 Hysteresis Loop 4.9.4 Coefficient of Restitution and Damping Factor (.) 4.9.5 Contact Duration 4.10 DAMPING FACTOR AND NATURAL FREQUENCY FROM TESTS 4.10.1 Conversions of the Stiffness and Damping Coefficient 4.10.2 Application to SUV and Sedan Frontal Structure Properties 4.11 EXCITATION OF THE KELVIN MODEL — OCCUPANT AND RESTRAINT 4.11.1 General Crash Pulse Excitation by Fourier Series 4.11.1.1 Testing the Haversine Excitation 4.11.2 Effect of Restraint Damping Control on Occupant Response 4.12 REFERENCES CHAPTER 5 RESPONSE PREDICTION BY NUMERICAL METHODS 5.1 INTRODUCTION 5.2 HYBRID MODEL — A STANDARD SOLID MODEL 5.2.1 E.O.M. for Hybrid Model 5.2.2 Dynamic Response and Principles of Superposition 5.2.3 Combination of Two Hybrid Models 5.2.4 Dynamic Equivalency between Two NonIsomorphic Hybrid Models 5.2.4.1 Dynamic Equivalency in Transient Kinematics and Crush Energy 5.3 TWO MASSSPRINGDAMPER MODEL 5.3.1 Solutions of the Characteristic Equation 5.3.2 Vehicle Displacement Responses in Fixed Barrier Impact 5.3.3 Application in PreProgram Vehicle Structural Analysis 5.3.4 Application in PostCrash Structural Analysis 5.4 NATURAL FREQUENCIES IN TWO–MASS SYSTEM 5.4.1 Formulas for the Natural Frequencies 5.4.1.1 Decoupling of a TwoMass System 5.4.2 Natural Frequency Ratio and Stiffness Computation 5.4.3 AddOn or Splitting of a SpringMass Model © 2002 by CRC Press LLC 5.4.3.1 DoubledUp of a SpringMass Model 5.4.3.2 Splitting of a SpringMass Model 5.5 NUMERICAL SEARCHING TECHNIQUES 5.5.1 Imbedded Random Search (IRS) 5.5.2 NewtonRaphson Search Algorithm 5.6 LOADING AND UNLOADING SIMULATION 5.6.1 Loading Phase Simulation 5.6.2 Unloading Phase Simulation 5.6.3 Model with Power Curve Loading and Unloading 5.6.3.1 Unloading Parameters k', n', and xi in Reloading Cycle 5.6.3.2 Deceleration Contributions of Spring and Damper 5.7 A LUMPEDPARAMETER MODEL — CRUSH II 5.7.1 Simple Structure ForceDeflection Table 5.7.2 Push Bumper ForceDeflection Data 5.7.3 Basic Operation of EA Types 5.7.4 Basic Operation of CV Factor (Velocity Sensitive Factor) 5.7.5 Coefficient of Restitution, Static, and Dynamic Crush Relationship 5.7.5.1 1mass Model with ElastoPlastic Spring 5.8 SIDEIMPACT AND FRONTAL OFFSET MODELS 5.8.1 Side Impact Model 5.8.2 Frontal Offset Impact 5.8.2.1 Basic Concepts in Offset Impact Modeling 5.8.2.2 Full Barrier and Frontal Offset Test Results 5.8.2.3 Modeling the Full Barrier and Frontal Offset Tests 5.8.2.4 Optimal Vehicle Structure for Both Full Frontal and Offset Tests 5.8.2.5 An Offset LumpedMass Model 5.9 REFERENCES CHAPTER 6 IMPULSE, MOMENTUM, AND ENERGY 6.1 INTRODUCTION 6.2 BACKGROUND 6.2.1 Impulse and Momentum for a Single Particle 6.2.2 Impulse and Momentum for a System of Particles 6.3 CENTER OF GRAVITY AND MOTION THEOREM 6.3.1 Location and Motion of Center of Mass 6.3.2 Conservation of Momentum and CG Formula 6.3.3 CG Motion Theorem 6.3.4 Use of CG Motion Theorem in a Three!Car Collision Analysis 6.4 IMPULSE AND CIRCLE OF CONSTANT ACCELERATION 6.4.1 Derivation of Acceleration at Point Q 6.4.2 Circle of Constant Acceleration (COCA) 6.4.3 Construction of COCA Given the Acceleration Ratio, c 6.4.4 COCA Case Studies 6.4.5 Determination of the Direction of Acceleration, aQ 6.4.6 COCA Evaluation of Impact Severity 6.4.7 Given the Coordinates of Point Q, Find the Acceleration Ratio c 6.4.8 Distributed Loading by Superposition 6.5 PRINCIPLE OF WORK AND ENERGY 6.5.1 Applications using Principle of Impulse, Momentum, and Energy © 2002 by CRC Press LLC 6.5.2 Drop Test and Impact Using a Spring Having Finite Weight 6.5.2.1 Drop Test on a Weightless Spring 6.5.2.2 Drop Test Using a Spring Having Finite Weight 6.5.2.3 Horizontal Impact on a Bar/Spring 6.5.2.4 Vertical Impact on a Beam/Spring 6.5.3 Rebound Criterion in a TwoMass Impact 6.5.4 Separation Kinematics in a MultiMass Impact 6.5.4.1 Separation Kinematics in a 3Vehicle Collision 6.5.5 COR, Times of Dynamic Crush, and Separation Time 6.5.6 Coefficient of Restitution and Stiffness in Vehicle Crashes 6.6 VEHICLE INERTIA PROPERTIES AND CRITICAL SLIDING VELOCITY 6.6.1 CG Height Determination 6.6.2 Moment of Inertia Using Trifilar Pendulum Method 6.6.3 Moment of Inertia Using Swinging Pendulum Method 6.6.4 Critical Sliding Velocity (CSV) 6.6.4.1 Derivation of CSV Formulas 6.6.4.2 Normalized CSV Equation and Applications 6.7 ROLLOVER CRASHES 6.7.1 Rollover Dynamics of a Rigid Vehicle in a Steady Turn 6.7.2 Rollover Detection and Threshold Criterion of a Rigid Vehicle 6.7.3 Transient Rollover Dynamics of a Rigid Vehicle 6.7.3.1 Transient Rollover Without Lateral Acceleration 6.7.3.2 Transient Rollover With Lateral Acceleration 6.7.4 Rollover and Yaw Detections 6.8 ECCENTRIC LOADING ON VEHICLE ROLLOVER 6.8.1 Vector Method for Eccentric Loading Analysis 6.8.2 Rollover Kinematics Using the Vector Method 6.8.3 Conditions for a Vehicle to Stop Rolling Following Rollover 6.9 REFERENCES CHAPTER 7 CRASH SEVERITY AND RECONSTRUCTION 7.1 INTRODUCTION 7.2 OCCUPANT MOTION UNDER IMPACT AND EXCITATION 7.2.1 TwoDegreeofFreedom Occupant Model 7.2.2 Effect of Seat Belt and Pretensioner on Occupant Kinematics 7.3 PRELOADING ON AN OCCUPANT 7.3.1 Modeling Pretensioning Effects in a System Test 7.3.2 Modeling Pretensioning Effects in a Component Test 7.3.3 Transient Analysis of a Preloaded Model — Impact and Excitation 7.4 CENTRAL COLLISIONS 7.4.1 A Collision Experiment 7.4.2 Relative Motion During Impact 7.4.3 Kelvin’s Theorem, Total Crush, and Dissipated Energies 7.4.4 Total Crush Energy 7.4.5 Individual Crush Energy 7.5 NONCENTRAL COLLISIONS 7.5.1 Case Study 1: Central Collision 7.5.2 Case Study 2: Noncentral or Offset Collision 7.6 USE OF )V AND BEV IN CRASH SEVERITY ASSESSMENT © 2002 by CRC Press LLC 7.6.1 Crash Severity Index 7.6.1.1 Compatibility by Equal Crash Severity Index 7.6.2 Crash Momentum Index 7.6.3 Crash Severity Assessment by a Power Curve Model 7.6.3.1 Power Curve Model and Methodology 7.6.3.2 Power Curve ForceDeflections 7.6.3.3 Computation of Barrier Equivalent Velocity (BEV) 7.7 VEHICLE ACCELERATION AND CRASH SEVERITY 7.7.1 Damage Boundary Curve 7.7.1.1 Construction Steps for DBC 7.7.1.2 Mechanic Principles of DBC 7.7.2 Crash Severity Assessment in VehicletoVehicle Compatibility Test 7.7.2.1 Vehicle Crush Characteristics 7.7.2.2 Vehicle Peak Responses 7.8 VELOCITY AND ENERGY DISTRIBUTIONS IN TWOVEHICLE IMPACT 7.8.1 Kelvin’s Theorem 7.8.2 Lumped Mass Modeling on Crash Severity 7.9 INTERMEDIATE MASS EFFECT 7.10 MODELING THE VEHICLETOVEHICLE COMPATIBILITY TEST 7.10.1 Models with Same Effective Stiffness 7.10.2 Models with Different Effective Stiffness 7.11 ACCIDENT RECONSTRUCTION METHODOLOGY 7.11.1 Background 7.11.2 Vehicle Size and Stiffness Coefficient Categories 7.11.2.1 Computing Stiffness Coefficients, Intercept and Slope 7.11.3 Stiffness Coefficient Comparison Between Data Base and Crash Tests 7.11.4 FourWay Plot of Stiffness Coefficients and Responses 7.11.5 NonLinear Crush Profile and Force Deflection Data 7.11.5.1 NonLinear Crush Profile 7.11.5.2 ElastoPlastic Force Deflection 7.11.5 Estimate of the Impact Severity and Sensor Performance in a Van Impact 7.11.5.1 Estimate of the Vehicle Impact Severity 7.11.5.2 Estimate of the Sensor Performance 7.12 REFERENCES LIST OF FIGURES UNIT CONVERSIONS © 2002 by CRC Press LLC LIST OF FIGURES CHAPTER 1 CRASH PULSE AND KINEMATICS 1.1 Unitized Body Vehicle 1.2 BodyonFrame Vehicle 1.3 A Typical Body Mount on a BodyonFrame Vehicle 1.4 Crash Test Sensor and Accelerometer Locations 1.5 Crash Test Sensor/Accelerometer Locations 1.6 Crash Test Mode ! 1 1.7 Crash Test Mode ! 2 1.8 Accelerometer Types and Schematic 1.9 Vehicle Coordinate System 1.10 Occupant Coordinate System 1.11 SAE J211 Frequency Response Corridor 1.12 Piano Keys Covering One Octave 1.13 Case Study: Frequency Response Corridor 1.14 Butterworth nth Order Filter 1.15 Chebyshev nth Order Filter 1.16 Butterworth nth Order Passband Response Function 1.17 Chebyshev nth Order Passband Response Function 1.18 Effects of Filter on Magnitude and Phase Delay 1.19 Filtered Response Comparison ! SingleStep Function and Channel Class 60 1.20 Filtered Response Comparison ! MultipleStep Function and Channel Class 60 1.21 CloseUp of Filtered and Wideband Crash Pulse Comparison 1.22 Vehicle Pulse Filtered by Channel Class 60 ! Butterworth and Chebyshev 1.23 Chest Decel. Filtered by Channel Class 180 ! Butterworth and Chebyshev 1.24 Revised Transition Bands for Channel Class #3 & #4 per SAE J211, March 1995 1.25 Crash Test Data (Acceleration) with Moving Window Averaging 1.26 A Truck Crash Test Raw Data with Moving Window Averaging 1.27 1st and 2nd Integrals of Crash Test Raw Data with Moving Window Averaging 1.28 Crash Pulse Comparison between Window Averaging and the Butterworth Filter 1.29 Body Pulse With Butterworth 100 Hz Cutoff and 61bite Averaging 1.30 Raw Data of Body Pulse of a MidSize Car Struck by a SUV at 58 mph 1.31 Velocity vs. Displacement Curve 1.32 Unbelted Occupant Relative a vs. t 1.33 Unbelted Occupant Relative a vs. d 1.34 Unbelted Occupant Relative Velocity vs. Relative Displacement in Two Scales 1.35 Car Jump & Particle Kinematics 1.36 Normalized TakeOff Velocity 1.37 A Quantity to Normalize Velocity 1.38 Min. Norm. Velocity To Go Over an Obstacle 1.39 Trajectories with Different Ramp Angles and Velocities To Clear an Obstacle 1.40 Perfect Landing Without a Crash 1.41 Geometric Relationship for Perfect Landing 1.42 Slipping on an Incline due to Side or Down Push 1.43 Slipping on an Incline due to Side Push 1.44 Slipping on an Incline due to Down Push 1.45 Normalized Push to Slide on an Incline 1.46 Safe Distance 1.47 Radar Braking Velocity Diagram © 2002 by CRC Press LLC 1.48 Rocker BPost Kinematics of a MidSize Car in a 14 mph Rigid Barrier Test 1.49 Sled Test Set Up 1.50 Unbelted Occupant Kinematics in a MidSize Sedan 14 mph Barrier Test 1.51 MidSize Sedan 14 mph Rigid Barrier and Sled Test Kinematics 1.52 VehicleBarrier 14 mph Crash and Sled (Occupant freeflight) Displacement 1.53 Dummy Seating Position 1.54 Unbelted Driver Motion from Crash Test Film 1.55 Vehicle Crush, Sled Displacement, and Centroid 1.56 Vehicle Deceleration in Two Barrier and One Pole (21mph) Crash Tests 1.57 Vehicle Velocity vs. Time in Three Crash Tests 1.58 Vehicle Displacement vs. Time in Three Crash Tests 1.59 Vehicle Energy Densities in Three Crash Tests 1.60 Vehicle Front Center Pole Test Setup 1.61 Vehicle Front Center Pole Post Test 1.62 Unbelted Occupant Velocity vs. Displacement in Three Crash Tests 1.63 Unbelted Occupant Displacement vs. Time in Three Crash Tests 1.64 Special Pulses 1.65 Sensor Activation Threshold Window 1.66 A Single Point Crash Sensing Algorithm 1.67 Air Bag Deployment Sequence 1.68 The Car and Right Front Damage 1.69 The tree 1.70 The Driver and Bags 1.71 Vehicle Deceleration vs. Time from Flight Recorder (Sampling Rate = 2k Hz) 1.72 Deceleration vs. Time for Low (Test #1) and High (Test #2) Speed Pole Tests 1.73 Deceleration Comparison of Case and Test #1 Vehicles 1.74 Deceleration Comparison of Case and Test #3 (High Speed CarCar) Vehicles 1.75 Unbelted Occupant Displacement vs. Time 1.76 Unbelted Occupant )V vs. Time 1.77 Unbelted Occupant )V vs. Displacement 1.78 Fourier Power Spectrum of the Case, Tests #3 and #4 1.79 Residual Energy Density (red) vs. displacement 1.80 A Simple OccupantVehicle Impact Model 1.81 Truck and Chest Decelerations vs. Time in a 35 mph Barrier Test 1.82 Restraint Slack and Stiffness of an Air Bag System in a Truck 35 mph Test 1.83 Chest Onset Rate (j) in a Truck 35 mph Fixed Barrier Test 1.84 A Sled Impact Model 1.85 A Constant Sled Excitation Pulse 1.86 Model Chest g as a Function of Restraint, f; Slack, *; and ESW 1.87 Model Chest g Time as a Function of Restraint, f; Slack, *; and ESW 1.88 VOR Chart #2 – Example 1.89 VOR Chart #1 – Example 1.90 VOR Chart #3 – Example 1.91 VOR Chart #4 – Example 1.92 VOR Chart #2 1.93 VOR Chart #1 1.94 VOR Chart #3 1.95 VOR Chart #4 1.96 Comparison of Responses: Model vs. Tests 1.97 Chest g as a Function of Restraint, f; Slack, *; and Four Crash Test Data 1.98 Chest g vs. Dynamic Crush in 31 mph Rigid Barrier Tests 1.99 Chest g vs. Restraint Natural Frequency in 31 mph Rigid Barrier Tests © 2002 by CRC Press LLC 1.100 Chest g vs. Restraint Slack in 31 mph Rigid Barrier Tests 1.101 Left Front Occupant Seating Position in a VehicletoBarrier Impact 1.102 Decelerations of Driver Left Femur and Truck in 31 mph Barrier Test 1.103 Displacements of Driver Left Femur and Truck in 31 mph Barrier Test 1.104 Restraint and Ridedown Curves of Left Femur in 31 mph Test 1.105 Restraint and Ridedown Energy Densities of Left Femur in 31 mph Test 1.106 A Crash Model 1.107 Test Decelerations of a Truck and L.F. Torso 1.108 Test Displacements of a Truck and L.F. Torso 1.109 Test Restraint and Ridedown Curves 1.110 Test Restraint and Ridedown Energy Densities 1.111 Test and Model Comparisons of Vehicle and Torso Decelerations 1.112 Test and Model Comparisons of Restraint and Ridedown Curves 1.113 Test and Model Comparisons of Restraint and Ridedown Energy Densities 1.114 Ridedown Efficiency Contour Plot (*=6 in) 1.115 Femur Deceleration Contour Plot (*=6 in) 1.116 Ridedown Efficiency Contour Plot (*=0 in) 1.117 Femur Deceleration Contour Plot (*=0 in) 1.118 Energy Densities versus Time for 3 Cases with Restraint Slack = 0 1.119 Occupant Acceleration vs. Restraint Deformation 1.120 Restraint Energy Density vs. Deformation 1.121 Constant Energy Knee Bolster Force vs. Deflection CHAPTER 2 CRASH PULSE CHARACTERIZATION 2.1 MomentArea Method and Displacement Equation 2.2 Centroid Time of a MidSize Passenger Car (slope of C vs. V line) 2.3 Centroid Location and Residual Deformation 2.4 Chest g vs. RD in 31 mph Rigid Barrier Tests 2.5 Derivation of x & y Coordinates of QuarterSine Centroid 2.6 Even, Extremely Rear, and FrontLoaded Pulses 2.7 TESW Parameters 2.8 Aavg as a Function of )Vm and tm 2.9 TESW Deceleration as a Function of Aavg and tc/tm 2.10 Front / RearLoaded Pulses and Integrals 2.11 Even / RearLoaded Pulses and Integrals 2.12 Crash Pulse Comparison (rearloaded) 2.13 Velocity Comparison (rearloaded) 2.14 Displacement Comparison (rearloaded) 2.15 Deceleration vs. Displacement Comparison 2.16 Energy Density vs. Displacement Comparison 2.17 Crash Pulse Discrete Data Points 2.18 Pulses of Truck Test at 31 mph and FEW with and without Modification 2.19 Velocities of a Truck Test at 31mph and FEW w/ and w/o Modification 2.20 Displacements of Truck Test at 31 mph and FEW w/ and w/o Modification 2.21 Deceleration vs. Displacement 2.22 Energy Density vs. Displacement of Truck Test at 31 mph and FEW 2.23 A Simple Halfsine Wave and its Power Rate Density (prd) 2.24 Test and FEW Decelerations at Tunnel of Two MustNotActivate Tests 2.25 Test and FEW Decelerations at Tunnel of Two MustActivate Tests 2.26 Power Rate Density With Original FEW Coefficients of Four MidSize Tests 2.27 Power Rate Density Curves With AllNegative FEW Coefficients 2.28 Effect of AllNegative FEW Coefficients on #1' and #4' Decelerations © 2002 by CRC Press LLC 2.29 Test and FEW Decelerations at Rocker/BPillar of Three FullSize Tests 2.30 Power Rate Density with Original FEW Coefficients of Three Tests 2.31 Power Rate Density Curves with AllNegative FEW Coefficients 2.32 Crash Pulse Data Points (from 2.17) 2.33 Pulse Curve Length vs. Time from Four Sets of Crash Data Filtered at 100 Hz 2.34 Pulse Curve Length vs. Time from Four Sets of Crash Data Filtered at 300 Hz 2.35 Truck Body on Frame 2.36 A Typical Body Mount on a BodyonFrame Vehicle 2.37 Frame Wideband Data 2.38 Frame Data Filtered by Butterworth at 100 Hz Cutoff Freq 2.39 FEW Coefficients for the Frame Data 2.40 Body Deceleration Wideband Data 2.41 Body Data Filtered by Butterworth at 100 Hz Cutoff Freq 2.42 FEW Coefficients for the Body Data 2.43 Frame and Body Filtered Crash Pulses (Butterworth rolloff freq. = 100 Hz) 2.44 Frequency Spectrum Magnitude of Frame and Body Filtered Crash Pulses 2.45 A Sensor Module bracket 2.46 Coefficients of Fourier Equivalent Wave at ECS Module 2.47 Bracket 100 Hz Resonance 2.48 Velocity and Displacement Changes of the Resonance A2(t) 2.49 Composition of Accelerometer Data at ECS: Signal A1 and Resonance A2 2.50 Velocity and Displacement of the FEW With and Without Resonance A2(t) 2.51 Trapezoidal Wave Approximation and Its Integrals 2.52 BiSlope Approximation (BSA) 2.53 Crash Pulse Comparison between Test and BSA (BiSlope Approximation) 2.54 Velocity Comparison between Test and BSA (BiSlope Approximation) 2.55 Displacement Comparison between Test and BSA (BiSlope Approximation) 2.56 Harmonic Pulses 2.57 Normalized Halfsine, Haversine and Velocity Changes 2.58 Halfsine 2.59 Haversine 2.60 Triangle 2.61 Harmonic Pulses with Same Magnitude 2.62 Harmonic Pulses with Same Velocity Change 2.63 Halfsine: Relative Centroid Location vs. b 2.64 Halfsine: Normalized Peak Deceleration 2.65 Halfsine Normalized Peak Deceleration 2.66 Halfsine: Normalized Deceleration at tm 2.67 Haversine: Relative Centroid Location vs. b 2.68 Haversine: Normalized Peak Deceleration 2.69 Haversine: Normalized Time at Peak 2.70 Halfsine: Normalized Deceleration at tm 2.71 Crash Pulse Comparison between Test, Halfsine, and Haversine Waves 2.72 Velocity Comparison between Test, Halfsine, and Haversine Waves 2.73 Displacement Comparison between Test, Halfsine, and Haversine Waves 2.74 An Air bag BallinTube (BIT) Crash Sensor 2.75 BallinTube Crash Sensor Components 2.76 Input Pulses to a BIT Model 2.77 Sensor Velocity Changes at Activation Times 2.78 Sensor Ball Displacements 2.79 Calibration of a 10 mph Crash Sensor 2.80 A Sensor Model © 2002 by CRC Press LLC 2.81 HIC Formula for Three Tolerance Regression Lines 2.82 Halfsine 2.83 Haversine 2.84 Triangle 2.85 Square 2.86 HIC Topograph of a Square Pulse 2.87 HIC Topograph of a Triangular Pulse 2.88 HIC Topograph of a Halfsine Pulse 2.89 HIC Topograph of a Haversine Pulse 2.90 HIC Topograph of a R/F Dummy in a Truck 35 mph Barrier Test 2.91 Relationship between HIC, Impact Velocity, and Crush Space CHAPTER 3 CRASH PULSE PREDICTION BY CONVOLUTION METHOD 3.1 A Transfer Function – A Convolution Process 3.2 A Dynamic System with Multiple Transfer Functions 3.3 Input Discrete Data Points and FIR Coefficients 3.4 FIR Prediction with M=5 and Constant Input 3.5 FIR Prediction with M=9 and Constant Input 3.6 FIR Coefficient Comparison with M=5 and 9 ()t=12.5 ms) 3.7 A springdamper model subjected to an input excitation, p(t) 3.8 Input P(s), Output Q(s), and Transfer Function H(s) in S Domain 3.9 Model FIR coefficients with f = 10 Hz and .= 0 and 0.3, respectively ()t=.8 ms) 3.10 Output Responses of a SpringDamper Model with Square Wave Input 3.11 Constant Input and Triangular Output 3.12 FIR Coefficients Transferring Constant Input to Triangular Output 3.13 Torso Restraint Curves (Decel. vs. Relative Disp.) for Two Trucks 3.14 Torso Ridedown Efficiencies in Trucks #1 and #2 3.15 A Steering Column and Local Coordinate System 3.16 Steering Column Transient Loading in Truck #1 and #2, 31 mph Test 3.17 Truck #1/Torso G: Test(x,y), Validation(x, y^), and FIR Prediction (xTWA, y^TWA) 3.18 Truck #1/Femur G: Test(x,y), Validation(x, y^), and FIR Prediction (xTWA, y^TWA) 3.19 Truck #1: FIR Coefficients of Air Bag Restraint System 3.20 Truck #1: FIR Coefficients of Knee Bolster Restraint System 3.21 ButterworthFiltered FIR Coefficients of Truck #1 3.22 Truck #2: Test, Validation, and Prediction of Torso Response using TWA 3.23 Filtered Signals of FIR Coeff. of Truck #1 and #2 with Air Bag Restraints 3.24 Truck #2: Test, Validation, and Prediction of Torso Response using (FEW) 3.25 Two Body Mount Locations 3.26 Type F Body Mount (= 2.36) 3.27 Type T Body Mount 3.28 Body Mount Deformation (Type F) 3.29 Material Damping Properties 3.30 Force Transmissibility as a Function of Damping Factor and Frequency Ratio 3.31 TT of the Body Mount in Truck F 3.32 Type F Body Mount Deformation in Truck #1 35 mph test 3.33 3D Plot of TT, a Function of f and ., Given Haversine with )T=10 ms 3.34 Contour Plot of TT, a Function of f and ., Given Haversine with )T = 10 ms 3.35 Body Mount FIR Coefficients and KC Parameters of Trucks F and T 3.36 Body Response of Truck T with Type F Body Mount 3.37 3D Contour Plot of TT in Terms of . and )T of Frame Impulse 3.38 Haversine Frame Crush versus Peak Deceleration amd Duration –Impact 3.39 Haversine Displacement versus Peak Deceleration and Duration – Excitation © 2002 by CRC Press LLC 3.40 Vehicle and Occupant Responses of Truck F in a 35 mph Barrier Impact 3.41 Vehicle and Occupant Responses of Truck T in a 35 mph Barrier Impact 3.42 Left Front Occupant Restraint and Ridedown Curves of Both Trucks F and T 3.43 Right Front Occupant Restraint and Ridedown Curves of Both Trucks F and T 3.44 Transfer Functions of Left and Right Torso Restraints of Truck F 3.45 Prediction of LF Chest g in Truck T using LF T.F. from Truck F 3.46 Prediction of Right Front Chest G in Truck T using RF T.F. from Truck F 3.47 Kinematics of Barrier and Sled Test Pulses with Initial Velocity 3.48 Kinematics of Barrier and Sled Test Pulses without Initial Velocity 3.49 Chest g Comparisons between the FIR Model and Barrier Test 3.50 A Driver Restraint Transfer Function in a 35 mph SedanBarrier Test 3.51 Barrier and FIR Model Chest g 3.52 31 MPH 30o Right Angular Barrier Test 3.53 3D Plot of Transfer Function [H] from Body to Frame 3.54 Validation on [H]: Transferring Frame (x) Pulse to Body (y) Pulse 3.55 [H] (xYy): N=35, M=29, )t=2.8ms 3.56 Validation on [H]N: Transferring Body (y) Pulse to Frame (x) Pulse 3.57 [H]N (yYx): N=35, M=29, )t=2.8ms 3.58 Validation on [IF], N=93, M=80, )t=1.28ms 3.59 [IF] Function: N=35, M=29, )t=2.8ms 3.60 Frame Pulse by Inverse Filtering on the Target Body Pulse #1 (Case I) 3.61 BodyFrame Displacements of Test and Target Body Pulse #1 (Case I) 3.62 Frame Pulse by Inverse Filtering on the Target Body Pulse #2 (Case II) 3.63 BodyFrame Displacements of Test and Target Body Pulse #2 (Case II) CHAPTER 4 BASICS OF IMPACT AND EXCITATION MODELING 4.1 Truck Kinematics in 35 mph Barrier and Sled Tests 4.2 Displacements of a Truck in 35 mph Barrier and Sled Tests 4.3 Vehicle and Sled Accelerations: Haversine and Triangular Pulses 4.4 Velocity vs. Time: Haversine and Triangular Pulses 4.5 Displacement vs. Time: Haversine and Triangular Pulses 4.6 Vehicle and Sled Displacements of a Truck in 35 mph Test 4.7 A SpringMass Vehicle Model 4.8 Displacements of a Sedan at Three Speeds in Rigid Barrier Tests 4.9 Normalized Vehicle Displacements: Model and Test at Three Speeds 4.10 Normalized Vehicle and Sled Displacements vs. Normalized Time 4.11 Unbelted Dummy and Vehicle Motion in a 14 mph Barrier Test 4.12 Vehicle and Occupant Kinematics in a Frontal Rigid Barrier Test 4.13 Normalized Contact Velocity vs. Restraint Slack (*) and Dynamic Crush (C) 4.14 Vehicle Kinematics in a 14 mph Rigid Barrier Test 4.15 Unbelted Occupant Kinematics in a 14 mph Rigid Barrier Test 4.16 Occupant Deceleration vs. Contact Velocity (v*) and ESW 4.17 Window of V* and ESW for Constant 40 g Occupant Deceleration 4.18 Restraint Slack Constraint by v* and ESW 4.19 Chest G Sensitivity vs. ESW and Slack (*) 4.20 Constant Chest G Sensitivity vs. ESW and * 4.21 Chest G Sensitivity vs. Dynamic Crush (C) and * 4.22 Constant Chest G Sensitivity vs. C and * (31 mph) 4.23 Constant Chest G Sensitivity vs. C and * (35 mph) 4.24 Vehicle/Occupant Acceleration in a 30 mph Test (*=5", f=7 Hz) 4.25 Vehicle/Occupant Velocity 4.26 Vehicle/Occupant Displacement © 2002 by CRC Press LLC 4.27 Plane View of Occupant Interior Travel at 3 Times to, t*, and tm 4.28 (a) Spring, (b) Damper, (c) Kelvin, and (d) Maxwell Elements 4.29 A MacPherson Strut – A Kelvin Element 4.30 Schematic Representation of Spring and Damper Elements 4.31 VehicletoVehicle Impact Model – Two Kelvin Elements in Series 4.32 VehicletoVehicle Impact Model – A Kelvin Model 4.33 A TwoMass System 4.34 An EffectiveMass System 4.35 3D Surface Plot of Peak Acceleration of a SpringMass Model 4.36 3D Surface Plot of Dynamic Crush as a Function of Weight and Stiffness 4.37 Time of Dynamic Crush for a SpringMass System 4.38 Tm as a Function of Weight and Stiffness 4.39 Mass Effect on Vehicle Acceleration Response 4.40 Mass Effect on Vehicle Velocity and Displacement Responses 4.41 a vs. t of a FullSize Car in 8 and 14 mph Barrier Tests 4.42 v vs. t of a FullSize Car in 8 and 14 mph Barrier Tests 4.43 d vs. t of a FullSize Car in 8 and 14 mph Barrier Tests 4.44 An Occupant Model Subjected to An Excitation 4.45 Occupant Relative Kinematics at Restraint Contact Time 4.46 A Sled Impact Model 4.47 Occupant Decelerations due to Front and RearLoaded Pulse Excitation 4.48 DAF vs. Frequency Ratio and Contact Time 4.49 Time at Maximum DAF (Restraint Freq.= 8 Hz) 4.50 DAF due to Sinusoidal Excitation (* = 0) 4.51 SpringMass Model Under Impact and Sinusoidal Excitation 4.52 A TrucktoCar Central Impact 4.53 Deceleration ! A TwoMass System 4.54 Deceleration ! An Effective Mass System 4.55 Halfsine: Relative Centroid Location vs. b 4.56 Halfsine: Normalized Peak Deceleration 4.57 Halfsine: Normalized Time at Peak Deceleration 4.58 Halfsine: Normalized Deceleration at tm 4.59 Velocity ! A TwoMass System 4.60 Velocity ! An Effective Mass System 4.61 Displacement ! A TwoMass System 4.62 Displacement ! An Effective Mass System 4.63 Chest G from Test, and kc Model w/ Halfsine Excitation (. = 0, * = 1.1") 4.64 F vs. D of ElastoPlastic Spring 4.65 Force/Deflection vs. Time of an Impact with Elastic and ElastoPlastic Springs 4.66 Deceleration vs. Time of an Impact with Elastic Spring 4.67 Deceleration vs. Time of an Impact with ElastoPlastic Spring 4.68 A DamperMass System 4.69 Normalized Responses of a DamperMass System 4.70 A Maxwell Model 4.71 Maxwell Model with Zero Mass 4.72 Maxwell Model Accelerations with Damping and Various Stiffness 4.73 Maxwell Model Deflection w/ Stiff Spring 4.74 Maxwell Model Deflection with Regular Spring 4.75 Maxwell Model Deflection with Soft Spring 4.76 Maxwell Model Velocity with Stiff Spring 4.77 Maxwell Model Velocity with Regular Spring 4.78 Maxwell Model Velocity with Soft Spring © 2002 by CRC Press LLC 4.79 Maxwell Model: Normalized Disp. vs. Tt 4.80 A Kelvin Model 4.81 Normalized Responses of an Underdamped System with .=0.2 4.82 Responses of a Critically Damped System with .=1 4.83 Normalized Responses of an Overdamped System with .=2 4.84 Normalized Displacement vs. Time w/ Four Damping Levels 4.85 Normalized Velocity vs. Time w/ Four Damping Levels 4.86 Normalized Deceleration vs. Time with Four Damping Levels 4.87 Underdamped Transient Displacement 4.88 Critically Damped Transient Displacement 4.89 Overrdamped Transient Displacement 4.90 Displ. vs. Time of a Kelvin Model with . 4.91 Idealized Power Curve Transient Response 4.92 Normalized Deceleration vs. Displacement with Various . 4.93 Hysteresis Loops w/ Elastic and Plastic Unloading Compared w/ Test Data 4.94 v vs. d of Kelvin Model w/ Elastic and Plastic Unloading and Test 4.95 gd of an Engine Mount Model and Test at 10 mph (.=.08, f=9 Hz) 4.96 Deformation and Restitution Phases of Kelvin Model with .=2, Te=20 4.97 Coefficient of Restitution as a Function of Damping Factor of a Kelvin Model 4.98 Velocity Profiles for Case Study 3 4.99 A Vehicle Impact (Kelvin) Model 4.100 Damping Factor as a Function of Relative Centroid Location 4.101 Natural Frequency Multiplied by Tm vs. Damping Factor 4.102 kc Model and Test Responses of a MidSize Sedan in a 31 mph Barrier Test 4.103 A Kelvin Model with Slack, * 4.104 Haversine Excitation and its Three Harmonics 4.105 Occupant Responses due to Three Harmonic Excitations 4.106 Chest G from Test, and kc Model w/ Halfsine Excitation (. > 0, * = 1.1") CHAPTER 5 RESPONSE PREDICTION BY NUMERICAL METHODS 5.1 Hybrid Model #1 5.2 Hybrid Model #2 5.3 Two EffectiveMass Systems (Hybrid Model #1) 5.4 a vs. t of Hybrid Models at Three Impact Speeds 5.5 d vs. t of Hybrid Models at Three Impact Speeds 5.6 a vs. d of Hybrid Models at Three Speeds 5.7 VehicletoVehicle Impact Model: A TwoHybrid Model 5.8 Acceleration Responses of the Two Masses in the One and TwoHybrid Models 5.9 Dynamic Equivalency of Hybrid #1 (bottom) and #2 (top) Models 5.10 Transient Kinematics of Hybrid Models #1 and #2 5.11 Total and Individual Crush Energies vs. Time for Hybrid Model #1 5.12 Total and Individual Crush Energies vs. Time of Hybrid Model #2 5.13 A Two MassSpringDamper Model 5.14 Mass 1 Displacement Components 5.15 Mass 2 Displacement Components 5.16 Total Displacements of Mass 1 and Mass 2 5.17 Fore and AftFrame Displacement in a PreProgram Vehicle 5.18 A 3D Plot of Dynamic Coupling Factor 5.19 Natural Frequencies of a DoubleUp SpringMass System 5.20 Natural Frequencies of a Split SpringMass System 5.21 Truck Model Responses Dominated by 1st Mode of Vibration 5.22 Truck Model Responses Dominated by 2nd Mode of Vibration © 2002 by CRC Press LLC 5.23 Firing Cannons to a Target 5.24 Imbedded Random Search 5.25 NewtonRaphson Iterative Method 5.26 Loading and Unloading Phases of Two Vehicles 5.27 ForceDeflection Computation in the Unloading Phase 5.28 Parametric Relationships in Loading/Unloading Cycles 5.29 Test Body and Optimal Model Responses w/ and w/o Damping in a 35 mph Test 5.30 Spring and Damper Contributions in Model Body Response 5.31 Body Responses for the Truck Test and PowerCurve Model 5.32 Relationship between ForceDeflection and StressStrain Curves 5.33 F vs. , for Structural Steel 5.34 F vs. , for Brittle Materials 5.35 Police Car Push Bumper 5.36 Push Bumper Force vs. Deflection 5.37 Beam and Spring Modeling of Push Bumper 5.38 ForceDeflection Data with Unloading Properties 5.39 Loading and Unloading Events 5.40 ForceDeflection Responses of Three Models with Elastic and/or Plastic EA 5.41 The Kinematics of a MidSize Car for a 31 MPH Barrier Impact 5.42 A Model with an ElastoPlastic Spring 5.43 TType Side Impact 5.44 LType Side Impact 5.45 TType Side Impact Model 5.46 LType Side Impact Model 5.47 Velocity Profiles of Impactor, Side Struck Vehicle, Crushed Door, and Torso 5.48 A Frontal Distributed Vehicle Model 5.49 An Offset Distributed Vehicle Model 5.50 An Effective SpringMass Offset Model 5.51 A Simplified Offset Model 5.52 Transient Decelerations of a Sedan in Full Frontal and 50% Offset Tests 5.53 Velocity vs. Displacements of a Sedan in Full Frontal and 50% Offset Tests 5.54 Transient Displacements of a Sedan in Full Frontal and 50% Offset Tests 5.55 Model and Test Displacements for the Full and Offset Barrier Impacts 5.56 Vehicle Deceleration and Intrusion in Full and Offset Barrier Tests (35mph) 5.57 A 12Mass 20EA Offset Impact Model CHAPTER 6 IMPULSE, MOMENTUM, AND ENERGY 6.1 Resultant Force and Acceleration 6.2 Vector Addition of Momentum and Impulse 6.3 Batting a Baseball 6.4 Vector Operation of Batting a Baseball 6.5 External and Internal Forces Acting on a Rigid Body 6.6 External and Internal Forces Acting on Particle i 6.7 Vector Operation of Linear Impulse and Momentum Changes 6.8 CG Motion of Two Particles 6.9 Two Vehicles with and without Collision 6.10 Car & Truck HeadOn 6.11 Displacement (Travel) versus Time 6.12 Intersection Collision 6.13 ThreeCar Collision 6.14 ThreeCar Collision Trajectory 6.15 ThreeCar Collision Trajectory and CG Locations © 2002 by CRC Press LLC 6.16 An Impulse Acting on a Rigid Body 6.17 A Point Q on a COCA 6.18 Acceleration Components at Point Q 6.19 Special Case when N and Q Coincides 6.20 Relationship between Impact Point and Location of COCA 6.21 Range of x coordinate of point Q 6.22 A Slender Rod 6.23 A Rectangular Ring 6.24 Size and Location of COCA and Impulsive Loading (c=.5, p = 2b) 6.25 Size and Location of COCA and Impulsive Loading (c=.5, p = b) 6.26 Size and Location of COCA and Impulsive Loading (c=.5,1.,2., p = 2b) 6.27 Size and Location of COCA and Impulsive Loading (c=.5,1.,2., p = b) 6.28 Direction of Acceleration, aQ 6.29 Right Front Impact and COCA with c=0.7, 1, and 1.3 (k2=3, p=1) 6.30 LType Side Impact and COCA with c=0.5, 1, and 2 (k2=3, p=2) 6.31 LType Side Impact and COCA with Q1(1, .5) and Q2(1, .5) 6.32 Right Front Impact and COCA with Q1(1, 1) and Q2(1, 1) 6.33 Left NearCenter Impact and COCA with Q1(1, 1) and Q2 (1, 1) 6.34 Right Front Impact with Resultant Force of Right Front and Front Center 6.35 Right Front Distributed Impact (p=p1+p2) and COCA 6.36 Work by a Force 6.37 Vehicle Skidding to Stop 6.38 Collision at an Intersection 6.39 A SpringMass Impact Model 6.40 A Drop Test on a Weightless Spring 6.41 Vertical Bar in Drop Test 6.42 Equivalent Spring in Drop Test 6.43 A Bar Struck By a Weight 6.44 Spring Equivalent of a Struck Bar 6.45 Impact on a Horizontal Beam 6.46 Spring Equivalent of a Struck Horizontal Bar 6.47 A Drop Tower Test 6.48 Two Car Collision and Separation 6.49 TwoParticle Impact Rebound Condition 6.50 Middle Car RearEnded Twice 6.51 Middle Car (with larger weight) RearEnded Once 6.52 Middle Car (with High Rear COR) RearEnded Once 6.53 Velocity Changes in the Deformation and Rebound Phases 6.54 Normalized e/e2 as a Function of k1/k2 , e1/e2 6.55 Vehicle CG Location 6.56 Vehicle in Horizontal Equilibrium 6.57 Vehicle in Inclined Equilibrium 6.58 Method 2: CG Height Measurement 6.59 Method 3: CG Height Measurement 6.60 A Trifilar Pendulum Method 6.61 Rotation of Trifilar Platform 6.62 A Pendulum Platform and a Car 6.63 A Swinging Platform and a Car 6.64 A Vehicle Sliding On A Tilt Table for Rollover Test 6.65 Vehicle Impulse and Momentum at Impact (event 1 – event 2) 6.66 Sensitivity of Critical Sliding Velocity (2=0o, t=1.72 m) 6.67 Normalized CSV as a Function of F and $ © 2002 by CRC Press LLC 6.68 Dynamic Equilirium of a Vehicle in a Steady Left Turn 6.69 Vehicle Rollover Attitude at a Roll Angle 2 6.70 Required Lateral Acceleration for Roll Dynamic Equilibrium at a Roll Angle 2 6.71 Roll Rate and CSV vs. Roll Angle for Truck F and SUV B 6.72 Vehicle Rollover due to Curb Impact 6.73 Roll Angle vs. Time for SUV B at Trip Speeds of 8, 10, and 12 mph 6.74 Roll Rate vs. Roll Angle for SUV B at Trip Speeds of 8, 10, and 12 mph 6.75 Roll Angle vs. Time for SUV B w/ & w/o Lateral Accel., 0.15 g at 8 mph 6.76 Roll Rate vs. Angle of SUV B w/ & w/o Lateral Accel., 0.15 g at 8 mph 6.77 Initial Angular Acceleration in Rolling 6.78 Initial Angular Acceleration in Pitching 6.79 Gimbaled Gyroscope for Roll Rate Detection 6.80 Gimbaled Gyroscope for Yaw Rate Detection 6.81 Moment of Force about a Point G 6.82 FMVSS 208 Tilt Table Rollover Test 6.83 Right Roof Rail Acceleration in a 23o Tilt Table Test 6.84 Left Roof Rail Acceleration in a 23o Tilt Table Test 6.85 Acceleration at Right Roof Rail in a 23o Tilt Table Test 6.86 Acceleration at Left Roof Rail in a 23o Tilt Table Test 6.87 Accelerations at L. and R. Roof Rail in a Level Impact 6.88 Accelerations at L. and R. Roof Rail in a Quarter Turn Impact 6.89 Vehicle Tripped Rollover and Airborne 6.90 Vehicle Rollover Before Hitting the Ground 6.91 Assuming Vehicle Stops Rolling after Impacting the Ground 6.92 Condition to Stop after One side of Vehicle hits the Ground CHAPTER 7 CRASH SEVERITY AND RECONSTRUCTION 7.1 A Two!Degree!of!Freedom Occupant Model 7.2 Chest ForceDeflection Data 7.3 Vehicle Contact Surface ForceDeflection Curve 7.4 Unrestrained and Restrained Occupant Kinematics in a Crash 7.5 Effects of Pretensioner on Occupant Responses 7.6 Restraint System w/ Pretensioner and F vs. D 7.7 A VehicleOccupant Model w/ and w/o Restraint Preload 7.8 Pretensioner Effect on Chest Response 7.9 Force vs. Deflection of Models w/ and w/o Preload 7.10 Component Impact Model without and with Preload 7.11 Force vs. Deflection of Impact Model w/ and w/o Preload 7.12 Force and Deflection Relationships (a) w/o, and (b) with Pretensioner 7.13 Peak Load and Deflection Ratios w/ and w/o Preload of A OneMass Model 7.14 Acceleration vs. Time for Models w/ and w/o Preload 7.15 TwoParticle Impact to Determine Impact Velocity 7.16 Normalized Impact Velocity of Mass m 7.17 Velocities of a SUV (#1) and MidSize Car in a 58 mph Central Collision 7.18 Velocity Plot of Relative Motion 7.19 Definition of Velocity Changes 7.20 VehicletoVehicle Impact 7.21 NonCentral Collision 7.22 Angular Acceleration in Vehicle 1 7.23 VehicletoVehicle Central Collision 7.24 3D Relationship Between BEV1 and Closing Speed 7.25 3D Plot of Crash Severity Indices for Vehicles 1 and 2 © 2002 by CRC Press LLC 7.26 Crash Severity Index as a Function of Mass Ratio with Condition RmRk = 1 7.27 Crash Momentum Index of Vehicle 1 in TwoVehicle Collision 7.28 A Nonlinear TwoMass Impact Model 7.29 NHTSA Moving Deformable Barrier (MDB) Force/Deflection Characteristics 7.30 Simulated ‘85 Merkur XR4 Frontal Structure Characteristics 7.31 Individual Crush Energy and BEV 7.32 A Damage Boundary Curve 7.33 Transient Velocity and Deceleration of a Product at Two Impact Speeds 7.34 DBC Curves for a Fuel Shutoff Switch of a FullSize Vehicle 7.35 Specific Stiffness vs. Characteristic Length 7.36 Closing Speed Comparison Based on )V and BEV 7.37 Truck to FullSize Car Compatibility Test — Case 1 7.38 FullSize to MidSize Car Compatibility Test — Case 2 7.39 Effective Spring (or Mass) in 2Vehicle Impact 7.40 Energy and Velocity Distribution Map for Central Impact (v=25, V=15 m/s) 7.41 Energy and Velocity Distribution Map for Central Impact (v=0, V=40 m/s) 7.42 Energy and Velocity Distribution Map for Offset Impact (v=25, V=15 m/s) 7.43 A VehicletoVehicle Impact Model 7.44 Other Vehicle Decelerations at Body and Engine for 3 Cases 7.45 Subject Vehicle Decelerations at Body and Engine for 3 Cases 7.46 Vehicle Body Velocities for 3 Cases 7.47 Vehicle Body Displacements for 3 Cases 7.48 Load Cell Plate Loadings, Frame Vehicle at 35 mph Test 7.49 Energy Distribution on a Vehicle Front 7.50 Vehicle Model with Intermediate Mass, m 7.51 Kinematics of Frame Vehicle, Engine and Loadings in a 35mph Test 7.52 Front End Design Considerations 7.53 Models I (top) and II (bottom) with Same Effective Structure Stiffness 7.54 Responses of Models I and II with Same Effective Structure Stiffness 7.55 VehicleRigid Barrier Test 7.56 VehicleRigid Block Impact 7.57 Barrier Impact Speed vs. Residual Crush 7.58 Unit Crush Force versus Residual Crush 7.59 Crush Energy vs. Residual Crush 7.60 TruckBarrier Speed vs. Residual Crush (Test Data) 7.61 CarBarrier Speed vs. Residual Crush (Test Data) 7.62 A FourWay Crash Severity Plot 7.63 NonLinear Damage Profiles 7.64 ElastoPlastic Force Deflection 7.65 Aerostar Front End Damage 7.66 Damage Boundary Curves of BallinTube Crash and Safing Sensors © 2002 by CRC Press LLC UNIT CONVERSIONS Example: Length (in) × 25.4 = Length ( mm); Length (mm) × .03937 = Length (in) Quantity Unit × Unit × Unit Length in 25.4 mm .03937 in ft .3048 m 3.281 2 Area 2 ft 3 in2 in 645.16 mm ft2 .0929 m2 10.76 ft2 ft2 2.3x105 Acre 43500 ft2 cubic yard 21.7 Bushels 1.244 ft3 in3 1.639 x 104 mm3 6.102 x 105 in3 Mass slug slug 14.594 32.2 kg lbm 2.205 .454 lbm kg Moment of Inertia ftlbs2 1.356 kgm2 .7375 ftlbs2 lb/in2 6.895 kPa .1450 lb/in2 lb/ft2 .0479 kPa 20.89 lb/ft2 psi 6.895 kPa .1450 psi ksi 6.895 MPa .1450 ksi psi 6.895 x 106 GPa 1.450 x 105 psi ksi 6.895 x 10 3 GPa 145 ksi lbm/ft3 16.02 kg/m3 .0624 lbm/ft3 Force lbf 4.448 N .2248 lbf Moment lbAin .1130 NAm 8.851 lbAin lbAft 1.356 NAm .7376 lbAft in@lb .113 Joule 8.85 Volume Pressure Energy Velocity Power 1550 x 10 in@lb 4 kwh 2655.18 klbsft 3.766x10 m/s 2.237 mph .447 m/s mph 1.467 ft/s .682 mph mph 17.6 in/s 5.68x102 mph watt .73755 ftlb/s 1.356 watt watt 1.341x103 hp 745.7 watt 2 kwh Acceleration of Gravity (1g) = 32.2 ft/s (at latitude 45° near Alpena or Gaylord, Michigan) © 2002 by CRC Press LLC CHAPTER 1 CRASH PULSE AND KINEMATICS 1.1 INTRODUCTION A basic characteristic of a vehicle structural response in crash testing and model simulation is the “crash signature,” commonly referred to as the crash pulse [1] (numbers refer to references at the end of each chapter). This is the deceleration time history at a point in the vehicle during impact. The crash pulse at a point on the rocker panel at the Bpillar is presumed to identify the significant structural behavior and the gross motion of the vehicle in a frontal impact. Other locations, such as the radiator and the engine, are frequently chosen to record the crash pulse for component dynamic analysis. The nature of the crash response depends on the mass, structural stiffness, damping at that location, and on external interactions from neighboring components. In this chapter, techniques for analyzing the basic vehicle, occupant and restraint system interactions, digital filtering, and the crash pulse are reviewed; also, applications of the kinematic relationships in the analysis of restraint coupling and ridedown efficiency [25] are covered. Case studies involving air bag crash sensing, deployment, and crash recorder data analysis are also presented. 1.2 VEHICLE IMPACT MODES AND CRASH DATA RECORDING Figs. 1.1 and 1.2 show two structure types commonly found in vehicles. These types are unitizedbody and bodyonframe structures. The unitizedbody vehicle has no separate frame or steel girders. It has comparatively thin pieces of body sheet metal which are stamped into complex shapes and welded together to provide the strength required for the chassis. The resultant structure is usually stiffer and lighter than one using separate frame and body construction. Unitized bodies are commonly found on small and compact vehicles. Fig. 1.1 Unitized Body Vehicle Fig. 1.2 BodyonFrame Vehicle The disadvantages of unitized body construction are that (1) more road noise and vibration are transmitted, (2) a serious safety problem is posed if rust attacks the attachment points for the engine, transmission, or suspension, (3) repair costs for body damage are usually higher because a large expanse of the body may have to be cut away and replaced in order to maintain structural integrity, and (4) manufacturing costs are higher due to the need for more sophisticated metal stamping and welding equipment. However, a unitized body using subframes supporting a powertrain or platform chassis, plus modern rustproofing, can overcome some of these disadvantages. Some large North American passenger cars and most trucks and sport utility vehicles (SUV) have a separate frame and body or cab. The frame is made of heavy rectangular, or box section, steel tubes that are welded together. The frame design includes cross members forming a series of open rectangles which provide rigidity and powertrain support. A separate frame is heavy and not especially rigid without the use of Xshaped bracing across the passenger compartment. Rust is not a serious safety concern with separate frame construction. In the bodyonframe vehicle, the body or cab is fastened to the frame by body mounts. A typical © 2002 by CRC Press LLC body mount is shown in Fig. 1.3. It consists of two rubber bushings (on top and bottom of the frame bracket), a bolt, and a retainer. Typically, there are four body mounts on each side frame and two front end sheet metal (FESM) mounts. Body mounts are designed to carry the horizontal impact load in an accident and to isolate the noise, vibration, and harshness (NVH) due to road surface excitation from entering the passenger compartment. Fig. 1.3 A Typical Body Mount on a BodyonFrame Vehicle The crash pulse, which describes the nature and severity of a vehicle crash, depends not only on the type of structure, but also on the measurement site and the impact mode. Figs. 1.4 and 1.5 depict typical crash sensor and accelerometer locations on a unitized body vehicle where the crash pulses are measured. Fig. 1.4 Crash Test Sensor and Accelerometer Locations 1. Upper radiator support bracket 2. Front left/right shotguns (inside fender) 3. Left/right shotguns at spring tower 4. Steering wheel Fig. 1.5 Crash Test Sensor/Accelerometer Locations 5. Centerline tunnel in passenger compartment 6. Front left/right frame rails 7. Left/right rockers at Apillar 8. Left/right rockers at Bpillar Figs. 1.6 and 1.7 show the frontal impact crash test configurations used in a project on the optimization of an advanced air bag sensor system [6]. The crash tests were chosen after a thorough review of previous air bag work worldwide, accident statistics, and experience with the Ford Tempo air bag fleet based on a real world crash investigation. Some of the tests were cartocar tests versus barrier and fixed pole tests. Crash data were collected on twentytwo vehicles; the tests selected represented a broad range of accident encounters at or near the expected threshold of air bag deployment. The threshold tests were designed to produce a vehicle barrier equivalent velocity © 2002 by CRC Press LLC (BEV)1/ of about 12 mph, which is the approximate crash severity threshold at which, in the judgment of the project engineers, an air bag should deploy. Twentytwo vehicles in sixteen tests, as shown in Figs. 1.6 and 1.7, were used to collect the crash pulse data. The impact speed in each test configuration was chosen so the system performance under the air bag sensor must or mustnot activate condition could be evaluated. The types of impact include the following: 1. A perpendicular (90degree) barrier 2. A low and high speed rigid pole 3. A fronttofront overlap 4. An oblique front to front (three types) 5. A vehicle front to an MDB (moving deformable barrier) perpendicular 6. A vehicle fronttoside 7. A fronttorear 8. A bumper override (or truck underride) Fig. 1.6 Crash Test Mode ! 1 Fig. 1.7 Crash Test Mode ! 2 1.2.1 Accelerometer Mounting and Coordinate Systems The crash test data are recorded by accelerometers. Shown in Fig. 1.8 are the accelerometer types and the schematic of an accelerometer model. A typical accelerometer uses either a strain gauge mounted on a beam surface or a piezoelectric crystal. Those used in the vehicle crush zone have a range of about ±2000 g, and those at the engine, transmission, passenger compartment, and dummies, ±750 g. During crash test preparation, accelerometers with the specified ranges and sensitivities are instrumented in the vehicle. A typical crush zone accelerometer has a sensitivity of 0.25 millivolts/g 1/ BEV is the vehicle speed in the rigid barrier test which yields the same crush energy absorbed by the structure as that in the nonrigid barrier test condition. © 2002 by CRC Press LLC with a 10volt excitation. Damage to accelerometers and pullingoff of wires from accelerometer blocks are not uncommon in the crash tests. Fig. 1.8 Accelerometer Types and Schematic The axes of a triaxial accelerometer, an assembly of three accelerometers on a mounting block, are oriented along the vehicle axes. The initial angles of the accelerometer axes from the reference axes should not exceed 5°. Each axis should pass within 10 mm of a prescribed mounting point, and the center of gravity of each accelerometer should be within 30 mm of that point [7]. The coordinate systems for the vehicle and occupant are shown in Figs. 1.9 and 1.10, respectively. The X, Y, and Z directions in the 3dimensional reference frame are referred to as longitudinal, lateral, and vertical directions. Fig. 1.9 Vehicle Coordinate System Fig. 1.10 Occupant Coordinate System 1.3 DIGITAL FILTERING PRACTICE PER SAE J211 AND ISO 6487 The crash test data, recorded by an accelerometer, is prefiltered before sampling at a rolloff frequency of 4,000 Hz. The prefiltered data, referred to as wideband data, contains the same signal as the raw data (the impact stress recorded by an accelerometer). This data is then sampled at a rate of 12,500 points per second (or 0.08 milliseconds per data point) and yields an input acceleration, Ain. To obtain the signal in its useful frequency range, a digital filtering technique which satisfies the frequency response corridor specified by SAE J211 (SAE Recommended Practice on the "Instrumentation for Impact Tests") [8] should be used. The filtered output acceleration, designated as Aout, satisfies the amplitude gain relationship shown below. Consider an instrumentation system that has an input power of Pin and an input voltage of Vin and produces an output power of Pout and an output voltage of Vout. Then, the gain G, in decibels (db), of the system is given by © 2002 by CRC Press LLC (1.1) If Zout and Zin, the output and input impedances, respectively, are equal, Eq. (1.1) becomes (1.2) This formula will be used later to compute the filtered output magnitude provided that the unfiltered input magnitude of a given frequency and the corresponding attenuation are specified. The purpose of SAE J211 is to provide guidelines for filtering specifications and the selection of a class of frequency response. The aim is to achieve uniformity in instrumentation practice and in reporting test results. The channel classes recommended by SAE J211 are shown in Table 1.1. A filter frequencyband plot for Channel Class 60 is shown in Fig. 1.11. The frequency response corridor and limit values in the pass band, transition band, and stop band are shown for each channel class. For example, if the vehicle structural acceleration is used as a test measurement for total vehicle comparison, Channel Class 60 is selected according to Table 1.1. The tolerances in the pass band for the Channel Class 60 are a = !.5 to .5 db at f = 10 Hz; b = !1 to .5 db at fH = 60 Hz; and c = !4 to .5 db at fN (rolloff or cutoff frequency) = 100 Hz. The upper and lower slopes in the transition band are d = !9 and e = !24 db/octave, respectively. The stop band extends downward from the ends of the transition band at g = !30 db. The International Standard, ISO 6487 (the International Organization for Standardization), titled “Road Vehicles – Measurement Techniques in Impact Tests – Instrumentation” was issued on May 1, 2000 as the third edition. The standard is basically the same as SAE J211, which was issued in March 1995. There are four channel classes in which frequency response values are specified for the passband, transition band, and stop band. The specifications for the channel classes (or CFC, channel frequency class) 60 and 180 are the same for both SAE J211 and ISO 6487. Table 1.1 Band Pass Frequency Response Values For Various Channel Classes Channel Class fL, Hz a, db fH, Hz b, db fN, Hz c, db d 1000 0.1 .5, !.5 1000 .5, !1 1650 .5, !4 !9 !24 !30 600 0.1 .5, !.5 600 .5, !1 1000 .5, !4 !9 !24 !30 180 0.1 .5, !.5 180 .5, !1 300 .5, !4 !9 ! 24 !30 60 0.1 .5, !.5 60 .5, !1 100 .5, !4 !9 !24 !30 © 2002 by CRC Press LLC e db/octave g, db Fig. 1.11 SAE J211 Frequency Response Corridor Table 1.2 Channel Class Selection ! SAE J211 Typical Test Measurement Vehicle structural acceleration for use in: Total vehicle comparison Collision simulation (for example, impact sled) input Component analysis Integration for velocity or displacement Barrier face force Belt restraint system load Occupant Head acceleration Chest acceleration deflection Pelvis acceleration forces moments Femur/knee/tibia/ankle forces moments displacements Sled acceleration Steering column load Headform acceleration Channel Class 60 60 600 180 60 60 1000 180 180 1000 1000 1000 600 600 180 60 600 1000 The channel class selected for a particular application in Table 1.2 does not imply that all the frequencies passed by that channel are always significant for that application. In the case of measurements of occupant head and headform accelerations and femur force, the channel class band pass may be higher than necessary in order to cover biomechanical uncertainties. © 2002 by CRC Press LLC Crash test data generally has highfrequency components above the frequency fH (e.g., Channel Class 1000, where fN = 1650 Hz). This can occur more often with undamped accelerometers. To prevent these components from causing aliasing errors in the sampling process, a presampling filter should be used. The minimal acceptable sampling rate should be at least five times the !3db frequency of the presampling filter. Since the !3db frequency is fN (rolloff) frequency (see Table 1.1), the minimum sampling rate for Channel Class 1000 should then be at least 5 × 1650 = 8250 Hz. In order to derive a mathematical relationship between any two points on the frequency response plot, the terms decibel and octave are introduced as follows: Alexander Graham Bell defined a unit, the Bel, to measure the ability of people to hear. The deciBel (db), one tenth of a Bel, is the most common unit used in frequency domain analysis. The combination of ear and brain is an excellent frequency domain analyzer. The brain processes the signal received from the ear, splits the audio frequency spectrum into different narrow bands and determines the power present in each band. The decibel, db, is a unit expressing the ratio of two signals of electric current, voltages, acceleration, or sound pressure. The gain Gdb is equal to 20 times the common logarithm of the ratio. From Eq. (1.2), Gdb in terms of acceleration is defined in Eq. (1.3). (1.3) where Ain: input or unfiltered acceleration, Aout: output or filtered acceleration. The octave, a term used in vibration analysis, is a frequency interval analogous to a musical octave. Fig. 1.12 shows the arrangement of piano keys in one octave. The frequency of a typical keynote C (C5) is 523.25 Hz. There are a total of twelve notes in one octave ranging from C, C#, D,...., B with the corresponding note number of j equal to 0,1,2,3,....,11. The frequency relationship between the jth note and keynote C is shown in Eq. (1.4). Fig. 1.12 Piano Keys Covering One Octave (1.4) © 2002 by CRC Press LLC For example, given the frequency of note C, 523.25 Hz, we like to compute the frequency of note F. One can use j = 5 for note F in Eq. (1.4), and its frequency is then 698.46 Hz. Since the process of filtering a crash pulse involves the attenuation of deceleration magnitudes at different frequencies, the basic frequency relationship between any two points on the frequency response plot should be understood. The formula given in Eq. (1.4) is derived in the next section. 1.3.1 Relationship Between Two Points in a Frequency Response Plot In a plot of decibel vs. log of frequency, the frequency relationship between two points depends on the number of octaves between them. To derive this relationship, let b and log f be the vertical and horizontal coordinates, respectively; then, a straight line equation can be defined as shown in Eq. (1.5) in the following derivation. Deriving the Frequency Relationship Between Two Points in a Frequency Response Plot (1.5) © 2002 by CRC Press LLC Case Study (exercise): Frequency Response Corridor The transition band specified by SAE J211 is shown in Fig. 1.13. Fig. 1.13 Case Study: Frequency Response Corridor (1) The lower bound of the band has a slope of !24 db/octave, the frequency and output/input ratio in decibels at point 1 being 100 Hz and !4 db. The output/input ratio in decibels at point 2 is !30 db. Compute the output/input deceleration ratio at point 1 and the frequency at point 2. (2) The upper bound has a slope of !9 db/octave, and the frequency and output/input ratio in decibels at the beginning point are 100 Hz and 0.5 db, respectively. The output/input ratio in decibels at the ending point is !30 db. Compute the output/input deceleration ratio at the beginning point and the frequency at the end point. [Ans. (1) 0.63, 212 Hz, (2) 1.1, 1048 Hz] 1.3.2 Chebyshev and Butterworth Digital Filters Two digital filters, commonly known as Chebyshev and Butterworth filters, are used in processing vehicle crash test data. These filters are described by their frequency response characteristics and compared to the frequency response corridors specified in SAE J211. The parametric relationships between the deceleration attenuation (output/input ratio, db), f (frequency content of the crash pulse), and frolloff (rolloff frequency) are shown below in Figs. 1.14 and 1.15 for the Butterworth and Chebyshev nth order digital filters, respectively. Since the frequency response curves fall within the specified frequency response corridor, both Butterworth and Chebyshev 2nd order digital filters satisfy the SAE J211 requirements. Although Butterworth 3rd and 4th order filters also satisfy the requirements, they tend to have higher signal attenuation at a given frequency component than that of the Butterworth 2nd order filter as shown in Fig. 1.14. Shown in Fig. 1.15, only the Chebyshev 2nd order filter fulfills the SAE J211 response corridor requirement. Note that the entire frequency plots shown in Fig. 1.14 and Fig. 1.15 are made by using the two frequency response equations, Eq. (1.6) and Eq. (1.7), respectively. The two equations provide the relationships between the acceleration attenuation and the normalized frequency. The normalized frequency is defined as the component frequency of the signal normalized with respect to (w.r.t.) either the fN frequency (rolloff) for the Butterworth filter or the fH frequency for the Chebyshev filter as shown in both Eqs. (1.6) and (1.7), respectively. The same equations are also used to plot the passband responses of the Butterworth and Chebyshev filters shown in the following. © 2002 by CRC Press LLC (1.6) Fig. 1.14 Butterworth nth Order Filter (1.7) Fig. 1.15 Chebyshev nth Order Filter Butterworth lowpass filters are designed to have an amplitude response characteristic that is as flat as possible at low frequency and decreases with increasing frequency (Fig. 1.16). Fig. 1.16 Butterworth nth Order Passband Response Function © 2002 by CRC Press LLC On the contrary, Chebyshev filters are designed to have a relatively sharp transition from the pass band to the stop band in the amplitude response characteristics plot as shown Fig. 1.17. This sharpness is accomplished at the expense of ripples (waves) that are introduced into the response. For a Chebyshev nth order filter, there are n ripples in the passband. Fig. 1.17 Chebyshev nth Order Passband Response Function 1.3.3 Filter Type, Deceleration Magnitude, and Phase Delay The attenuation of magnitude and phase delay (angle) of deceleration depend on the filter type. If the wideband data are to be filtered at a low rolloff frequency using a 2nd order Chebyshev filter, such filtering alters information content, including phase angle. The resulting data filtered by Chebyshev are then commonly scaled and/or shifted so that the observed vehicle kinematics is compatible with that obtained from photographic film analysis. However, the use of the 2nd order Butterworth filter does not present any phase delay problem. Five sinusoidal pulses with a duration of 100 ms, magnitude of plus/minus 10 g and frequencies of 40, 100, 200, 300, and 500 Hz were sampled at a rate of 12,500 Hz. The pulses were filtered using both Chebyshev channel class 60 and Butterworth 2nd order filters with a rolloff frequency of 100 Hz. The deceleration attenuation (output/input ratio) and phase delay were then obtained from the filtering output and plotted on a frequency response plot as shown in Fig. 1.18. Note that the phase delay is 360°, if the filtered output is one cycle off from the sinusoidal input. It is observed that there is no phase delay for all the pulses filtered by Butterworth. The phase delay caused by the Chebyshev filter varies considerably depending on the component frequency of the excitation signal. In addition to phase delay, the deceleration attenuation by the 2nd order Butterworth is greater than that produced by the 2nd order Chebyshev filter. The difference in attenuation between the two filter types depends on the component frequency. As shown in the upper plot in Fig. 1.18, the higher the component frequency, the larger the attenuation difference between the two filter types. Unless otherwise noted, the default filter type and filtering rolloff frequency are the 2nd order Butterworth filter and 100 Hz (Channel Class 60 shown in Table 1.2), respectively. In the special case where the input deceleration has a frequency component the same as the rolloff frequency, the deceleration attenuation is then !3 db and the magnitude ratio of output deceleration (filtered) to the input is 0.707. © 2002 by CRC Press LLC Fig. 1.18 Effects of Filter on Magnitude and Phase Delay For example, given a sinusoidal pulse with a frequency of 100 Hz filtered by the Butterworth filter with a rolloff frequency of 100 Hz, 70.7% of the peak magnitude still passes through the filtering. Using the formulas for the Butterworth nth order filter, the computation is shown in Eq. (1.8): (1.8) © 2002 by CRC Press LLC The effects of the filter type on the deceleration magnitude and phase delay of the filtered responses are compared using the three input pulses described in the following: A. Singlestep and multiplestep function inputs The data set shown in Fig. 1.19 contains a step (or fast rise) input and is filtered with Channel Class 60. The output from the Butterworth filter contains data prior to and after the actual step event in the unfiltered data, and has no phase shift. Fig. 1.19 Filtered Response Comparison ! SingleStep Function and Channel Class 60 The data filtered by the Chebyshev filter, shown in Fig. 1.20, matches the initiation point of the step input closely, but with considerable phase delay. Comparing the location of the peak magnitude of the filtered pulse with the midpoint of the rectangular step input, the Chebyshev peak magnitude is delayed by half of the duration of the step input, and the Butterworth peak occurs right at the midpoint of the step input. Note that the peak magnitude of the Butterworth filtered output is slightly less than that of the step input, while the peak magnitude of the Chebyshev filtered output is slightly higher than that of the step input. Fig. 1.20 Filtered Response Comparison ! MultipleStep Function and Channel Class 60 © 2002 by CRC Press LLC Applying the same analysis to the raw data of the vehicle crash pulse, the differences in the magnitude attenuation, phase delay, and initiation point between the filtered data and raw data become clear. Shown in Fig. 1.21, the first impulse, between 4 and 8 ms, can be approximated by a unit step input. The relationships between the filter type, initiation point, attenuation magnitude, and phase delay applied to the step input can also be applied to the test data analysis. Fig. 1.21 CloseUp of Filtered and Wideband Crash Pulse Comparison B. Vehicle crash pulse and Driver chest deceleration The wideband crash pulse from an accelerometer on the left rocker at Bpillar of a midsize passenger car struck by a truck in a 58 mph full frontal test is shown in Fig. 1.22. The data set is filtered with Channel Class 60, with a rolloff frequency of 100 Hz according to Table 1.2 on Channel Class Selection  SAE J211. Peaks in the data sets filtered by Butterworth occur at the same time as the peaks in the unfiltered data. However, there are considerable phase delays between the Chebyshevfiltered and the wideband data sets. Since the deceleration attenuation by Butterworth (2nd order) is more than that by Chebyshev (2nd order), the peak magnitudes of the Butterworth filtered data are smaller than those by Chebyshev. The peak magnitude filtered by the Butterworth, about 12.5 g in the region of T = 30 ms, is about 1g less than that of the Chebyshev. Fig. 1.22 Vehicle Pulse Filtered by Channel Class 60 ! Butterworth and Chebyshev © 2002 by CRC Press LLC The wideband data of driver chest deceleration in the same trucktocar test is shown in Fig. 1.23. The data set of chest deceleration is filtered with Channel Class 180 with a rolloff frequency of 300 Hz according to Table 1.2 on Channel Class Selection  SAE J211. The outputs filtered by both Butterworth and Chebyshev algorithms are practically the same compared to the unfiltered data. This is due to the fact that the frequency content in the wideband data of the chest deceleration is low and the rolloff frequency is high. Therefore, the attenuation by the filter becomes small (see Eqs. 1.6 and 1.7), and the output to input deceleration ratio approaches one. Fig. 1.23 Chest Decel. Filtered by Channel Class 180 ! Butterworth and Chebyshev According to SAE J211, March 1995, Section 9.4.1 on digital filtering, the Butterworth filter should be used for the Channel Class 180 or 60. In the same section, it also states that any filtering algorithm can be used for Channel Class 1000 or 600 as long as the results conform to the data channel performance requirements shown in Fig. 1.24. For simplicity, NHTSA uses the Butterworth filtering for all four channel classes (see Table 1.3 for the Fortran subroutine), even though it is not mandatory for channel class 600 and 1000. Fig. 1.24 Revised Transition Bands for Channel Class #3 & #4 per SAE J211, March 1995 © 2002 by CRC Press LLC Table 1.3 FORTRAN SUBROUTINE – BUTTERWORTH FILTER C C C C C C C C C C C C C C C C C C C COMMON/I/YF(10:2505),YY(2505) (portion of) MAIN program: 2nd order Butterworth digital filter... FILTER THE ENTIRE WIDEBAND DATA READ FROM THE DATA FILE NFP = 1 ! NFP: FIRST DATA POINT NUMBER DO I = 1, NPT ! NPT: NO. OF DATA POINTS YF(I) = YY(I) ! YY(I): ORIGINAL WIDEBAND ACCELERATION AT POINT I ENDDO DO I=0,9 YF(NFP1I) = YF(NFP+I) ENDDO DEL = .08 ! DEL: DATA STEP, 0.08 MS. (SAMPLING RATE = 12,500 POINTS/SEC) FCUT: (Fn) CUTOFF FREQUENCY, Hz CALL FILTER(YF(NFP10),DEL,NFP10,NUM,FCUT*1.25) YF(I): FILTERED DATA POINTS, I=1,2,.., NUM STOP END SUBROUTINE FILTER(Y,DEL,N1,N2,FM6DB) SOURCE: NHTSA Crashworthiness Research, DOT. Function: Filters data forward and backward with a second order Butterworth algorithm, giving zero phase shift and 3dB at FM6DB/1.25. Principal Variables: A1,A2,B0,B1,B2  Difference equation coefficients Y(I)  Data array (pre and postfiltered) FM6DB  Filter (6 dB) frequency (Hz) DEL  Time increment of data (sec) N1  Index of first data point N2  Index of last data point Programmer: ASGI  C. Louie 11/78 ASGI  S. Mentzer 7/83 DIMENSION Y(N1:N2) Compute filter coefficients PI = 3.141592654 WD = 2*PI*FM6DB WA = SIN(WD*DEL/2.)/COS(WD*DEL/2.) B0 = WA**2/(1.+SQRT(2.)*WA+WA**2) B1 = 2.*B0 B2 = B0 A1=2.*(WA**21.)/(1.+SQRT(2.)*WA+WA**2) A2=(1.+SQRT(2.)*WAWA**2)/(1.+SQRT(2.) *WA+WA**2) C Filter forward Y1 = 0.0 DO 201 I=N1,N1+9 Y1 = Y1+Y(I) 201 CONTINUE Y1 = Y1/10.0 X2 = 0.0 X1 = Y(N1) X0 = Y(N1+1) Y(N1) = Y1 Y(N1+1) = Y1 DO 202 I=N1+2,N2 X2 = X1 © 2002 by CRC Press LLC X1 = X0 X0 = Y(I) Y ( I ) = B0*X0+B1*X1+B2*X2+A1*Y(I1)+A2*Y(I2) 202 CONTINUE C Filter backwards Y1 = 0.0 DO 203 I=N2,N29,1 Y1 = Y1+Y(I) 203 CONTINUE Y1 = Y1/10.0 X2 = 0.0 X1 = Y(N2) X0 = Y(N21) Y(N2) = Y1 Y(N21) = Y1 DO 204 I=N22,N1,1 X2 = X1 X1 = X0 X0 = Y(I) Y ( I ) = B0*X0+B1*X1+B2*X2+A1*Y(I+1)+A2*Y(I+2) 204 CONTINUE RETURN END 1.3.4 Moving Window Averaging and Equivalent Cutoff Frequency The purpose of using window averaging in processing the crash test wideband (raw) data is to reduce the total number of discrete data points and to average (filter) out the noise. A simple moving window averaging is used for this purpose and the techniques used to integrate the new set of averaged data points are presented. The window length, the number of data points in a window, controls the component frequency of the crash pulse, equivalent to the effect of cutoff frequency in signal attenuation. 1.3.4.1 Moving Window Averaging Moving window averaging is a simple digital filtering technique that filters out noise by averaging the data within the window length. The averaged deceleration value is then placed at the midpoint of the window. By moving this window through the entire crash pulse, the total number of data points is reduced nfold, where n is the window length, the number of points (bites) in the window. Note that in the raw data, the length of each bite is 0.08 ms, corresponding to a sampling rate of 12,500 points (bites) per second. Using an accelerometer on the left frame at the Apillar, the crash data for a light truck in a 31 mph rigid barrier impact was obtained. The data are plotted and shown in Fig. 1.25. Assume that the window length is 50 bites and 1 bite = 0.08 ms (sampling rate=12,500 Hz); the step size (window length), *, is then 4 ms (50 bites/window × 0.08 ms/bite). However, the distance between the midpoint of the first window and time zero is only half of the window step size. Any window after the first one has the full step size. This is the distance between the midpoints of the consecutive windows. Consequently, the first window has only a halfstep, which is 2 ms. Fig. 1.25 Crash Test Data (Acceleration) with Moving Window Averaging The duration of the wideband data shown is 20 ms, and each bite is .08 ms; therefore, the total number of points is 20/0.08 + 1 = 251. Since the window length (step) is 50 bites or 4 ms/step, the total number of points is reduced from 251 to 6 (251/50 + 1 = 6, or 20/4 + 1 = 6) which includes the first point at time zero. The integration scheme using moving window averaging for crash data should therefore use only a half step size when integrating the first data point at time zero. The full step size should then be used when integrating points after the first one as shown in the integration scheme in Eq. (1.9). Note that vo and do denote the initial velocity and initial displacement (at time zero) and *, the integration time step. © 2002 by CRC Press LLC (1.9) The entire wideband crash pulse of the truck frame at Apillar in the rigid barrier 31 mph test is shown in Fig. 1.26 along with the window averaged curve. Since the pulse duration is 150 ms, and the data step is 0.08 ms, the total number of data points is 1,876 points. The dotted solid curve is the window averaged curve with window length of 50 bites. Fig. 1.26 A Truck Crash Test Raw Data with Moving Window Averaging Although the number of points of the wideband data is reduced by 50 times, the first and second integrals (relative velocity and displacement) of the windowaveraged data points still retain the integration accuracy. For practical purposes, they are identical to the integrals of the wideband data as shown in Fig.1.27. Fig. 1.27 1st and 2nd Integrals of Crash Test Raw Data with Moving Window Averaging © 2002 by CRC Press LLC 1.3.4.2 Equivalent Cutoff Frequency The moving average operation can be regarded as a low pass filtering process. It averages out the high frequency noise. The filtering effect by window averaging depends on the window length and is equivalent to a low pass filter with a cutoff frequency [14] given by Eq. (1.10). (1.10) In the previous section, a truck crash pulse with a duration of 20 ms was used for the moving window averaging. A window length of 50 bites (4 ms step) was used and the total number of points after averaging was six. Since the sampling rate, fs is 12,500 Hz, and window length, M=50, the equivalent cutoff frequency, fc is therefore equals to 12,500/[2×(50+1)] = 123 Hz. A comparison of three curves, window averaging, the Butterworth filter at 123 Hz, and the raw test data, is shown in Fig. 1.28. Note that the initiation point for both window averaging and the Butterworth filtered curves occurs at about the same time, a halfstep of 2 ms. Throughout the 20 ms duration, the window averaging tends to average out the Butterworth filtered responses. Fig. 1.28 Crash Pulse Comparison between Window Averaging and the Butterworth Filter Instead of fixing the window length, one can compute the window length based on the required cutoff frequency. Let the equivalent cutoff frequency, fs, be 100 Hz, for filtering the vehicle structural deceleration. Since the sampling rate is 12,500 Hz, from Eq. (1.10), one can compute M, the window length: (1.11) Using the window length of 61 bites, the body crash pulse of a midsize vehicle struck by a truck at 58 mph is filtered by the window averaging method. Fig. 1.29 shows the raw data of the crash pulse overlapped with the two filtered outputs by the window averaging of 61 bites and Butterworth filter with cutoff frequency of 100 Hz. © 2002 by CRC Press LLC Fig. 1.30 Raw Data of Body Pulse of a MidSize Car Struck by a SUV at 58 mph Fig. 1.29 Body Pulse With Butterworth 100 Hz Cutoff and 61bite Averaging Fig. 1.30 shows a comparison between the two filtered curves over the entire 150 ms duration. Both curves are fairly close to each other in terms of magnitude and timing in the peaks and valleys. Just like the Butterworth filter, the window averaging method does not possess a phase delay problem. To properly overlay the wideband and window average data points, the averaged data point should be placed at the midpoint of the window and the halfstep should be used in the first point integration. The location of the first halfstep data point in time is critical in terms of the accuracy of timing and magnitude of integrals. This is especially true when the window length is long. Due to the reduced number of data points and given the accuracy of the integral, the window averaging technique is especially useful in obtaining a transfer function. Topics on transfer functions are covered in Chapter 3. The forthcoming sections describe the basic kinematic relationships, principles, and their applications in analyzing the crash pulse data for crashworthiness study. © 2002 by CRC Press LLC 1.4 BASIC KINEMATIC RELATIONSHIPS Using the three basic kinematic relationships relating the deceleration, velocity, and displacement shown below, crash test data can be further processed to yield the particle kinematics of a vehicle or occupant in the time and displacement domain. (1.12) (1) and (2) of Eq. (1.12) represent the basic kinematic relationships in the time domain. By combining (1) and (2) and eliminating the time variable, a kinematic relationship in the displacement domain is obtained as shown in (3). Define e: energy density (or specific energy, defined as energy per unit mass). Then (1.13) Note that the slope at a point on the energy versus displacement plot is the acceleration at that point in time. The four basic kinematic variables, a, v, d, and e in the time domain are the transient response variables. The same response variables expressed in the displacement domain offer a new perspective in evaluating the impact severity of a crash. Although the transient acceleration data, a(t), and displacement, x(t), are readily available from the recorded accelerometer data and their integrals, the energy density at any given time, t, is not equal to the product of the two quantities as shown below. (1.14) To obtain the correct energy density, the differential energy density should be integrated over a specified range of displacement on the acceleration versus displacement curve. The energy density can also be computed using the velocity information shown in Eq. (1.15) where vo is the initial velocity of a vehicle in an impact. (1.15) 1.4.1 Computing Acceleration from a VelocityDisplacement Curve Given a velocity (v) vs. displacement (x) curve as shown in Fig. 1.31, the acceleration (a) corresponding to any point on the curve can be computed using kinematic relationship (3) of Eq. (1.12). Note if v and x are relative quantities, then the computed a is also a relative quantity. Similarly, if v and x are absolute, so is a. © 2002 by CRC Press LLC Fig. 1.31 Velocity vs. Displacement Curve The slope at point P on the vx curve is (1.16) If conventional units (a, g; v, mph; x, in) are used, then the formula becomes (1.17) Note that if the angle is to be measured off the curve and the tangent function is to be used to compute the slope, the scales on both x and v axes have to be the same. As an example, if the scale on the x axis is 2 inches per division, the scale on the v axis would have to be 2 mph per division, and the measured lengths per division along x and v would have to be the same. Case Study: A fullsize passenger car was tested in a 14 mph rigid barrier test. The magnitude of the longitudinal deceleration at the left rocker/Bpillar is shown in Fig. 1.32. (1) Plot the velocity versus displacement curve for an unbelted occupant with respect to the vehicle, and (2) Compute the relative acceleration of the unbelted occupant at the relative displacement of 8 inches. Fig. 1.32 Unbelted Occupant Relative a vs. t In the case of a freeflying (unrestrained) occupant in the passenger compartment during a crash, © 2002 by CRC Press LLC it can be shown that the slope (deceleration) at a point on the relative occupant velocity vs. relative displacement curve is the corresponding vehicle deceleration. Let us define: ao = freeflying occupant acceleration (= 0), av = vehicle acceleration ao/v = deceleration of occupant relative to vehicle then ao/v = ao ! av = 0 ! av ; therefore, av = !ao/v The vehicle deceleration is equal to the negative of the freeflying occupant relative to vehicle acceleration. The first and second integrals of the unbelted occupant acceleration with zero initial velocity yield the relative velocity and displacement of the unbelted occupant, respectively. The relative acceleration versus relative displacement (a vs. d), and the relative velocity versus relative displacement (v vs. d) are shown in Figs. 1.33 and 1.34, respectively. Fig. 1.33 Unbelted Occupant Relative a vs. d To illustrate the proper use of the relationship between a and v in the displacement domain plot, two different displacement scales shown in Fig. 1.34 are used. Fig. 1.34 Unbelted Occupant Relative Velocity vs. Relative Displacement in Two Scales To compute the relative acceleration at point P from the velocity vs. displacement curve, the relative displacement and velocity at point P are found to be 8 inches and 13.7 mph, respectively. Since the first x scale is different from y scale, the slope at point P is determined by taking the ratio of the opposite side to the adjacent side: 6 mph/8.3 in = 0.72 mph/in. Therefore, the relative acceleration, a, at point P, is © 2002 by CRC Press LLC (1.18) If the angle is to be measured from the plot to compute the tangent function, the x and y scales would have to have the same magnitude (same length per division). A scale factor of 2.5 is used to change the x scale such that it has the same magnitude as the y scale as shown in Fig. 1.34. The angle of the tangent at point P from the scaleddown curve is about 36°. Therefore, the relative acceleration at point P is then (1.19) The value of the relative acceleration at point P computed from the velocity versus displacement plot is checked against that from the relative acceleration versus relative displacement plot as shown in Fig. 1.33. The plot of v vs. d, the relative velocity versus relative displacement, shown in Fig. 1.34 is useful in comparing the impact severity of an unbelted occupant in different crash modes in air bag development tests. Section 1.6.3, Occupant Kinematics in Different Test Modes, is devoted