DYNAMICS MODEL FOR PREDICTING MAXIMUM AND TYPICAL ACCELERATION RATES OF PASSENGER VEHICLES. Matthew C. Snare

Transcription

1 DYNAMICS MODEL FOR PREDICTING MAXIMUM AND TYPICAL ACCELERATION RATES OF PASSENGER VEHICLES Mahew C. Snare Thesis submied o he Faculy of he Virginia Polyechnic Insiue and Sae Universiy in parial fulfillmen of he requiremens for he degree of Maser of Science in Civil Engineering Dr. Hesham Rakha, Chair Dr. Francois Dion Dr. Pushkin Kachroo Augus 6, Blacksburg, Virginia Keywords: Vehicle Acceleraion, Vehicle Dynamics Models, Traffic Simulaion, Car-Following Behavior Copyrigh, Mahew C. Snare

2 Absrac DYNAMICS MODEL FOR PREDICTING MAXIMUM AND TYPICAL ACCELERATION RATES OF PASSENGER VEHICLES Mahew C. Snare Absrac Effecively modeling he acceleraion behavior of vehicles is an imporan consideraion in a variey of ransporaion engineering applicaions. The acceleraion profiles of vehicles are imporan in he geomeric design of roadways and are used o model vehicle behavior in simulaion sofware packages. The acceleraion profile of he vehicle is also a criical parameer in fuel consumpion and emissions models. This paper develops and validaes a vehicle dynamics model o predic he maximum acceleraion raes of passenger vehicles. The model is shown o be superior o oher similar models in ha i accuraely predics speed and acceleraion profiles in all domains and for a variey of vehicle ypes. The paper also modifies he model by inroducing a reducion facor, which enables he model o predic he ypical acceleraion paerns for differen driver ypes. The reducion facors for he driving populaion are shown o follow a normal disribuion wih a mean of.6 and a sandard deviaion of.8. The paper also provides new daa ses conaining maximum and ypical acceleraion profiles for hireen differen vehicles and weny differen drivers. ii

3 Acknowledgemens Acknowledgemens I would like o ake his opporuniy o hank he many people ha helped me hroughou he process of creaing his hesis. Firs and foremos, I hank my advisor Dr. Hesham Rakha, who guided me hrough he research process and was also insrumenal in my decision o choose raffic engineering as a career. I would also like o hank he oher members of my commiee, Dr. Francois Dion and Dr. Pushkin Kachroo, for heir suppor. I would also like o acknowledge he es drivers who voluneered heir ime owards my daa collecion. In addiion, I hank he Charles Edward Via family for he generous fellowship I received ha enabled me o pursue my graduae educaion. Finally, I am graeful owards my fuure wife, Miriam, who lifed my spiris when imes go rough. iii

4 Table of Conens Table of Conens ABSTRACT. ACKNOWLEDGEMENTS TABLE OF CONTENTS LIST OF TABLES.. LIST OF FIGURES. ii iii iv vi vii CHAPTER ONE: INTRODUCTION. BACKGROUND INFORMATION... PROBLEM DEFINITION.. THESIS OBJECTIVES.. THESIS CONTRIBUTIONS.. THESIS LAYOUT. CHAPTER TWO: LITERATURE REVIEW... INTRODUCTION.. MODELING USING KINEMATICS. MODELING USING VEHICLE DYNAMICS. 9. MAXIMUM VS. TYPICAL ACCELERATION... COMPARISON OF EXISTING MODELS...6 CONCLUSIONS CHAPTER THREE: RESEARCH METHODOLOGY... INTRODUCTION... ESTABLISHING A NEW DATABASE..... Smar Road Tesing 6... Sudy Secion Descripion Speed Measuremen Vehicle Descripions Tes Run Descripion. 8.. Driver Behavior Tesing. 9. PREDICTING MAXIMUM ACCELERATION RATES Truck Dynamics Model Tracive Force... Aerodynamic Resisance... Rolling Resisance..... Grade Resisance...6 Applying he Truck Model o Cars...7 Variable Power Model..8 Calibraing he Variable Power Model. PREDICTING TYPICAL ACCELERATION BEHAVIOR... COMPARISON OF MODELS 6.6 SUMMARY... 6 iv

5 Table of Conens CHAPTER FOUR: VEHICLE DYNAMICS MODEL FOR ESTIMATING MAXIMUM AUTOMOBILE ACCELERATION LEVELS 7. INTRODUCTION 7. BACKGROUND. 7.. Sae-of-Pracice Vehicle Acceleraion Models Vehicle Kinemaics Models Vehicle Dynamics Models... Sae-of-Pracice Field Daa Ses.. CONSTRUCTION OF FIELD DATA SET. 6.. Smar Road Tes Faciliy. 6.. Daa Collecion Procedures. 8.. Speed Measuremens 8.. Tes Vehicle Characerisics. 8. MODEL CONSTRUCTION AND COMPARISON 9.. Model Parameers. 9.. Model Applicaion.. Consan Power Assumpion... Vehicle Dynamics Model Predicions.... Comparison of Sae-of-Pracice Models Comparison Resuls Advanages of he Vehicle Dynamics Model 78. CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH. 78 CHAPTER FIVE: MODELING TYPICAL ACCELERATION BEHAVIOR INTRODUCTION 79. MAXIMUM MODEL FIELD TESTS Tes Procedure. 8.. Tes Drivers. 8. RESULTS 8.. Driver Facors and Classificaion 8.. Driver Summary... Disribuion. 7.. Age and Gender Variabiliy. LINEAR DECAY MODEL COMPARISON..6 IMPORTANCE OF MODELING A RANGE OF TYPICAL BEHAVIOR.7 CONCLUSIONS AND RECOMMENDATIONS.. 7 CHAPTER SIX: SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS SUMMARY CONCLUSIONS RECOMMENDATIONS. REFERENCES.. VITA... v

6 Lis of Tables Lis of Tables TABLE -: TEST VEHICLES. 8 TABLE -: SUMMARY OF TEST RUNS.. 9 TABLE -: SAMPLE VEHICLES TESTED... 9 TABLE -: MODEL INPUT PARAMETERS 9 TABLE -: SAMPLE MODEL SPREADSHEET (SATURN SL).. TABLE -: NATIONAL DRIVER STATISTICS 8 TABLE -: TEST DRIVER CHARACTERISTICS. 8 TABLE -: CHI-SQUARED TEST CALCULATION 9 TABLE -: ANOVA TABLE, MALE VERSUS FEMALE TABLE -: ANOVA TABLE, YOUNG VERSUS OLD. TABLE -6: ANOVA TABLE, BOTH FACTORS TABLE -7: FUEL CONSUMPTION AND EMISSIONS RATES FOR VARIOUS DRIVER TYPES... 7 vi

7 Lis of Figures Lis of Figures FIGURE -: ACCELERATION MODELS FLOW CHART.. FIGURE -: SMART ROAD VERTICAL PROFILE. 7 FIGURE -: CONSTANT POWER MODEL FIT TO FIELD DATA FOR THE DODGE INTREPID FIGURE -: COMPARISON OF CONSTANT POWER MODEL TO VARIABLE POWER MODEL... FIGURE -: PRELIMINARY TEST DATA FROM VARIOUS DRIVERS 6 FIGURE -: ACCELERATION VS. TIME DATA.. 9 FIGURE -: MODEL AND FIELD MEASURED POWER VS. SPEED RELATIONSHIP (ACURA INTEGRA).... FIGURE -: SMART ROAD VERTICAL PROFILE... 7 FIGURE -: MODEL PREDICTIONS VERSUS FIELD DATA (GEO METRO).. FIGURE -: MODEL PREDICTIONS VERSUS FIELD DATA (PLYMOUTH NEON)... FIGURE -6: MODEL PREDICTIONS VERSUS FIELD DATA (ACURA INTEGRA). 6 FIGURE -7: MODEL PREDICTIONS VERSUS FIELD DATA (SATURN SL) 7 FIGURE -8: MODEL PREDICTIONS VERSUS FIELD DATA (MAZDA PROTÉGÉ) 8 FIGURE -9: MODEL PREDICTIONS VERSUS FIELD DATA (HONDA ACCORD). 9 FIGURE -: MODEL PREDICTIONS VERSUS FIELD DATA (FORD TAURUS) FIGURE -: MODEL PREDICTIONS VERSUS FIELD DATA (BMW).. FIGURE -: MODEL PREDICTIONS VERSUS FIELD DATA (DODGE INTREPID).. FIGURE -: MODEL PREDICTIONS VERSUS FIELD DATA (CROWN VICTORIA) FIGURE -: MODEL PREDICTIONS VERSUS FIELD DATA (CHEVY BLAZER) FIGURE -: MODEL PREDICTIONS VERSUS FIELD DATA (FORD WINDSTAR). FIGURE -6: MODEL PREDICTIONS VERSUS FIELD DATA (CHEVY S-) 6 FIGURE -7: DUAL-REGIME MODEL FIT TO FIELD DATA 8 FIGURE -8: DUAL-REGIME MODEL FIT TO FIELD DATA 9 FIGURE -9: DUAL-REGIME MODEL FIT TO FIELD DATA 6 FIGURE -: DUAL-REGIME MODEL FIT TO FIELD DATA 6 FIGURE -: DUAL-REGIME MODEL FIT TO FIELD DATA 6 FIGURE -: LINEAR DECAY MODEL FIT TO FIELD DATA.. 6 FIGURE -: LINEAR DECAY MODEL FIT TO FIELD DATA.. 6 FIGURE -: LINEAR DECAY MODEL FIT TO FIELD DATA.. 6 FIGURE -: LINEAR DECAY MODEL FIT TO FIELD DATA.. 66 FIGURE -6: LINEAR DECAY MODEL FIT TO FIELD DATA.. 67 FIGURE -7: POLYNOMIAL MODEL FIT TO FIELD DATA. 68 vii

8 Lis of Figures FIGURE -8: POLYNOMIAL MODEL FIT TO FIELD DATA. 69 FIGURE -9: POLYNOMIAL MODEL FIT TO FIELD DATA. 7 FIGURE -: POLYNOMIAL MODEL FIT TO FIELD DATA. 7 FIGURE -: POLYNOMIAL MODEL FIT TO FIELD DATA. 7 FIGURE -: SEARLE MODEL FIT TO FIELD DATA 7 FIGURE -: SEARLE MODEL FIT TO FIELD DATA 7 FIGURE -: SEARLE MODEL FIT TO FIELD DATA 7 FIGURE -: SEARLE MODEL FIT TO FIELD DATA 76 FIGURE -6: SEARLE MODEL FIT TO FIELD DATA 77 FIGURE -: CROWN VIC MAXIMUM ACCELERATION DATA ON SMART ROAD WITH RAKHA MODEL.. 8 FIGURE -: RAKHA MODEL FOR CROWN VIC MAXIMUM ACCELERATION ON LEVEL ROADWAY. 8 FIGURE -: EXAMPLE PROFILES OF THREE DIFFERENT DRIVER TYPES FIGURE -: EXAMPLE EFFECT OF DRIVER FACTORS ON ACCELERATION PROFILE 8 FIGURE -: DRIVER # DATA WITH RAKHA MODEL, REDUCTION FACTOR = FIGURE -6: DRIVER # DATA WITH RAKHA MODEL, REDUCTION FACTOR = FIGURE -7: DRIVER # DATA WITH RAKHA MODEL, REDUCTION FACTOR = FIGURE -8: DRIVER #7 DATA WITH RAKHA MODEL, REDUCTION FACTOR = FIGURE -9: DRIVER #8 DATA WITH RAKHA MODEL, REDUCTION FACTOR = FIGURE -: DRIVER #9 DATA WITH RAKHA MODEL, REDUCTION FACTOR =.6. 9 FIGURE -: DRIVER # DATA WITH RAKHA MODEL, REDUCTION FACTOR =. 9 FIGURE -: DRIVER # DATA WITH RAKHA MODEL, REDUCTION FACTOR =.6 9 FIGURE -: DRIVER # DATA WITH RAKHA MODEL, REDUCTION FACTOR =.8 9 FIGURE -: DRIVER # DATA WITH RAKHA MODEL, REDUCTION FACTOR =.8 9 FIGURE -: DRIVER #6 DATA WITH RAKHA MODEL, REDUCTION FACTOR =. 96 FIGURE -6: DRIVER #7 DATA WITH RAKHA MODEL, REDUCTION FACTOR =. 97 FIGURE -7: DRIVER #8 DATA WITH RAKHA MODEL, REDUCTION FACTOR =. 98 FIGURE -8: DRIVER #9 DATA WITH RAKHA MODEL, REDUCTION FACTOR =.7 99 FIGURE -9: DRIVER # DATA WITH RAKHA MODEL, REDUCTION FACTOR =.7 FIGURE -: DRIVER # DATA WITH RAKHA MODEL, REDUCTION FACTOR =.7 FIGURE -: DRIVER # DATA WITH RAKHA MODEL, REDUCTION FACTOR =.7 FIGURE -: DRIVER # DATA WITH RAKHA MODEL, REDUCTION FACTOR =.7. FIGURE -: DRIVER # DATA WITH RAKHA MODEL, REDUCTION FACTOR =.7. FIGURE -: DRIVER #6 DATA WITH RAKHA MODEL, REDUCTION FACTOR =.7. 6 FIGURE -: OBSERVED TYPICAL SPEED PROFILES VERSUS MAXIMUM PROFILE.. 7 FIGURE -6: DISTRIBUTION OF OBSERVED DRIVER FACTORS.. 8 viii

9 Lis of Figures FIGURE -7: NORMAL DISTRIBUTION FIT TO OBSERVED DRIVER FACTORS 8 FIGURE -8: CUMULATIVE NORMAL DISTRIBUTION FIT TO DATA. 8 FIGURE -9: DRIVER RUN DATA WITH DISTRIBUTION MEAN AND SELECTED PERCENTILES. 9 FIGURE -: SAMPLE GRADUAL ACCELERATION DRIVER VERSUS LINEAR DECAY MODEL.. FIGURE -: SAMPLE STANDARD ACCELERATION DRIVER VERSUS LINEAR DECAY MODEL.. FIGURE -: SAMPLE HARD ACCELERATION DRIVER VERSUS LINEAR DECAY MODEL.. FIGURE -: SAMPLE SPEED PROFILES FOR DIFFERENT DRIVER FACTORS.. 6 ix

10 Chaper : Inroducion Chaper One: Inroducion. Background Informaion Vehicle behavior is one of he mos imporan -- ye surprisingly one of he mos ofen overlooked -- facors relaing o safe and efficien flow of raffic. Donald Drew poins ou ha "he raffic engineer, paradoxically, has done lile o sudy and undersand he very elemen ha is responsible for his professional exisence [he vehicle]" (968). Vehicle acceleraion in paricular has a significan impac on several facors in raffic engineering. These include he analysis of signalized inersecions, he applicaion of microsimulaion raffic modeling, and he design of roadway characerisics. The acceleraion capabiliy of a vehicle is also an imporan facor in he invesigaion of cerain road accidens. In addiion, accurae models of ypical acceleraion behavior of vehicles are an essenial componen for some of he sae-of-he-ar emissions models being developed (Rakha and Ahn, ). However, due o he wide variey of vehicle ypes on he road and he wide variey of driver behavior exhibied on our roadways, i is difficul o model he acceleraion behavior of vehicles when considering all hese facors. This paper serves o address some of hese issues.. Problem Definiion Exising acceleraion models in he lieraure ypically use kinemaics o describe acceleraion behavior and herefore end o generalize all personal vehicles ino one caegory. Simply dividing raffic ino rucks and cars is no longer sufficien o properly model he differences beween he wide variey of vehicles on he road. Also, many of he models only describe maximum acceleraion behavior or do no accuraely describe ypical driver acceleraion behavior in he field, oher han o give i a fixed percenage of he maximum acceleraion value. Furhermore, no complee model has been sysemaically developed and verified o describe acceleraion behavior of personal auomobiles based on he forces acing on he vehicle, i.e. vehicle dynamics.. Thesis Objecives The objecive of his hesis is o develop a vehicle dynamics model o predic acceleraion paerns of vehicles ranging from compac cars o SUV's for boh maximum acceleraion behavior and ypical driver behavior, and o es he model hrough sudies in boh a conrolled environmen (Smar Road es faciliy) and in acual driving condiions.. Thesis Conribuions The hesis makes several significan conribuions:

11 Chaper : Inroducion Firs, he hesis validaes a consan power and a variable power vehicle dynamics model o predic he acceleraion behavior of personal auomobiles. The models are calibraed o describe boh maximum acceleraion behavior and ypical acceleraion behavior. The models will be useful for several purposes. They will help o define vehicle acceleraion paerns in microsimulaion sofware packages, and can also be used in oher models -- such as emissions models -- whose variables are relaed o acceleraion raes. Second, he hesis examines he effec of various variables on he accuracy of he models. These variables include vehicle ype, vehicle age, vehicle weigh, driver gender, and driver age. The paper addresses how he vehicle and driver characerisics affec he dispersion of he vehicle speed daa. Third, he hesis provides a fresh se of vehicle acceleraion daa. Much of he daa used o verify exising models is oudaed or exrapolaed o reflec curren condiions. The hesis provides vehicle speed and posiion daa colleced by GPS unis every second for vehicles ha were acually in-use during he year. The daa is also caegorized by vehicle ype, vehicle properies, and driver characerisics. This new daa can be used o compare model effeciveness, and es he validiy of previously exrapolaed daa.. Thesis Layou The hesis consiss of six chapers. I firs presens a lieraure review, followed by a descripion of he research mehodology applied. Subsequenly, consan power and variable power vehicle dynamics models are presened and verified for maximum vehicle acceleraion and for ypical acceleraion behavior. Finally, conclusions are drawn, and recommendaions for furher research are presened. Chaper discusses he exising models for predicing vehicle acceleraion behavior in he lieraure and documens where exising research is lacking. Chaper describes he research approach aken. The secion oulines he model developmen procedure, summarizes he vehicle esing ha was performed, and discusses how he models were validaed. Chaper presens a consan power model and a variable power model for predicing maximum vehicle acceleraion behavior and compares he models o field daa colleced for a variey of vehicles. Chaper presens variaions o he consan power and variable power models presened in Chaper o reflec ypical driver behavior, and compares he resuls o field daa obained for differen driver ypes. Chaper 6 provides a summary of he findings and recommends fuure research acions.

12 Chaper : Lieraure Review Chaper Two: Lieraure Review. Inroducion Effecively modeling he acceleraion behavior of vehicles is an imporan componen in a variey of raffic engineering applicaions. The AASHTO Geomeric Design of Highways and Srees poins ou ha he acceleraion raes of vehicles are ofen criical parameers in deermining highway design (99). They are imporan in dimensioning inersecions, ramps, climbing lanes, and oher roadway feaures. Acceleraion models are also an inegral par of microscopic raffic simulaion models. The fundamenal movemen of he vehicles hroughou a simulaion nework is based on acceleraion models. Deermining vehicle acceleraion capabiliy is also someimes imporan when invesigaing road accidens (Searle, 999). In addiion, predicing fuel consumpion is dependen on he vehicle's acceleraion rae (Akcelik, 987). The vehicle's acceleraion rae also has a significan impac on vehicle emissions raes, and herefore effecive acceleraion models are needed in he developmen of sae-of-he-ar emissions models (Rakha and Ahn, ). Consequenly, several researchers have developed models in an aemp o predic he speed and acceleraion profiles of he differen vehicles on he roadway o be used for he various purposes described above. Essenially, hese models can be broken down ino wo ypes -- models ha predic behavior based on kinemaics of moion, and models ha incorporae vehicle dynamics. The following secions describe hese exising models, heir inended applicaions, and heir srenghs and weaknesses, followed by a conclusions secion ha documens where he exising research is lacking. Acceleraion Models Uses Types Simulaion Modeling Geomeric Design Fuel Consumpion Vehicle Emissions Kinemaics Models Dynamics Models Equaions of Moion Empirical Relaionships Tracive Engine Force Opposing Resisances Figure -: Acceleraion Models Flow Char

13 Chaper : Lieraure Review. Modeling Using Kinemaics Kinemaics models ake ino accoun he mahemaical relaionship beween acceleraion, speed, and disance raveled of any moving objec. The mos basic of hese models is he consan acceleraion model. In his model, he acceleraion of he vehicle is assumed o be consan for he enire period in which he vehicle is acceleraing. Because acceleraion is he derivaive of speed, and speed is he derivaive of disance, he following relaionships can be generaed: (Drew, 968) dv = a [-] d v dv = v a d [-] v = v + a [-] x dx = v + ( a) d [-] x + a Where: a = acceleraion v = speed v = iniial speed x = disance = ime = v [-] While his is he oldes and mos commonly used model, i is eviden from field research ha vehicle behavior does no exhibi consan acceleraion, and ha he assumpion of consan acceleraion in a model produces erroneous resuls when compared o field daa. However, because of is simpliciy, he consan acceleraion model is sill used in several raffic simulaion packages. Field sudies have shown ha vehicles can accelerae a a higher rae when raveling a lower speeds. Therefore, Bham and Benekohal developed a dual-regime model (). This model is similar o he consan acceleraion model, excep ha wo differen acceleraion raes are used -- one for low speeds and anoher for high speeds. The wo acceleraion raes can be found from he average speed profile of vehicles from he field daa. Based on wo ses of field daa, i was deermined ha he second regime begins a a speed of approximaely m/s. I is unclear how his value was deermined or if differen cuoff speeds would be calculaed for differen daa ses. This model is used in he raffic simulaion model ILLISIM/CELLSIM. In an aemp o more accuraely describe acual vehicle behavior, a linear decay model was developed (also known as he non-uniform acceleraion model). This model assumes ha acceleraion raes vary inversely wih speed (Drew, 968). In his model, vehicles obain heir maximum acceleraion a a speed of, and he acceleraion rae decreases

14 Chaper : Lieraure Review linearly o a value of a he maximum speed. The vehicle s acceleraion behavior is represened by he following relaionship. a = α βv [-6] In his formulaion, α is a consan represening he maximum acceleraion rae, and he raio α/β represens he maximum speed ha can be aained. Using he fundamenal equaions of kinemaics, he following relaionships can be deermined from he linear decay model ha relae speed and disance raveled o ime. α β β v = ( e ) + ve [-7] β α α β v β x = ( e ) + ( e ) [-8] β β β Long has compiled a lis of ypical values for α and β from various sources as hey relae o differen vehicle ypes and driver behavior (). He found ha values for β are similar for each vehicle ype, and he values for α increase wih decreasing weigh-ohorsepower raios. His research was primarily focused on geomeric design consideraions. A similar model o he linear decreasing model, known as he linear acceleraion model, has also been developed. In his model, acceleraion decreases as a funcion of ime raher han as a funcion of speed (Lee, 977). Acceleraion of he vehicle is described in his model as: a = a + β [-9] Where: β = acceleraion slope The speed and disance relaionships derived from he model are: v = v + a +. β [-] x = x + v +.a +. 67β [-] This model is used in he raffic simulaion models INTELSIM and TEXAS. However, Bham and Benekohal have found a much sronger relaionship beween acceleraion and speed han acceleraion and ime, and do no recommend he use of his model (). Furher sudy of he acceleraion paerns of vehicles revealed some addiional characerisics of he speed profile. Sudies showed ha he rae of acceleraion was no maximum a ime, as he linear decreasing and linear acceleraion models sugges, bu raher equaled a ime and increased rapidly o is maximum value a a ime afer = (Dockery, 966). Afer addiional sudy of speed-ime daa colleced by he Sydney Universiy, Akcelik noed he following hree requiremens for an acceleraion model (987). - The speed profiles should indicae an S-shape. - Acceleraion rae mus equal a he sar and end of acceleraion. - Jerk (da/d) should equal a he sar and end of acceleraion.

15 Chaper : Lieraure Review Akcelik developed hree kinemaics acceleraion models primarily o assis in fuel consumpion modeling based on hese crieria (987). The firs of he hree models developed is he wo-erm sinusoidal model. This model presens an empirical mahemaical relaionship for acceleraion as a funcion of ime as described by he series of equaions below: a( ) = Cam (sin πθ + B sin πθ) [-] [ ( v v ) /( v )] B = a f v [-] / C = sin πθ + Bsin πθ [-] m m m θ m = = cos [( + ( + B ) ) / 8B] [-] π f Where: a m = maximum acceleraion v = average speed during acceleraion a v = final speed f = ime o reach maximum acceleraion m = oal acceleraion ime f To ensure ha acceleraion is greaer han zero, he value of B is resriced o values beween / and /. This condiion also resrics he value of θ m o values beween / and /. This means ha he maximum acceleraion can only occur a imes beween / and / of he oal acceleraion ime. Speed and disance can be calculaed from he model as follows: v( ) = v +.( v f v )[(+.B) cos πθ +.B cos πθ] [-6] x( ) = v +.( v f v)[( +.B).8 f sin πθ.7996b f sin πθ] [-7] This model does saisfy he condiion of zero acceleraion a he sar and end of acceleraion, bu does no accoun for he condiion of zero jerk. To accoun for his, Akcelik developed a hree-erm sinusoidal model. This model describes acceleraion as: a( ) = Ra m (. P cosπθ.cos πθ+ P cosπθ) [-8] P = 9π / )(. + ( v v )/( v v )) [-9] θ θ m m ( f a f = cos ( ( + 8P ) π =., P = ) /P), P [-] R = ( cos γ ) /(sin γ ) [-] γ = πθ m [-] Like he wo-erm sinusoidal model, his model is subjec o several resricions and is herefore limied in is applicaion. The value of P is resriced o values beween. and.. This equaes o a resricion on θ m o values beween.9 and.68. This resricion is more severe han he resricion in he wo-erm model. Equaions for speed and disance associaed wih he hree-erm sinusoidal model are as follows: 6

16 Chaper : Lieraure Review Ra m f P v( ) = v +.πθ Psin πθ.sin πθ + sin πθ π [-] Ra m f.78 x( ) = v + +.8P f cosπθ +.98 f cos πθ.p f cosπθ π f [-] Bham and Benekohal found ha he disance calculaed in he hree-erm sinusoidal model becomes negaive when P values oher han.68 are used (). The model herefore needs modificaion o allow differen values of P o be used. Akcelik feels his bes model is he polynomial model (987). This model generaes a peak in he acceleraion profile near he beginning of he acceleraion period, which more closely maches he field resuls. Acceleraion using he polynomial model is defined as: m a( ) = ra θ ( θ ) ( m >.) [-] m. 7ρ + (8ρ 8ρ + 7) m = ρ [-6] va v ρ = v v [-7] f ( m + )( m + ) ram = a avg [-8] m Relaionships for speed and disance raveled are calculaed as follows: m θ m θ v( ) = v + f ramθ. + [-9] m + m + x( ) = v + f m m θ θ ramθ + [-] 6 ( m+ )( m+ ) (m + )(m + ) The hree models developed by Akcelik require ha he oal disance raveled be known in order o calculae he acceleraion raes. This resrics he use of he model when his value is no known or canno be approximaed easily (Bham, ). Vara and Husher have developed hree models ha also aemp o replicae acceleraion curves observed in field daa by using mahemaical funcions (). These hree models include he haversine model, he sinusoidal model, and he riangular model. Each of hese models predics ha maximum acceleraion will occur a he cener of he acceleraion profile, bu each describes he acceleraion behavior of he vehicle in a slighly differen way. Acceleraion in he haversine model is described as: π a( ) = α sin θ = α sin [-] In his model, α represens he maximum acceleraion rae as i did in he linear decreasing model. From his relaionship, values for speed and disance are calculaed as: 7

17 Chaper : Lieraure Review 8 + = f f v v π π π α sin ) ( [-] + + = cos 8 ) ( ) ( f f v x π π α [-] Similarly, he sinusoidal model describes acceleraion, speed, and disance as: = a π α sin ) ( [-] f f v v ) ( = π α [-] + = f f a v v π π α cos ) ( [-6] + = π π π α ) / sin( ) ( f f f v x [-7] The riangular model models acceleraion in he shape of an isosceles riangle. The peak represening maximum acceleraion occurs a ime f /, and he acceleraion profile is broken ino wo regimes as follows:, ) ( f f a = α [-8] f f f a =, ) ( α α [-9] Equaions for speed and disance are defined as:, ) ( f f v v + = α [-] f f f f v v + =, ) ( α α α [-], ) ( f f v x + = α [-] f f f f a v x + + =, ) ( α α α [-] The assumpion ha he maximum acceleraion occurs in he middle of he acceleraion profile, as required in he hree models described by Vara and Husher, is no consisen wih field daa. As a resul, hese models produce unrealisic acceleraion and speed

18 Chaper : Lieraure Review profiles (Bham, ). Bham and Benekohal have ried o address his problem by modifying he riangular model o vary he locaion of he maximum acceleraion. Insead of inroducing he second regime a ime f /, hey use he observed ime of maximum acceleraion as he break poin. However, i is very difficul o accuraely predic he ime of maximum acceleraion due o he variey of vehicle capabiliies. Therefore, he applicaion of he modified riangular model is limied. Bham and Benekohal also developed heir own model based on he gamma saisical densiy funcion (). They chose his funcion because i had a similar shape o he FHWA daa acceleraion profile. The gamma model describes acceleraion from a sopped posiion as follows: i α α β β ( ) = i e a * η, α >, β > [-] Γ( α) n ln i i= ln α + ψ ( α) = n [-] αβ = (n) [-6] Where: ψ(α) = digamma funcion Γ(α) = gamma funcion n = oal number of observaions in field daa i = ime in seconds (n) = average duraion of ime field daa was observed While he gamma model has been shown o perform well for cerain daa ses, he exensive calibraion required and he complexiy of he model are limiaions. The kinemaics models presened in his secion aemp o empirically develop mahemaical expressions o describe he acceleraion paerns of he vehicle. In doing so, he acual componens ha go ino providing he moion of he vehicle he racive force provided by he engine and he opposing resisance forces affecing he vehicle have essenially been ignored. Therefore, a reasonable fi o field daa canno be expeced from hese kinemaics models in each of he plos of acceleraion versus ime, speed versus ime, and speed versus disance. To beer accoun for he acual physics of moion involved wih an acceleraing vehicle, vehicle dynamics models have been developed. These models are described in he following secion.. Modeling Using Vehicle Dynamics Vehicle dynamics describes he forces acing on he vehicle ha resul in is moion. Tracive effor and resisance are he wo primary opposing forces ha deermine he performance characerisics of road vehicles (Mannering, 99). The engine in he vehicle supplies he racive-effor force, and he magniude of his force is resriced by 9

19 Chaper : Lieraure Review inernal fricion losses. This force is opposed by he forces of air resisance, rolling resisance, grade resisance, and fricion resisance (Drew, 968). Only a few acceleraion models have been developed ha incorporae vehicle dynamics. Rakha e al. pu forh a consan power model and a variable power model o deermine he performance of rucks (). Their research is he basis for his hesis, and hese models are described fully in he following chaper. Searle also uilized vehicle dynamics when he prepared equaions for speed, ime, and disance of vehicles under maximum acceleraion o help wih he invesigaion of road accidens (999). The model does no accoun for he specific effecs of he resisances on he vehicle, bu raher predics vehicle performance based on he raio of engine oupu o he weigh of he vehicle by generaing a power consan, k. 7.9 * η * Pmax k = [-7] M where: k = power consan, bhp/on (kilowa/on) η = acceleraion efficiency P max = maximum engine power, bhp (kilowas) M = mass of vehicle, onnes The acceleraion efficiency erm (η) accouns for losses in he ransmission as well as resisances o moion, such as air resisance. The power consan, k, is hen used o predic he speed of he vehicle and disance raveled, as given in he following equaions. v = v + k [-8] v = v + kx [-9] [( v + k) v ] x = [-] k The Searle model provides a reasonable approximaion of speeds ha can be obained by vehicles wihin a specific ime frame or over a cerain disance. This is useful when rying o recreae accidens for invesigaions. However, he model is no as useful in describing he overall acceleraion behavior of vehicles, as is required in simulaion sofware packages. I does no accuraely describe he moion of he vehicle a he beginning of acceleraion from a sop or a high speeds. Bham and Benekohal also found ha he model overesimaes speed during he iniial sages of acceleraion because i was formulaed o represen maximum acceleraion capabiliy (). They also found ha he model overesimaes he disance profile. Therefore, hey recommended a modificaion o he Searle model ha would be a beer fi o field daa. In heir model, hree differen values of he power consan, k, are used during differen ime periods. The k values can be calculaed from field daa as follows: v v 9x v k = = = [-] 8 x

20 Chaper : Lieraure Review Using his model provided a good fi o he field daa. However, i is difficul o deermine appropriae breaking poins for he differen regimes, and hese breaking poins vary for differen daa ses. This model also requires exensive calibraion and is no very easy o use.. Maximum vs. Typical Acceleraion The majoriy of hese models only predic he maximum acceleraion capabiliies of a vehicle. However, drivers rarely use he full capabiliies of he vehicle while driving in heir everyday lives. The maximum acceleraion characerisics are no useful in design excep as bounding values (Long, ). Therefore, i is imporan o predic he ypical driver acceleraion behavior exhibied by drivers. However, due o large number of variables -- vehicle ype, driver gender, driver age, driver's mood, ec. -- a limied amoun of research has been done in his area. Searle recognized ha drivers use only a porion of he poenial capabiliy of heir vehicle under normal driving condiions (999). He also noed ha because he driver conrols he acceleraion paern, here is no guaranee ha he acceleraion curve will follow he same shape as ha for maximum performance. He concluded ha he only way o properly accoun for acceleraion a everyday levels was o observe and measure acual vehicles. Searle observed subjec vehicles saring a an inersecion and heading along a long sraighaway wih a speed limi of 6 mph. The vehicles did no know hey were paricipaing in he experimen. Searle used a radar gun o rack he vehicle s speed as i acceleraed away, and recorded is make and model. Resuls from six subjec vehicles are presened in his work. I is unclear if hese were he only six vehicles esed or if hese were seleced as ones ha bes fi his model. From hese ess, he concluded ha normal driving behavior followed a similar pah as maximum performance, bu was less aggressive. This resuled in a linear relaionship beween velociy squared and ime, as his model suggess, bu wih smaller slopes han for he maximum acceleraion behavior. However, he paper did no ry o develop a mahemaical relaionship beween he maximum and ypical acceleraion values. Several researchers have aken he ypical acceleraion behavior of drivers o be some fixed percenage of he maximum acceleraion capabiliy hroughou he acceleraion period. Based on a series of road ess in he 9 s, Louzenheiser concluded ha he normal acceleraion rae for drivers was abou wo-hirds of he maximum value (98). The 9 Blue Book declared ha he acceleraion rae of he average driver was abou 6% of he maximum rae for he vehicle (AASHTO). Daa presened by Long showed ha his approximaion was inconsisen wih field daa (). The slope of he acceleraion curve was oo fla and he maximum acceleraion rae is underesimaed. In general, Long recommends using a linearly decreasing acceleraion model wih α = and β =. o esimae normal driver behavior.

21 Chaper : Lieraure Review. Comparison of Exising Models Several sources in he lieraure have aemped o compare differen exising acceleraion models using various daa ses. Akcelik compared he hree models he developed o he consan and linear decreasing acceleraion models using he daa exraced from he Sydney field ess (987). He compared he models based on heir abiliy o predic he acceleraion disance and fuel consumpion of he vehicles, based on heir corresponding relaive errors. He broke down he comparisons ino hree differen driving environmens, including cenral business disric, urban, and non-urban. He compared he models under hree ses of condiions: - Acceleraion ime and disance known - Acceleraion ime known bu disance unknown - Acceleraion ime and disance unknown The percenage errors in disance and fuel consumpion were found for each of he models. The mean and sandard deviaion of he percenage errors were also calculaed. Akcelik noed ha he sandard deviaion for percenage errors varied only slighly, and could herefore be aribued o driver behavior. The mean percenage error herefore represens he bias in he model. Afer conducing he saisical evaluaion of he models, Akcelik concluded ha he polynomial model was he bes overall and ouperformed he oher models in predicing acceleraion disance and fuel consumpion. Bham and Benekohal performed an exensive comparison of foureen models (). These models included exising kinemaics and dynamics models, models hey developed, and modificaions o exising models ha hey proposed. A lis of he models analyzed is given below: - Gamma Model - Single-Regime Model (consan acceleraion) - Dual-Regime Model - Searle Model - Modified Searle Model - Non-Uniform Acceleraion Model (linearly decreasing acceleraion) - Linear Acceleraion Model - Polynomial Model - Two-Term Sinusoidal Model - Three-Term Sinusoidal Model - Haversine Model - Sinusoidal Model - Triangular Model - Modified Triangular Model

22 Chaper : Lieraure Review A variey of saisical ess were used o compare he differen acceleraion models based on heir compliance wih daa colleced by FHWA in 98. The paired -es, and six error ess including roo mean square percen error, roo mean square error, mean percen error, posiive and negaive mean percen errors, and maximum absolue error were applied as hey relaed o he speed profile. The auhors chose o perform a variey of ess because he srengh of each model varied depending on he es applied. The goal was o find he models ha performed consisenly beer han he oher models. Based on heir ess, he hree models ha performed he bes overall were he modified Searle, gamma, and dual-regime models. The auhors noe ha he modified Searle model and he gamma model require deailed calibraion and are very difficul o use. Therefore, despie heir abiliy o mach field daa, hese models are no recommended. The auhors recommend he use of he dual regime model for use in raffic simulaion models. I should be noed ha he polynomial model, which was deermined o perform bes by Akcelik agains he single-regime model, he non-uniform model, he wo-erm sinusoidal model, and he hree-erm sinusoidal model, consisenly ranked beween fifh and sevenh ou of he foureen models for he various error ess performed by Bham and Benekohal. The polynomial model was ouperformed by he single-regime model in four of he six error ess using he FHWA daa, and was ouperformed by he non-uniform, wo-erm sinusoidal, and hree-erm sinusoidal models in he paired -es. This demonsraes ha he effeciveness of each of hese models is dependan on he daa se used and he comparison es applied, which suggess ha empirical mahemaical models based on one daa se may no be he mos effecive way o model vehicle acceleraion behavior..6 Conclusions Several aemps have been made o model he acceleraion paerns of vehicles and drivers. The models range from simplisic approximaions of consan acceleraion o elaborae saisical disribuions designed in an aemp o duplicae he inricae paern of acceleraion profiles observed in field daa. Comparison sudies show ha he effeciveness of hese models ofen varies depending on he se of field daa used or he comparison es applied. This is parly because he majoriy of he models aemp o use empirical kinemaics relaionships o model a sysem ha is governed by physics. The lieraure is lacking a complee dynamics model for vehicles ha calculaes acceleraion based on he acual forces and resisances ha are responsible for providing he moion. Insufficien sudy has also been done o relae he ypical acceleraion paerns of drivers o he maximum capabiliies of heir vehicles. A survey of he lieraure also shows ha many of he daa ses used o validae or compare models are oudaed. Daa is lacking ha is curren, horough, uses modern vehicles, and conains informaion on vehicle and driver ype. Much of he daa colleced is from a raffic sream, where acceleraion is limied by car following. Using his daa, here is also no way o deermine he ype of cars involved, he roadway condiions, or oher feaures ha conribue o he acceleraion behavior of he vehicle, such as roadway

23 Chaper : Lieraure Review grade. Fresh daa, complee wih vehicle and driver informaion, colleced in a conrolled environmen wih unopposed acceleraion is necessary o validae hese acceleraion models. I is eviden from everyday driving ha he acceleraion paerns of he various vehicles and driver ypes raveling on he roadways have large variaions. However, no exising model has aemped o address dispersion in he speed profile. An acceleraion model ha can readily generae a disribuion of acceleraion and speed profiles for various vehicle and driver ypes ha is based on updaed daa from modern vehicles would be a valuable ool in simulaion modeling and in oher applicaions.

24 Chaper : Research Mehodology Chaper Three: Research Mehodology. Inroducion The following research approach has been developed in an aemp o creae a vehicle acceleraion model ha overcomes he shorcomings of prior research effors. The previous chaper addressed some of he problems wih exising acceleraion models -- heir reliance on oudaed daa, heir dependence on empirical relaionships, and heir inabiliy o predic he ypical acceleraion paerns of drivers. The research mehodology of his hesis involves four primary asks ha have been designed o address some of he problems seen in exising research effors. The firs ask involves esablishing a new daabase hrough field ess. The second ask involves creaing a model o predic he maximum acceleraion raes of vehicles based on he vehicle and roadway characerisics. The hird ask involves modifying he model o replicae he ypical acceleraion behavior of various drivers. The final ask involves comparing he models developed wihin his hesis o oher exising models using various daa ses. These four asks are described compleely in he following secions.. Esablishing a New Daabase The firs ask in his research effor was o esablish a new daabase of vehicle acceleraion daa. A fresh se of daa was necessary for several reasons. Firs, he available daa in he lieraure is ou-daed. In heir comparison sudy of exising models, Bham and Benekohal () were forced o use daa ses ha were colleced in 968 and 98. These ses were used because more recen daa ses were no exensive enough and did no capure he acceleraion behavior of vehicles for a sufficien amoun of ime. Obviously, he acceleraion capabiliies of vehicles have changed dramaically since hese daa ses were colleced. Bham and Benekohal ried o overcome his by exrapolaing he rend from he 968 daa hrough he 98 daa o accoun for modern vehicle capabiliies. The field ess developed for his hesis will creae a new daa se for modern vehicles and herefore eliminae he need o exrapolae resuls from old daa. Anoher limiaion of exising daa is ha here is rarely informaion available on he vehicle ypes, roadway characerisics, and drivers involved in he daa collecion. The speed and acceleraion daa is jus grouped ogeher for all vehicles raveling along various grades on differen pavemen ypes. Common sense suggess ha he acceleraion behavior of a vehicle is grealy affeced by he power he vehicle can generae, he grade he vehicle is raveling on, and he resisance encounered by he vehicle. However, he exising daa ses do no address hese issues. The speed and acceleraion daa colleced for his research will be complee wih vehicle daa, pavemen daa, and driver daa. Finally, he exising daa ses are ofen colleced from a raffic sream, where acceleraion is limied by car following behavior. The field ess performed here will enable he collecion of speed daa from vehicles in a conrolled environmen, where he acceleraion is no limied by exernal facors oher han he

25 Chaper : Research Mehodology vehicle capabiliy and he driver s desire o accelerae a a given level. This is appropriae because he models developed in his hesis are designed o represen he lead vehicle acceleraing from a sop line. Many car-following models have been developed o predic he acceleraion behavior of vehicles behind he lead vehicle, which is ouside he scope of his paper... Smar Road Tesing The following secion describes he procedures followed during he firs se of field ess of vehicles for he purposes of daa collecion.... Sudy Secion Descripion The firs se of esing was performed during summer on he Smar Road es faciliy a he Virginia Tech Transporaion Insiue in Blacksburg, Virginia. Currenly, he Smar Road is a.-km (.9 mile) roadway ha will be expanded o a.-km experimenal highway in souhwes Virginia ha spans varied errain, from in-own o mounain passes. The horizonal layou of he es secion is fairly sraigh wih some minor horizonal curvaure ha does no impac vehicle speeds. The verical layou of he secion demonsraes a subsanial upgrade ha ranges from 6% a he lefmos end o.8% a he righmos end, as illusraed in Figure -. In consrucing he verical profile of he es secion he elevaion of saions were surveyed, as indicaed in Figure -. The verical profile of he es secion was hen generaed by inerpolaing beween saion elevaions using a cubic spline inerpolaion procedure a -m (.8-f) incremens. The cubic spline inerpolaion ensures ha he elevaions, he slopes, and he rae of change of slopes are idenical a he boundary condiions (in his case every meer). The grade was compued for each -m (.8-f) secion and was found o vary considerably, as illusraed in Figure - (hin line). A polynomial regression relaionship was fi o he grade daa (R of.9) for wo reasons. Firs, his ensured a smooh ransiion in he roadway grade while mainaining he same verical profile. Second, i also faciliaed he soluion of he ODE because i ensures ha he grade funcion is coninuous. The modified grade and verical elevaion, which are illusraed in Figure - (hick line), demonsrae an almos idenical verical profile wih much smooher grade ransiions when compared o he direc inerpolaion. Apar from a -m (9-f) segmen of he roadway ha was a rigid pavemen, he enire roadway surface was asphal. Consequenly, a rolling resisance coefficien for asphal pavemen only was uilized. The qualiy of he road surface was fair a he ime he es runs were conduced. These facors were imporan in idenifying he road surface rolling resisance coefficiens. 6

26 Chaper : Research Mehodology.7.6 Grade (percen) Elevaion (m) Inerpolaed Elevaion Regression Elevaion Saion Elevaions Figure -: Smar Road Verical Profile... Speed Measuremens Before each run, es vehicles were equipped wih a Global Posiioning Sysem (GPS) uni ha measured he vehicle speed o an accuracy of. m/s. (. f/sec). GPS is a worldwide, saellie-based radio-navigaion sysem ha can deermine wih cerain accuracy he posiion and velociy of any objec equipped wih a GPS receiver. Typical oupu daa from GPS receivers include laiude, longiude, aliude, speed, heading, and ime. The GPS receivers used were able o updae hese parameers once every second. Nominal posiion accuracy is specified wih a -m (8-f) spherical error probabiliy, while nominal velociy accuracy is specified wihin. m/s (. f/sec) error probabiliy. These inaccuracies are aribued o a number of sources of error. The majoriy of hese errors are linked o he way he disance beween a saellie and a GPS receiver is measured. Wihin he sysem, disances are measured by calculaing he ime i akes for a signal o ravel beween a saellie and a receiver. Consequenly, any delay in he signal ransmission hen resuls in disance overesimaion and inaccuracies in he esimaed posiion of he objec. 7

27 Chaper : Research Mehodology... Vehicle Descripions Foureen vehicles were chosen for use in our field ess (See Table -). These vehicles were chosen o reflec he populaion of passenger vehicles on he roads in he year. Vehicles were chosen from faculy and sudens a he Virginia Tech Transporaion Insiue, and herefore consiue he various ypes of auomobiles ha were in use a he ime. The vehicles represen a wide range of sizes and a variey of EPA vehicle classes including subcompac cars, compac cars, midsize cars, large cars, spor uiliy vehicles, pickup rucks, and minivans. Many characerisics of each vehicle were recorded including weigh, lengh, fronal area, ype of ires, engine ype and power, and air drag coefficien. This daa, as well as he performance daa from he vehicle esing, will be discussed furher and analyzed in Chapers and. Table -: Tes Vehicles 99 Acura Inegra SE 99 Chevy Blazer 99 Chevy S 99 Dodge Inrepid 99 Saurn SL 99 BMW 7I 996 Geo Mero Hachback 998 Ford Windsar 998 Ford Taurus 998 Honda Accord 999 Ford Crown Vicoria Mazda Proégé LX. Plymouh Neon... Tes Run Descripion Each of he foureen vehicles was subjeced o he same se of ess. The es runs involved acceleraing he vehicles from a sop a he sar of he es secion up o various ending speeds. The ess were performed wice for each se of speeds. Firs, he drivers acceleraed a he maximum possible acceleraion rae up o he desired speed. Nex, he drivers performed he es while acceleraing a ypical raes as if hey were driving heir own vehicles in own. The hree speed ranges involved acceleraing he vehicle from mph o mph, mph o mph, and mph o he maximum speed ha could be achieved by he end of he es secion. Depending on he ype of vehicle, maximum speeds aained by he end of he es secion were beween 6 mph and mph. In conducing he sudy, a minimum of repeiions were execued for each es se in order o provide a sufficien sample size for he validaion analysis. This generaed a oal of es runs for each vehicle. Please see Table - for a summary of he es runs. 8

28 Chaper : Research Mehodology Table -: Summary of Tes Runs Maximum Acceleraion Typical Acceleraion - mph rials rials - mph rials rials Max rials rials.. Driver Behavior Tesing A second se of esing was performed in spring o es he effec of driver characerisics on ypical acceleraion behavior. For his es, he same vehicle (999 Ford Crown Vicoria) was used for all he rial runs o serve as he conrol in he experimen. Approximaely weny differen drivers were seleced for he es, and he drivers were broken down ino he following age brackes: o 9 years old, o 9 years old, o 9 years old, and years of age and older. Several drivers were chosen from each age group, and an even spli of men and women was chosen. The es secion used was a relaively fla and sraigh srip of roadway in Blacksburg, Virginia ha is conrolled by a sop sign and subjec o very ligh raffic volumes, herefore eliminaing he effec of car following behavior. The vehicle was equipped wih he same GPS receiver as was used during he Smar Road esing. Drivers were insruced o come o a sop a he sop sign and hen accelerae from ha poin, as hey would normally unil he end of he es secion marked by a cone. The es secion was approximaely fee in lengh and ypical speeds reached a he end of he es secion ranged from mph o 7mph, depending on he driver. The driver would hen come back o he sar line and coninue performing weny-five rial runs so ha sufficien sample daa could be obained. The oal esing ime was approximaely one hour. The daa from hese ess are analyzed in Chaper.. Predicing Maximum Acceleraion Raes Afer he new daabase is esablished, he nex ask was o develop a model o predic he maximum acceleraion raes of differen vehicles. The maximum acceleraion rae for a vehicle can be deermined readily based on several vehicle and roadway characerisics, as proposed by Rakha e al. () and described in he following secion. As was discussed in he previous chaper, drivers rarely accelerae a he maximum rae available o hem. However, i is sill imporan o deermine he maximum acceleraion rae as a saring poin before aemping o apply ypical driver behavior o he model... Truck Dynamics Model The acceleraion model presened in his repor is an adapaion of a previous model developed a Virginia Tech (Rakha e al., ). The Rakha model, as i will be referred o in his ex, is a vehicle dynamics model designed o predic he acceleraion 9

29 Chaper : Research Mehodology characerisics of rucks. This model compues he maximum acceleraion based on he resulan force, as indicaed in Equaion -. Given ha acceleraion is he second derivaive of disance wih respec o ime, Equaion - resolves o a second-order Ordinary Differenial Equaion (ODE) of he form indicaed in Equaion -. The ODE is a funcion of he firs derivaive of disance (vehicle speed) because he racive effor, he rolling resisance, and aerodynamic resisance forces are all funcions of he vehicle speed. In addiion, he ODE may be a funcion of he disance raveled if he roadway grade changes along he sudy secion. I should be noed a his poin ha because he racive effor includes a minimum operand, he derivaive of acceleraion becomes a non-coninuous funcion. F R a = [-] M x & = f ( x&, x) [-] Where: F: residual force acing on he ruck (N), M: vehicle mass (kg), and R: oal resisance force (N)... Tracive Force The sae-of-pracice vehicle dynamics models esimae he vehicle racive effor using Equaion - wih a maximum value based on Equaion -, as demonsraed in Equaion -. Equaion - accouns for he maximum fricion force ha can be mainained beween he ires of he vehicle s racive axle and he roadway surface. The use of Equaion - ensures ha he racive effor does no approach infiniy a low vehicle speeds. Equaion - indicaes ha he racive force F is a funcion of he raio beween he vehicle speed u and he engine power P. The model assumes he vehicle power o be consan and equal o he maximum poenial power. The model considers wo main sources of power loss ha degrade he racive effor produced by he ruck engine. The firs source of power loss is caused by engine accessories including he fan, generaor, waer pump, magneo, disribuor, fuel pump and compressor. The second source of power loss occurs in he ransmission sysem. Typical ransmission efficiencies of rucks are assumed o be consan and range from.89 o.9 depending on he ype of ransmission (SAE J88, 996). The maximum racive force is a funcion of he proporion of he vehicle mass on he racive axle. Typical axle mass disribuions for differen ruck ypes and ypical axle mass disribuions were described in Rakha e al. () o range beween o percen. Alernaively, he percenage mass on he racive axle ypically ranges from o 6 percen for ligh duy vehicles. P F = 6 η [-] u

30 Chaper : Research Mehodology F = M a µ [-] max F = min ( F, F ) [-] max Where: F: racive force acing on he ruck (N), F max : maximum racive force (N), F : racive force (N), M a : vehicle mass on racive axle, M perc a (kg), perc a : percenage of vehicle mass on racive axle (%), M: vehicle mass (kg), P: engine power (kw), u: vehicle speed (km/h), η: power ransmission efficiency (ranges from.89 o.9), and µ: coefficien of fricion beween ires and pavemen... Aerodynamic Resisance Sae-of-he-ar vehicle dynamics models consider hree major ypes of resisance forces, including aerodynamic, rolling, and grade resisances as suggesed in he lieraure (Mannering and Kilareski, 99; Fich, 99; Archilla and De Cieza, 999; Rakha e al., ). The oal resisance force is compued as he sum of he hree resisance componens, as summarized in Equaion -6. The aerodynamic resisance, or air drag, is a funcion of he vehicle fronal area, he aliude, he ruck drag coefficien, and he square of speed of he vehicle, as indicaed in Equaions -7 and -8. The consan c accouns for he air densiy a sea level a a emperaure of o C (9ºF). Typical values of vehicle fronal areas for differen vehicle ypes and ypical drag coefficiens are provided in he lieraure (Rakha e al., ). Equaion -8 is a linear approximaion ha was derived from a more complex formulaion (Waanada e al., 987). The linear approximaion was found o provide similar resuls o he more complex formulaion for aliudes in he range of o m. R = R + R + R [-6] R a r g = c C C Au [-7] a d h C h = 8. H [-8] Where: R a : aerodynamic resisance (N), R r : rolling resisance (N), R g : grade resisance (N), A: fronal area (m ), C d : air drag coefficien, C h : aliude coefficien, H: aliude (m), and c : consan =.78.

31 Chaper : Research Mehodology.. Rolling Resisance The rolling resisance is a linear funcion of he vehicle speed and mass, as indicaed in Equaion -9. Typical values for rolling coefficiens (C r, c, and c ), as a funcion of he road surface ype, condiion, and vehicle ires, are provided in he lieraure (Rakha e al., ). Generally, radial ires provide a resisance ha is percen less han ha for bias ply ires. M R = C ( c u + c ) [-9] r r Where: M: vehicle mass (kg). u: vehicle speed (km/h). C r, c, c : rolling resisance consans... Grade Resisance The grade resisance is a consan ha varies as a funcion of he vehicle s oal mass and he percen grade ha he vehicle ravels along, as indicaed in Equaion -. The grade resisance accouns for he proporion of he vehicle weigh ha resiss he movemen of he vehicle: R g = M i [-] Where: M: vehicle mass (kg). i: grade magniude (m/ m)...6 Applying he Truck Model o Cars In heir research, Rakha e al. () showed ha he relaionships presened herein would give a reasonable approximaion of he maximum acceleraion raes of rucks for a variey of weigh o power raios. The firs goal of his hesis will be o deermine if he maximum acceleraion behavior of passenger vehicles can also be deermined using his model by comparing he speed and acceleraion values prediced by he model o he daa colleced in our field ess for he foureen es vehicles. Figure - demonsraes he good fi beween he consan power model and field daa colleced for he Dodge Inrepid during his research effor. Alhough he model developed by Rakha e al. showed reasonable correlaion o field daa, i did end o overesimae speeds a he beginning of he acceleraion profile for rucks. They deermined ha his was due o he loss of power a he beginning of acceleraion caused by gear shifing. To accoun for his, Rakha and Lucic developed a variable power model, which is described in he following secion ().

32 Chaper : Research Mehodology Spd (km/h) Dis (m) Figure -: Consan Power Model Fi o Field Daa for he Dodge Inrepid..7 Variable Power Model Rakha and Lucic () proposed he use of a variable power efficiency facor ha is dependen on he vehicle speed, as opposed o a consan ransmission efficiency facor ha is currenly uilized in sae-of-he-ar vehicle dynamics models. The power is assumed o increase linearly from an inercep of zero o a maximum value of. a a speed u, as demonsraed in Equaion -. In order o ensure ha he vehicle has sufficien power o accelerae from a speed of zero, a vehicle power lower bound is compued using a speed of km/h, as demonsraed in Equaion -. The adjusmen facor is hen muliplied by he vehicle power and incorporaed in Equaion - o compue he racive force, as demonsraed in Equaion -. The esimaion of he variable power facor β requires he calibraion wo parameers, namely, he minimum power and he speed a opimum power. Given ha he proposed model assumes he minimum power o be a funcion of he opimum speed, as demonsraed in Equaion -, only one parameer is calibraed, namely he speed a which he vehicle aains is opimum power. These parameers have been calibraed for rucks and research is currenly underway o calibrae similar relaionships for ligh duy vehicles. Specifically, he parameers were calibraed for engine powers ranging from 6 o 7 kw ( o hp), each involving en weigh configuraions. The calibraion demonsraed ha higher weigh-o-power raios required a lower opimum speed and a higher minimum power. The proposed power model addresses his need for a variable lower bound while mainaining a single calibraion parameer. Figure - illusraes he variaion in he vehicle power and he resuling vehicle acceleraion by incorporaing he power adjusmen facor ha is presened in Equaion -. As illusraed in he figure, he modificaion reduces he vehicle acceleraion levels a

33 Chaper : Research Mehodology lower speeds wih a minor aleraion of vehicle speeds a high speeds. I should be noed ha he use of exponenial smoohing ensures ha he acceleraion increases gradually as he vehicle acceleraes from a complee sop. Vehicle Power (kw) Acceleraion (m/s ) Consan Power Variable Power 6 8 Vehicle Consan Power Variable Power 6 8 Figure -: Comparison of Consan Power Model o Variable Power Model The calibraion of he variable power ransmission efficiency involves deermining he speed a which he vehicle power reaches is maximum (opimum speed). The opimum speed was found o vary as a funcion of he weigh-o-power raio. The specifics of how his relaionship was derived are discussed in he lieraure (Rakha and Lucic, ); however, i is sufficien o noe a his poin ha he relaionship is a power relaionship, as demonsraed by Equaion -. The linearly increasing power/speed relaionship resuls in a consan acceleraion as a funcion of vehicle speed given ha force (produc of he vehicle mass and acceleraion) is he firs derivaive of power wih respec o speed and ha he vehicle mass is a consan. u < u u [-] β = u u u. u > u F P = 6 β [-] u

34 Chaper : Research Mehodology u = 6w.7 [-] Where: β: Variable power ransmission efficiency (uniless). u: Vehicle speed (km/h). u : Speed a which vehicle aains maximum power (km/h). w: Vehicle weigh-o-power raio (kg/kw)...8 Calibraing he Variable Power Model This hesis will also examine he accuracy of he variable power model wih respec o predicing he acceleraion capabiliy of he foureen vehicles uilized in he field ess. This will require more effor, because he speed a which maximum power occurs mus firs be calibraed for each vehicle before he model can be applied. For rucks, his speed was relaed o he vehicle weigh-o-power raio, as described above. Rakha and Lucic found a beer correlaion beween ruck speeds and he speeds prediced by he variable power model in comparison o he consan power model (). This research aemps o deermine if i is appropriae or necessary o apply he variable power model o cars. Passenger cars carry much less mass han rucks and also have fewer gears, so he gearshifing effec may no be as subsanial as i is for rucks. If he consan power model could be shown o be as effecive as he variable power model for cars, i would be beneficial because he variable power model is more complex and requires addiional calibraion. These are he issues ha his hesis aemps o address in regards o predicing maximum acceleraion raes of passenger vehicles.. Predicing Typical Acceleraion Behavior The hird ask of his research effor involves modifying he consan power and variable power models o accoun for ypical acceleraion raes applied by drivers during heir normal driving aciviies. I is clear ha he acual acceleraion raes applied by drivers will be some fracion of he maximum acceleraion capabiliy. However, i is unclear wheher his is some fixed percenage hroughou he acceleraion of he vehicle or wheher he percenage of acceleraion power used varies as a funcion of speed or ime. I is also unclear how much variabiliy here is beween differen drivers or even wihin runs for he same driver. Therefore, he following procedure was developed for his ask: Firs, a consan reducion facor will be calibraed and ried based on he resuls of he field daa. This could be a fixed value or a range of values, depending on he dispersion of he resuls. The reducion facor would be muliplied by he maximum acceleraion prediced by he original models o ge he ypical acceleraion raes. The accuracy of his model will be checked agains he field daa. Second, a reducion facor ha varies as a funcion of speed will be incorporaed ino he model and checked for accuracy. As was discussed in he previous chaper, consan reducion facors ended o underesimae maximum acceleraion levels when applied o he linear decreasing model. By creaing a reducion facor ha varies wih speed, we hope o creae a beer fi o field daa.

35 Chaper : Research Mehodology A recommendaion for fuure acion would be o incorporae dispersion ino he model in an aemp o accoun for he variabiliy observed wihin a single driver and beween drivers. Resuls from he firs field ess performed seemed o indicae ha he variabiliy was minimal a low speeds and hen increased as speeds increased. See Figure -. A randomness facor could also be incorporaed ino he model ha would generae a disribuion of driver behavior wihin a raffic sream Speed (mph) 6 8 Disance (f) Figure -: Preliminary Tes Daa from Various Drivers. Comparison of Models The final ask in his hesis will be o compare he models presened in his repor o some of he bes-performing models currenly available in he lieraure. This will involve a saisical analysis of each of he models' performance agains various daa ses, including he one developed herein. We hope o show he benefi of using our models hrough his comparison..6 Summary This chaper described he research mehodology ha is being applied o he problem of developing a vehicle dynamics model o predic he maximum and ypical acceleraion raes of passenger vehicles. To accoun for some of he shorcomings of exising research effors on he opic, four asks have been idenified o enable he developmen and validaion of such a model. These asks include esablishing a new daabase, which involves performing field ess on various vehicles and drivers, applying exising maximum acceleraion models developed for rucks o cars, modifying he model o accoun for ypical driver behavior, and comparing he new models o curren sae-ofhe-ar models used in pracice. 6

36 Chaper : Maximum Acceleraion Chaper Four: Vehicle Dynamics Model for Esimaing Maximum Auomobile Acceleraion Levels. Inroducion Microscopic simulaion sofware packages use car-following models o capure he ineracion of a vehicle and is preceding vehicle raveling in he same lane. The process of car-following is modeled as an equaion of moion under seady-sae condiions plus a number of consrains ha govern he behavior of vehicles while moving from one seady-sae o anoher (deceleraing and acceleraing). Typically, up o wo consrains are considered. The firs consrain governs he vehicle acceleraion behavior, which is ypically a funcion of he vehicle dynamics. The second and final consrain ensures ha vehicles have a safe posiion relaive o he lead vehicle in order o decelerae o a complee sop wihou colliding wih he preceding vehicle in he even ha he preceding vehicle deceleraes o a complee sop. In addiion, vehicle acceleraion behavior is criical o he accurae modeling of vehicle fuel consumpion and emissions, as was demonsraed in previous research (Rakha and Ahn, ). This chaper exends he research of Rakha and Crowher (), which examined seady-sae car-following behavior by characerizing and modeling maximum vehicle acceleraion behavior when a vehicle is no consrained by oher surrounding raffic. A forhcoming chaper will characerize ypical vehicle acceleraion behavior for a single and plaoon of vehicles. The paper iniially presens sae-of-pracice vehicle acceleraion models and sae-ofpracice daa ses for he modeling of vehicle acceleraion behavior. Subsequenly, he experimenal design and he procedures for collecing new vehicle acceleraion daa are presened. The sae-of-pracice vehicle acceleraion models are hen applied o he daa and he validiy of he various models are characerized. Finally, he conclusions of he paper and recommendaions for furher research are presened.. Background.. Sae-of-Pracice Vehicle Acceleraion Models Several researchers have developed models ha predic vehicle speed and acceleraion profiles for inclusion in raffic simulaion models. Essenially, hese models can be broken down ino wo caegories, namely models ha predic acceleraion behavior based on kinemaics of moion and models ha consider vehicle forces in esimaing vehicle acceleraion (i.e. vehicle dynamics models). 7

37 Chaper : Maximum Acceleraion... Vehicle Kinemaics Models Vehicle kinemaics models ake ino accoun he mahemaical relaionship beween acceleraion, speed, and disance raveled for any moving objec. They generally sar wih an empirical mahemaical relaionship beween acceleraion and speed or acceleraion and ime. The values of speed are calculaed by inegraing acceleraion wih respec o ime while disance raveled is compued by inegraing speed wih respec o ime. Alhough similar in principle, kinemaics models can range from very simple o very complex. The mos basic of hese models is he consan acceleraion model. As he name suggess, his model uses a consan acceleraion value for he vehicle hroughou is acceleraion maneuver. Because of is simpliciy, his model is used in several raffic simulaion packages. However, research has shown ha he assumpion of consan acceleraion is erroneous. Specifically, vehicles end o achieve higher acceleraion raes while raveling a low speeds han hey a high speeds. To accoun for his behavior, several models have been suggesed. For example, a dual-regime model similar o he consan acceleraion model was proposed (Bham, ). The model considers wo consan acceleraion raes, a high rae for low speeds and a lower rae for high speeds. Anoher commonly used model is he linear decay model. In his model, he acceleraion rae of he vehicle sars a some maximum value a a zero speed and decreases linearly as a funcion of speed. Furher research suggesed ha he maximum acceleraion rae migh no acually occur a ime zero, bu raher a some ime shorly afer he sar of acceleraion. Field daa were gahered o characerize acual vehicle acceleraion paerns, such as he profile shown in Figure -. New models based on elaborae saisical disribuions and mahemaical funcions were designed in an aemp o duplicae he inricae paern of hese acceleraion profiles observed in field daa. Examples of hese models include a riangular model in which acceleraion increases linearly o is maximum before decreasing linearly (Vara, ), a Gamma model based on he gamma saisical densiy funcion (Bham, ), a polynomial model (Akcelik, 987), and several models based on he rigonomeric sine funcion (Vara, ; Akcelik, 987). More complee descripions of all hese models can be found in he lieraure. Previous sudies have demonsraed ha he dual-regime, linear decay, and polynomial models bes fi field daa and were herefore chosen for analysis in his paper. These hree models are described in more deail in he following secions. 8

38 Chaper : Maximum Acceleraion Acceleraion (m/s) Figure -: Acceleraion vs. Time Daa (Source: Ohio Sae Universiy Sudy, 968) a. Dual-Regime Model The dual-regime model is similar o he consan acceleraion model, excep ha wo separae acceleraion raes are used, one for low speeds and one for high speeds, as shown in he following equaions (Bham, ): v = v + a, v < m/s [-] i i v = v + a, v > m/s [-] i i In his formulaion, a and a represen he acceleraion raes for he firs and second regimes, respecively, and can be calculaed based on he resuls of field daa. Bham and Benekohal recommended a ransiion poin of m/s for making he swich from he high acceleraion rae o he lower rae when applying he model o heir daa. For his comparison sudy, a bes fi was approximaed for each vehicle, which did no necessarily have a ransiion poin a m/s. b. Linear Decay Model This linear decay acceleraion model assumes ha he acceleraion rae varies inversely wih speed (Drew, 968). In his model, vehicles aain heir maximum acceleraion a a speed of zero, and he acceleraion rae decreases linearly o a value of a he maximum speed. The vehicle s acceleraion behavior is represened by he following relaionship. a = α βv [-] 9

39 Chaper : Maximum Acceleraion In his formulaion, α is a consan represening he maximum acceleraion rae, and he raio α/β represens he maximum speed ha a vehicle can aain. By inegraing Equaion -, he following relaionships can be deermined from he linear decay model ha relae speed and disance raveled o ime. α β β v = ( e ) + v e [-] β α α β v β x = ( e ) + ( e ) [-] β β β c. Polynomial Model The polynomial model was designed o saisfy he condiions of zero acceleraion and zero jerk (firs derivaive of acceleraion wih respec o ime) a he sar and end of he acceleraion maneuver. This model generaes a peak in he acceleraion profile near he beginning of he acceleraion maneuver, which more closely maches he field resuls han oher similar models. Acceleraion a any insan is compued using Equaion -6 in conjuncion wih Equaions -7 hrough -9 (Akcelik, 987). m a( ) = ra θ ( θ ) ( m >.) [-6] m. 7ρ + (8ρ 8ρ + 7) m = [-7] ρ v a ρ = [-8] v f v v ( m + )( m + ) ra = a m avg [-9] m where: a m = maximum acceleraion v a = average speed v o = iniial speed v f = final speed a avg = average acceleraion By inegraing Equaion -6 he speed and disance raveled a any insan can be compued using Equaions - and -, respecively. m θ m θ v( ) = v + f ramθ. + [-] m + m + x( ) = v + f m m θ θ ramθ + [-] 6 ( m + )( m + ) (m + )(m + )

40 Chaper : Maximum Acceleraion... Vehicle Dynamics Models The problem wih he kinemaics models, including he ones described above, is ha by empirically developing mahemaical expressions o describe he acceleraion paerns of he vehicle, he acual componens ha go ino providing he moion of he vehicle he racive force provided by he engine and he opposing resisance forces resising he vehicle s moion are no modeled explicily. Therefore, in general he vehicle kinemaics models do no provide a good fi o field daa for each of he acceleraion, speed, disance, and ime domains. Furhermore, vehicle kinemaics models do no accoun for differen vehicle ypes, roadway grades, and oher facors ha affec he acceleraion paerns of vehicles. To beer accoun for he acual physics of moion of an acceleraing vehicle, acceleraion models based on vehicle dynamics have been developed. Vehicle dynamics describes he forces acing on he vehicle ha resul in is moion. Tracive effor and resisance are he wo primary opposing forces ha deermine he performance characerisics of road vehicles. The engine in he vehicle supplies he racive-effor force, and he magniude of his force is resriced by inernal fricion losses. The forces of air resisance, rolling resisance, grade resisance, and fricion resisance oppose he engine racive force and limi he acceleraion capabiliy of he vehicle. This secion describes wo vehicle dynamics models, namely he Searle and Rakha models. a. Searle Model The Searle model is an example of a vehicle dynamics model. Searle uilized vehicle dynamics when he prepared equaions for speed, ime, and disance of vehicles under maximum acceleraion o help wih he invesigaion of road accidens (999). The model does no accoun for he specific effecs of he resisances on he vehicle, bu raher predics vehicle performance based on he raio of engine oupu o he weigh of he vehicle by generaing a power consan, k. 7.9 η Pmax k = [-] M where: k = power consan, bhp/on (kilowa/on) η = acceleraion efficiency P max = maximum engine power, bhp (kilowas) M = mass of vehicle, ones The acceleraion efficiency erm (η) accouns for losses in he ransmission as well as resisances o moion, such as air resisance. The power consan, k, is hen used o predic he speed of he vehicle and disance raveled, as given in he following equaions. v = v + k [-] v = v + kx [-]

41 Chaper : Maximum Acceleraion x. [( v k) v ] + = [-] k The Searle model provides a reasonable approximaion of speeds ha can be obained by vehicles wihin a specific ime frame or over a cerain disance. This is useful when rying o recreae accidens for invesigaions. However, he model is no as useful in describing he overall acceleraion behavior of vehicles, as is required in simulaion sofware packages. I does no accuraely describe he moion of he vehicle a he beginning of acceleraion from a sop or a high speeds. A complee comparison of he Searle model o oher models is presened laer in he paper. b. Rakha Model Rakha e al. described and applied a sae-of-pracice consan power vehicle dynamics model o he modeling of ruck acceleraion behavior (), while Rakha and Lucic () developed a variable power vehicle dynamics model for predicing maximum ruck acceleraion levels. This paper seeks o apply hese models o ligh duy vehicles and validae he models agains field daa. The following secions describe he vehicle dynamics models pu forh by Rakha and Lucic as hey can be used o predic he maximum acceleraion raes of various passenger vehicles. The model is based on he basic principle of physics ha force equals mass imes acceleraion. If he ne force on he vehicle and he vehicle mass are known, he acceleraion of he vehicle can be deermined by he relaionship in Equaion -6. Noe ha he ne force on he vehicle is he difference beween he force applied by he vehicle and he various resisance forces he vehicle encouners as i ravels. The mass of he vehicle is consan, bu he magniude of he applied force and he resisance forces are variables. As you will see in he following secions, hese values change as a funcion of vehicle speed and disance raveled. Therefore, given ha acceleraion is he second derivaive of disance wih respec o ime, Equaion -6 resolves o a second-order Ordinary Differenial Equaion (ODE) of he form indicaed in Equaion -7. I should be noed a his poin ha because he racive effor includes a minimum operand, he derivaive of acceleraion becomes a non-coninuous funcion. F R a = [-6] M x & = f ( x&, x) [-7] Where: F: residual force (N), M: vehicle mass (kg), and R: oal resisance force (N). i. Tracive Force The firs variable in he fundamenal equaion of he Rakha model (Equaion -6) is he force applied by he engine in he vehicle. As Equaion -8 indicaes, he racive force F is a funcion of he raio beween he vehicle speed v and he engine power P. The

42 Chaper : Maximum Acceleraion model assumes he vehicle power o be consan and equal o he maximum poenial power. The model considers wo main sources of power loss ha degrade he racive effor produced by he vehicle engine. The firs source of power loss is caused by engine accessories including he fan, generaor, waer pump, magneo, disribuor, fuel pump and compressor. The second source of power loss occurs in he ransmission sysem. Values for he engine efficiencies of he hireen es vehicles used for daa collecion are provided laer. Equaion -8 suggess ha he force applied by he vehicle engine approaches infiniy as he speed of he vehicle approaches zero. However, he maximum aainable racive force is consrained by he fricion beween he vehicle ires on he racive axle and he roadway pavemen. Higher racive forces would resul in wheel spin. Therefore, Equaion -9 is used o cap he force value o an aainable level. This force is a funcion of he mass of he vehicle on he racive axle and he coefficien of fricion beween he ires and he pavemen. Typical values for he percenage of mass on he racive axle are presened laer in he paper. Equaion - demonsraes he overall equaion for he force applied. P F = 6η [-8] v F = M a µ [-9] max F = min ( F, F ) [-] max Where: F: racive force (N), F max : maximum racive force (N), F : racive force (N), M a : vehicle mass on racive axle, M perc a (kg), perc a : percenage of vehicle mass on racive axle (%), M: vehicle mass (kg), P: engine power (kw), v: vehicle speed (km/h), η: power ransmission efficiency, and µ: coefficien of fricion beween ires and pavemen. ii. Resisance Forces The nex variable of imporance in he fundamenal equaion (Equaion -6) is he resisance force. Three major ypes of resisance forces need o be considered, including aerodynamic, rolling, and grade resisances as suggesed in he lieraure (Fich, 99; Mannering, 99). The oal resisance force is simply compued as he sum of he hree resisance componens, as summarized in Equaion -. Each of he hree resisance forces is dependan on several parameers, as discussed in he following secions. R = R + R + R [-] a r g Where: R a : aerodynamic resisance (N), R r : rolling resisance (N),

43 Chaper : Maximum Acceleraion R g : grade resisance (N). The aerodynamic resisance, or air drag, is a funcion of he vehicle fronal area, he aliude, he drag coefficien, and he square of speed of he vehicle, as indicaed in Equaions - and -. The consan c accouns for he air densiy a sea level a a emperaure of o C (9ºF). Typical values of vehicle fronal areas for differen vehicle ypes and ypical drag coefficiens are provided laer in he paper. Equaion - is a linear approximaion ha was derived from a more complex formulaion (Waanada, 987). The linear approximaion was found o provide similar resuls o he more complex formulaion for aliudes in he range of o m. R = c C C Av [-] a d h C h = 8. H [-] Where: A: fronal area (m ), C d : air drag coefficien, C h : aliude coefficien, H: aliude (m), and c : consan =.78. The rolling resisance is a linear funcion of he vehicle speed and mass, as indicaed in Equaion -. Typical values for rolling coefficiens (C r, c, and c ), as a funcion of he road surface ype, condiion, and vehicle ires, are provided in he lieraure (Rakha e al., ). Generally, radial ires provide a resisance ha is percen less han ha for bias ply ires. M R = 9.866C ( c v + c ) [-] r r Where: M: vehicle mass (kg). u: vehicle speed (km/h). C r, c, c : rolling resisance consans. The grade resisance varies as a funcion of he vehicle s oal mass and he percen grade ha he vehicle ravels along, as indicaed in Equaion -. The grade resisance accouns for he proporion of he vehicle weigh ha resiss he movemen of he vehicle: R g = M i [-] Where: M: vehicle mass (kg). i: grade magniude (m/ m). iii. Variable Power I should be noed ha he use of he maximum racive force consrain ha is imposed on he consan power model resuls in a vehicle power ha increases linearly as a funcion of vehicle speed given ha power is equal o he produc of force and speed. For

44 Chaper : Maximum Acceleraion example, Figure - illusraes for a sample vehicle field observed and model esimaed power as a funcion of vehicle speed. Rakha and Lucic () demonsraed ha in he case of rucks vehicle acceleraions prediced by he consan power model over-esimae ruck acceleraions a low speeds. Rakha and Lucic demonsraed ha his over-esimaion of ruck acceleraions was aribued o he fac ha he effecive engine power is less during low vehicle speeds as a resul of gear shif effecs. Consequenly, Rakha and Lucic proposed a reducion facor o accoun for he reduced power a low speeds in order o more capure ruck acceleraion behavior more accuraely. This paper will invesigae wheher such a power reducion facor is required for ligh duy vehicle vehicles, as was he case for heavy duy rucks, using field observed daa, as will be described laer in he chaper. Power (kw) 8 6 Figure -: Model and Field Measured Power vs. Speed Relaionship (Acura Inegra).. Sae-of-Pracice Field Daa Ses In order o creae and validae he acceleraion models, daa ses compiled from field research are required. Unforunaely, much of he daa used o verify exising models are oudaed or have been exrapolaed o reflec curren condiions. For example, in heir comparison sudy of exising models, Bham and Benekohal were forced o use daa ses ha were colleced in 968 and 98 (). These daa ses were used because more recen daa ses were no exensive enough and did no capure he acceleraion behavior of vehicles for a sufficien amoun of ime. Obviously, he acceleraion capabiliies of vehicles have changed dramaically since hese daa ses were colleced. Bham and Benekohal ried o overcome he limiaion of he old daa ses by exrapolaing vehicle acceleraion rends from he 968 daa hrough he 98 daa o accoun for modern vehicle capabiliies. However, i would be more appropriae o use curren daa if i were available.

45 Chaper : Maximum Acceleraion Anoher limiaion of exising daa is ha here is rarely informaion available on he vehicle ypes, roadway characerisics, and drivers involved in he daa collecion effor. Specifically, he speed and acceleraion daa are aggregaed regardless of roadway grade, surrounding raffic, and pavemen surface condiions. Furhermore, he daa ses do no provide informaion on he ype of vehicles involved in he ess, he roadway condiions, or oher feaures ha conribue o he acceleraion behavior of he vehicle, such as roadway grade. Common sense suggess ha he acceleraion behavior of a vehicle is grealy affeced by he power he vehicle can generae, he grade he vehicle is raveling on, and he resisance encounered by he vehicle. However, he exising daa ses do no address hese issues. Finally, he exising daa ses are ofen colleced from a raffic sream, where acceleraion is limied by vehicle-o-vehicle ineracion and hus migh no reflec maximum vehicle acceleraion behavior. Consequenly, conemporary daa are required in which vehicle acceleraion behavior is conduced in a conrolled environmen wihou vehicle-o-vehicle ineracion. In addiion, hese daa ses require ha vehicle, driver, and roadway characerisics be documened... Consrucion of Field Daa Se The field ess developed for his repor were designed o creae a new daa se for modern vehicles and herefore eliminae he need o exrapolae resuls from old daa. The speed and acceleraion daa colleced for his research are complee wih vehicle, pavemen, and driver characerisics. The field ess performed allowed for he collecion of speed daa from vehicles in a conrolled environmen, where he acceleraion is no limied by exernal facors oher han vehicle capabiliies. The daa are designed o represen he maximum acceleraion of a lead vehicle acceleraing from a sop. The modeling of vehicle acceleraion in a plaoon of vehicles is an area of research ha is beyond he scope of his paper and hus requires furher invesigaion. The following secions describe he procedures applied o consruc he field daa se... Smar Road Tes Faciliy Tesing was performed during he summer of on he Smar Road es faciliy a he Virginia Tech Transporaion Insiue in Blacksburg, Virginia. Currenly, he Smar Road is a.-km (-mile) experimenal highway in souhwes Virginia ha spans varied errain, from in-own o mounain passes. The horizonal layou of he es secion is fairly sraigh wih some minor horizonal curvaure ha does no impac vehicle speeds. The verical layou of he secion demonsraes a subsanial upgrade ha ranges from 6% a one end o.8% a he oher end, as illusraed in Figure -. Apar from a -m (9-f) segmen of he roadway ha is a rigid pavemen, he enire roadway surface is asphal. Consequenly, a rolling resisance coefficien for asphal pavemen only is uilized. The qualiy of he road surface was good a he ime he es runs were conduced. These facors were imporan in idenifying he road surface rolling resisance coefficiens used in he Rakha model. Conducing he ess on he Smar Road allowed he vehicles o accelerae wihou being impeded by oher raffic. 6

46 Chaper : Maximum Acceleraion In consrucing he verical profile of he es secion he elevaion of saions were surveyed, as indicaed by he diamond symbols in Figure -. The verical profile of he es secion was hen generaed by inerpolaing beween saion elevaions using a cubic spline inerpolaion procedure a -m (.8-f) incremens. The cubic spline inerpolaion ensures ha he elevaions, he slopes, and he rae of change of slopes are idenical a he boundary condiions (in his case every meer). The grade was compued for each -m (.8-f) secion and was found o vary considerably, as illusraed in Figure - (hin line). A polynomial regression relaionship was fi o he grade daa (R of.9) for wo reasons. Firs, o ensure a smooh ransiion in he roadway grade while mainaining he same verical profile. Second, o faciliae he soluion of he ODE because i ensures ha he grade funcion is coninuous. The modified grade and verical elevaion, which are illusraed in Figure - (hick line), demonsrae an almos idenical verical profile wih much smooher grade ransiions when compared o he direc inerpolaion. The final equaion for grade as a funcion of disance is given as Equaion i = x.79 x +. x [-6] Where: x: disance (m). i: grade magniude (m/ m)..7.6 Grade (percen) Elevaion (m) 68 Inerpolaed Elevaion 67 Regression Elevaion 66 Saion Elevaions Figure -: Smar Road Verical Profile 7

47 Chaper : Maximum Acceleraion.. Daa Collecion Procedures Thireen differen vehicles were chosen for esing, and each was subjeced o he same se of ess. The es runs involved acceleraing he vehicles from a complee sop a he maximum acceleraion rae for he enire.-km rip. Three speed ranges were esed, namely acceleraing he vehicle from o 6 km/h ( mph), o 88 km/h ( mph), and km/h o he maximum aainable speed wihin he es secion. Depending on he ype of vehicle, maximum speeds aained by he end of he es secion varied beween 8 and 6 km/h (8 and mph). In conducing he sudy, a minimum of five repeiions were execued for each es se in order o provide a sufficien sample size for he validaion analysis. This generaed a oal of es runs for each vehicle... Speed Measuremen Before each run, he es vehicles were equipped wih a Global Posiioning Sysem (GPS) uni ha measured he vehicle speed o an accuracy of. m/s. (. f/sec). GPS is a worldwide, saellie-based radio-navigaion sysem ha can deermine wih cerain accuracy he posiion and velociy of any objec equipped wih a GPS receiver. Typical oupu daa from GPS receivers include laiude, longiude, aliude, speed, heading, and ime. The GPS receivers used were able o updae hese parameers once every second. Nominal posiion accuracy is specified wih a -m (8-f) spherical error probabiliy, while nominal velociy accuracy is specified wihin. m/s (. f/sec) error probabiliy. These inaccuracies are aribued o a number of sources of error. The majoriy of hese errors are linked o he way he disance beween a saellie and a GPS receiver is measured. Wihin he sysem, disances are measured by calculaing he ime i akes for a signal o ravel beween a saellie and a receiver. Consequenly, any delay in he signal ransmission hen resuls in disance over-esimaion and inaccuracies in he esimaed posiion of he objec... Tes Vehicle Characerisics The hireen es vehicles were seleced o cover a wide range of ligh duy vehicles, as demonsraed in Table -. Specifically, he vehicles were seleced o reflec he populaion of ligh duy vehicles on he roads in he year. Vehicles were seleced from faculy and sudens a he Virginia Tech Transporaion Insiue, and herefore consiue he various ypes of auomobiles ha were in use a he ime. The vehicles represen a wide range of sizes and a variey of EPA vehicle classes including subcompac cars, compac cars, midsize cars, large cars, spor uiliy vehicles (SUV), pickup rucks, and minivans. Many characerisics of each vehicle were recorded including weigh, lengh, fronal area, ype of ires, engine ype and power, and air drag coefficien. These parameers serve as inpus o he Rakha model, and he values used are described in he following secion. 8

48 Chaper : Maximum Acceleraion Table -: Sample Vehicles Tesed Vehicle Vehicle Class 99 Acura Inegra SE 996 Geo Mero Hachback Subcompac Car 99 Saurn SL Plymouh Neon Compac Car Mazda Proégé LX. 99 BMW 7I 998 Ford Taurus Midsize Car 998 Honda Accord 99 Dodge Inrepid 999 Ford Crown Vicoria Large Car 998 Ford Windsar LX Minivan 99 Chevy S- Pickup Truck 99 Chevy Blazer Spor Uiliy Vehicle.. Model Consrucion and Comparison As menioned earlier, his research effor involved applying he Rakha model, originally developed for rucks, o a variey of passenger vehicles. Specifically, he parameers for each vehicle was deermined and inpu ino he model equaions o generae acceleraion, speed, and disance values. This process is described in he following secions... Model Parameers The equaions ha go ino he model were described earlier. To apply he model successfully, i is necessary o generae he values of he parameers for each vehicle and discuss how hey can be obained. The following is a lis of vehicle and roadway characerisics ha are required for he model. The fac ha here are numerous inpu parameers emphasizes he complexiy of vehicle acceleraion behavior and demonsraes why i is difficul o model acceleraion paerns. However, while here are numerous inpu parameers, hese parameers are fairly easy o obain eiher from vehicle manuals or from he World Wide Web. The values of all he parameers used for he hireen vehicles in our ess are summarized in Table -. Table -: Model Inpu Parameers Vehicle Accura Chevy Chevy Dodge Geo Ford Ford Honda Ford Mazda Plymouh BMW Saurn Inegra Blazer S Inrepid Mero Taurus Windsar Accord Crown Vic Proégé Neon 7I SL Power (hp) % Mass Tracive Axle Aliude (m) Grade Variable Variable Variable Variable Variable Variable Variable Variable Variable Variable Variable Variable Variable Engine Efficiency Coefficien Fricion Cd Ch C C C Cr Fronal Area (m^) Power (kw) Mass (kg)

49 Chaper : Maximum Acceleraion. Vehicle Engine Power: The engine power can easily be obained from he vehicle specificaions and is usually given in he unis of horsepower. To conver from horsepower o kilowas, muliply he value by.76.. Engine Efficiency: Power losses in he engine due o inernal fricion and oher facors generally accoun for beween -% losses of he engine losses for ligh duy vehicles. Therefore, ypical efficiency values range beween Vehicle Mass: The vehicle mass is an imporan parameer in he model. The curb weigh is available wihin he vehicle specificaions. Noe ha you should include he weigh of passengers in he vehicle.. Percenage of Vehicle Mass on he Tracive Axle: For a four-wheel drive vehicle, his value is %. However, mos ligh duy vehicles are wo-wheel drive. Typical values for wo-wheel drive vehicles are in he range of -6%, because mos vehicles are fronwheel drive. Conversely, rear-wheel drive vehicles have a racive axle mass of -% he oal mass. As par of he esing, he axles were weighed separaely o obain his value.. Pavemen: The pavemen ype and condiion are required o deermine several consans. 6. Coefficien of Fricion: The value of he coefficien of fricion is dependen on he pavemen ype and condiion. For concree pavemens, he coefficien of fricion varies from.8 o.7 o.6 depending for an excellen, good, or poor condiion, respecively. For asphal pavemens, he values are.6,., and. for good, fair, and poor condiions, respecively. The Smar Road es faciliy was a good asphal surface a he ime of esing. 7. Aliude: This is he aliude above sea level for he esing locaion, in meers. In he case of he Smar Road he aliude was 6m. 8. Air Drag Coefficien: The air drag coefficien is given in he vehicle specificaions. Typical values for ligh duy vehicles range from. o., depending on he aerodynamic feaures incorporaed ino he body of he vehicle. 9. Fronal Area: The fronal area of he vehicle can be approximaed as 8% of he heigh imes he widh of he vehicle if i is no given direcly in he vehicle specificaions.. Rolling Resisance Consans: Three rolling resisance consans are used in he model. The consan C r is a funcion of he pavemen ype and condiion. For concree pavemens, he coefficien varies from. o. o. depending for excellen, good, or poor pavemens, respecively. For asphal pavemens, he values are.,.7, and. for good, fair, and poor pavemens, respecively. The consans C and C depend on ire

50 Chaper : Maximum Acceleraion ype. Mos ligh duy vehicles use radial ires. The consans for radial ires are.8 and.7, respecively. If bias ply ires are used, he consans are.8 and 6... Grade: The grade of he roadway is given as a decimal (m/ m). The grade is ofen considered a consan for a given secion of roadway. However, he grade of he Smar Road es faciliy was compued using Equaion Model Applicaion The previous secion described he inpu parameers o he various vehicle acceleraion models. This secion describes how o solve he equaions of moion and apply he model o predic he speed and acceleraion raes for differen cars. The second order ordinary differenial equaion (ODE) shown in Equaion -7 can be recas as wo firs order ODE's and a numerical soluion can be reached using an Euler approximaion as shown below in Equaions -7, -8, -9, and -: F( ) R( ) i i a( ) = [-7] i M v &( i ) a( i ) = x &( i ) v ( i ) v ) ( i = x [-8] v ( i ) + a( i ) [-9] x( i ) + v( i ) [-] ( i ) = Where: a = acceleraion (m/s) v = velociy (m/s) x = disance raveled (m) i = + i for i =,,.,n The smaller he value of used, he more accurae he model will be. For our formulaions, a of. seconds was used. By insering hese equaions ino a spreadshee, as well as equaions for he forces and resisances and he parameers for he vehicle and roadway, he values for disance, speed, and acceleraion are generaed for a vehicle a any given ime. See Table - for a sample spreadshee.

51 Chaper : Maximum Acceleraion Table -: Sample Model Spreadshee (Saurn SL) Dis (m) Spd (km/h) Acc (m/s ) F Grade Ra Rr Rg R Consan Power Assumpion As was menioned earlier Rakha and Lucic had observed a reducion in vehicle power a low speeds o accoun for he build-up of power ha occurs as he rucks accelerae hrough he gears (). However, i was unclear wheher his behavior would occur for ligh duy vehicles, because hese vehicles have fewer gears, carry less mass, and generally use auomaic ransmissions. To es he validiy of he consan power model, power was ploed versus speed using he daa obained from he field ess for each of he hireen es vehicles. The resuls indicaed a srong linear relaionship beween power and speed, as was demonsraed in he plo for he Acura Inegra in Figure -. An R value of.99 was obained for his linear regression. The plos for en of he es vehicles had an R value above.9. The Neon and BMW also exhibied reasonable correlaion wih R values of.86 and.88, respecively. The Geo Mero had he lowes correlaion,.7, which can be parly aribued o he fac ha i is a manual ransmission vehicle. In addiion o checking he validiy of a consan power assumpion, he power versus speed plo was also used o improve he calculaion of he percenage of mass on he racive axle. The slope of he power versus speed plo was used o deermine he acual

52 Chaper : Maximum Acceleraion maximum force applied by he engine. Recall ha Equaion -9 gives he heoreical value of his maximum force. Sligh adjusmens o he percenage of mass on he racive axle values were made (less han percen) so ha he model would fi o he field daa more closely... Vehicle Dynamics Model Predicions The vehicle dynamics model (Rakha e al., ) demonsraed a srong correlaion o he hireen vehicle field daa. Specifically, he model demonsraed a good fi in he acceleraion versus speed, acceleraion versus ime, acceleraion versus disance, speed versus ime, and speed versus disance domains, as demonsraed in Figure - hrough Figure -6. In addiion, he model was able o predic vehicle speed and acceleraion profiles accuraely for vehicles ranging from subcompac cars o large cars o SUV's and pick-up rucks. Consequenly, he resuls clearly demonsrae he flexibiliy and validiy of he model in predicing maximum vehicle acceleraion levels. While he vehicle dynamics model did offer accurae modeling of vehicle acceleraion behavior, he model did end o over-esimae vehicle speeds for wo es vehicles. Specifically, he field speed profiles for he Neon (Figure -) and Blazer (Figure -) vehicles seem o demonsrae a sligh drop in vehicle acceleraion owards he end of he es run, which is no capured by he vehicle dynamics model. However, hese drops in vehicle acceleraion may be aribued o he es vehicle no properly shifing ino he final gear. Also, he model seems o underesimae acceleraion raes a low speeds for some of he vehicles. I is unclear wheher hese higher acceleraion raes are a resul of a build-up of power wihin he vehicle, or if here is jus high variabiliy in vehicle acceleraion capabiliies a low speeds. I should be noed ha he daa from he es vehicles in Figure - hrough Figure -6 represen a series of runs.

53 Chaper : Maximum Acceleraion Acceleraion (m/s ) 7 - Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed 6 8 Figure -: Model Predicions versus Field Daa (Geo Mero)

54 Chaper : Maximum Acceleraion Acceleraion (m/s ) 7 - Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -: Model Predicions versus Field Daa (Plymouh Neon)

55 Chaper : Maximum Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -6: Model Predicions versus Field Daa (Acura Inegra) 6

56 Chaper : Maximum Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -7: Model Predicions versus Field Daa (Saurn SL) 7

57 Chaper : Maximum Acceleraion Acceleraion (m/s ) 7 - Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -8: Model Predicions versus Field Daa (Mazda Proégé) 8

58 Chaper : Maximum Acceleraion Acceleraion (m/s ) 7 - Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -9: Model Predicions versus Field Daa (Honda Accord) 9

59 Chaper : Maximum Acceleraion Acceleraion (m/s ) 7 - Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -: Model Predicions versus Field Daa (Ford Taurus)

60 Chaper : Maximum Acceleraion Acceleraion (m/s ) 7 - Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -: Model Predicions versus Field Daa (BMW)

61 Chaper : Maximum Acceleraion Acceleraion (m/s ) 7 Acceleraion (m/s ) 6 8 Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -: Model Predicions versus Field Daa (Dodge Inrepid)

62 Chaper : Maximum Acceleraion Acceleraion (m/s ) 7 - Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -: Model Predicions versus Field Daa (Crown Vicoria)

63 Chaper : Maximum Acceleraion Acceleraion (m/s ) 7 Acceleraion (m/s ) 6 8 Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -: Model Predicions versus Field Daa (Chevy Blazer)

64 Chaper : Maximum Acceleraion Acceleraion (m/s ) 7 Acceleraion (m/s ) 6 8 Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -: Model Predicions versus Field Daa (Ford Windsar)

65 Chaper : Maximum Acceleraion Acceleraion (m/s ) 7 Acceleraion (m/s ) 6 8 Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -6: Model Predicions versus Field Daa (Chevy S-) 6

66 Chaper : Maximum Acceleraion.. Comparison of Sae-of-Pracice Models A comparison sudy was performed involving he Rakha model and sae-of-he-ar acceleraion models. The four exising models chosen fo r comparison include he Dual- Regime model, he Linear Decay model, he Polynomial model, and he Searle model. The dual-regime model was chosen because i was recommended by Bham and Benekohal in a comprehensive comparison sudy hey performed involving foureen models (). The Linear decay model is recommended for use in a paper by Long () and also appears in several exbooks, including one by Drew (968). In addiion, he Polynomial model was recommended in a comparison sudy performed by Akçelik (987). Finally, he Searle model was chosen for comparison because i represens anoher example of a vehicle dynamics model (999).... Comparison Resuls Daa from five of he hireen es vehicles were seleced for comparison purposes. The five vehicles chosen were he Mazda Proégé, Dodge Inrepid, Chevy Blazer, Ford Windsar, and Chevy S-. These vehicles were chosen because hey represen a variey of vehicle ypes, including small, large, SUV, minivan, and pick-up ruck vehicles, respecively. The parameers for each of he models were chosen in an aemp o creae a bes fi beween model esimaes and field daa. Figures -7 hrough Figure - illusrae he dual-regime model predicions superimposed on he field measuremens for each of he five es vehicles. The calibraed inpu parameers for he dual-regime model included he final speed a he end of acceleraion, he oal acceleraion ime, he speed breakpoin beween he wo regimes, and ime o reach he second regime. These parameers were calibraed o provide a bes fi in he speed versus ime regime for each vehicle. The dual-regime model appears o provide a reasonable fi o he speed profiles of he vehicles wih respec o disance and ime. The model does end o overesimae speeds in he middle of he disance profile, owards he end of he firs regime. Furhermore, he model ends o underesimae vehicle speeds a he beginning of he ime profiles for he Blazer and Windsar es vehicles. The acceleraion plos, however, do no provide a reasonable fi o field daa. The major disadvanage of he model is he disconinuiy in he acceleraion profile. Specifically, he insananeous drop in he vehicle acceleraion rae appears o be unrealisic. However, he simpliciy of he model and is abiliy o generae reasonable speed profiles makes he model useful in cerain applicaions. 7

67 Chaper : Maximum Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) 7 Acceleraion (m/s ) 6 Figure a: Speed versus disance Figure b: Speed versus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed 7 Figure -7: Dual-Regime Model Fi o Field Daa (Mazda Proégé) 8

68 Chaper : Maximum Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed versus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed 7 Figure -8: Dual-Regime Model Fi o Field Daa (Dodge Inrepid) 9

69 Chaper : Maximum Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed versus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed 7 Figure -9: Dual-Regime Model Fi o Field Daa (Chevy Blazer) 6

70 Chaper : Maximum Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed versus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed 7 Figure -: Dual-Regime Model Fi o Field Daa (Ford Windsar) 6

71 Chaper : Maximum Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed versus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed 7 Figure -: Dual-Regime Model Fi o Field Daa (Chevy S-) For he linear decay model, he parameers of maximum acceleraion and maximum speed were calibraed o obain an opimum fi beween model esimaes and field daa in he acceleraion-speed regime. The linear decay model shows a good fi for boh he speed profiles, as indicaed by Figure - hrough Figure -6. The linear decay model also improves on he dual-regime model by providing a coninuous acceleraion funcion ha more closely resembles field daa. The linear decay model also accouns for he apparen curvaure in he daa for he plos of acceleraion versus ime and disance. However, he daa in he acceleraion versus speed plo also seems o exhibi some curvaure, while he model predics a linear funcion. Therefore, he model ends o underesimae he acceleraion rae of he vehicle when he vehicle is raveling a low speeds. This model appears o be he bes kinemaics model, bu i requires accurae field daa before i can be applied. 6

72 Chaper : Maximum Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) 7 Acceleraion (m/s ) 6 Figure a: Speed versus disance Figure b: Speed versus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed 7 Figure -: Linear Decay Model Fi o Field Daa (Mazda Proégé) 6

73 Chaper : Maximum Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed versus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed 7 Figure -: Linear Decay Model Fi o Field Daa (Dodge Inrepid) 6

74 Chaper : Maximum Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed versus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed 7 Figure -: Linear Decay Model Fi o Field Daa (Chevy Blazer) 6

75 Chaper : Maximum Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed versus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed 7 Figure -: Linear Decay Model Fi o Field Daa (Ford Windsar) 66

76 Chaper : Maximum Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed versus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed 7 Figure -6: Linear Decay Model Fi o Field Daa (Chevy S-) Four parameers were required for he calibraion of he polynomial model, including iniial and final speeds, acceleraion ime, and acceleraion disance. The final speed, ime, and disance in he model relaes o he maximum possible speed he vehicle can aain. Since he vehicles in he daa se did no necessarily reach heir maximum speed over he.-km es secion, his parameer was more difficul o calibrae. Figure -7 hrough Figure - indicae ha he polynomial model provides a fairly reasonable fi o he speed profiles of he vehicles, alhough i does end o overesimae speeds owards he end of he runs. The model was designed o follow he shape of he acceleraion versus ime plo, and herefore generally shows he proper rend in his domain. However, he model seems o inver he rend in he acceleraion versus speed domain. Specifically, while he field daa sugges a curve approaching a slope of zero for high speeds, he polynomial model suggess a curve whose slope approaches infiniy as speed increases. This model was shown o accuraely predic disance raveled and fuel consumpion by Akçelik (987), however, his sudy demonsraes ha he model does no reflec field daa accuraely. 67

77 Chaper : Maximum Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed versus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed 7 Figure -7: Polynomial Model Fi o Field Daa (Mazda Proégé) 68

78 Chaper : Maximum Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed versus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed 7 Figure -8: Polynomial Model Fi o Field Daa (Dodge Inrepid) 69

79 Chaper : Maximum Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed versus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed 7 Figure -9: Polynomial Model Fi o Field Daa (Chevy Blazer) 7

80 Chaper : Maximum Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed versus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed 7 Figure -: Polynomial Model Fi o Field Daa (Ford Windsar) 7

81 Chaper : Maximum Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed versus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed 7 Figure -: Polynomial Model Fi o Field Daa (Chevy S-) The power consan, k, used in he Searle model was deermined for each of he vehicles in he sudy. The Searle model appears o break down a very low speeds and speeds ouside he normal driving range, as indicaed by Figure - hrough Figure -6. The model predics acceleraion raes approaching infiniy for low speeds, because i does no accoun for he poenial of wheel spinning a low speeds. The vehicle dynamics model presened by Rakha e al. () inroduces a maximum force value ha is dependen on fricion beween he vehicle ires and pavemen surface, as indicaed by Equaion -9. The Searle model also grealy overesimaes speeds beyond speeds of km/h (7 mph). Because of hese limiaions, he Searle model is only useful in limied applicaions. 7

82 Chaper : Maximum Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) 7 Acceleraion (m/s ) 6 Figure a: Speed versus disance Figure b: Speed versus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed 7 Figure -: Searle Model Fi o Field Daa (Mazda Proégé) 7

83 Chaper : Maximum Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed versus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed 7 Figure -: Searle Model Fi o Field Daa (Dodge Inrepid) 7

84 Chaper : Maximum Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed versus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed 7 Figure -: Searle Model Fi o Field Daa (Chevy Blazer) 7

85 Chaper : Maximum Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed versus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed 7 Figure -: Searle Model Fi o Field Daa (Ford Windsar) 76

86 Chaper : Maximum Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed versus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed 7 Figure -6: Searle Model Fi o Field Daa (Chevy S-) 77

87 Chaper : Maximum Acceleraion..6 Advanages of Vehicle Dynamics Model Based on his comparison sudy, hree primary advanages of he Rakha dynamics model can be observed:. Good Fi in all Domains: The vehicle dynamics model offers a good fi o field daa in each of he domains, namely speed versus disance, speed versus ime, acceleraion versus speed, acceleraion versus ime, and acceleraion versus disance. Alernaively, he oher models, especially he empirical models, are designed o fi a single domain, and herefore do no fi well over all domains.. Model Flexibiliy: The vehicle dynamics model is able o adap o differen vehicle ypes, differen pavemen condiions, and ravel over differen errain.. Simple Calibraion: Anoher advanage of he vehicle dynamics model is ha i does no require field daa ses o calibrae inpu parameers. Specifically, inpu parameers can be obained from vehicle manuals, car magazines, and/or he World Wide Web. Alernaively, he oher models require field daa for he calibraion of inpu parameers... Conclusions and Recommendaions for Furher Research The paper consrucs a daabase of unconsrained vehicle acceleraion daa for hireen ligh duy vehicles and rucks. In addiion, he paper validaes a vehicle dynamics model for predicing maximum ligh duy vehicle acceleraion raes. The model is compared o four of sae-of-ar acceleraion models using he developed daa se. Advanages of he vehicle dynamics model are presened, including is abiliy o accuraely predic vehicle behavior wih readily available inpu parameers and is flexibiliy in esimaing acceleraion raes of boh large and small vehicles on varied errain. I is recommended ha furher research be conduced in a number of areas. Firs, here is a need o collec field daa on ypical lead vehicle acceleraion (no ineracing wih oher preceding vehicles). Second, daa are required on vehicle acceleraion behavior wihin a plaoon (ineracing wih oher vehicles). Third, daa are required on ypical vehicle acceleraion during criical evens, including merging wih freeway raffic a an on-ramp and acceleraing o overake a vehicle. For all hese scenarios vehicle acceleraion models should be developed o capure driver behavior in he non-seady sae car-following mode of operaion. The nex chaper serves o address some of hese issues. 78

88 Chaper : Typical Acceleraion Chaper Five: Modeling Typical Acceleraion Behavior. Inroducion Our roadways are filled wih a broad range of vehicles driven by a wide variey of drivers. The drivers vary from old o young, male o female, aggressive o passive. The acceleraion behavior of hese drivers varies grealy. The aggressiveness of he drivers can be influenced by a large number of facors, including he oher cars on he road or even he mood of he driver on a given day. Consequenly, i s difficul o accuraely predic acual driver behavior and be able o model condiions ha are acually presen on our roadway infrasrucure. However, simulaion models, fuel consumpion models, and emissions models, among ohers, are essenial ools for engineers rying o solve realworld problems. These ools require accurae represenaions of acual siuaions in order o be effecive. Therefore, a model ha accouns for differen driver ypes and vehicle behavior is essenial. However, limied effor has been made o ry o caegorize he ypical acceleraion behavior of drivers. Only a few generalizaions abou ypical driver behavior have been proposed in he lieraure as i relaes o he maximum performance of he vehicle. In a 97 sudy by he Bureau of Public Roads, Louzenheiser suggesed ha drivers accelerae a a rae approximaely wo-hirds of he vehicle capabiliy (98). In 9, he Blue Book recommended using a value of 6% when comparing ypical acceleraion o maximum acceleraion for an average driver (AASHTO). However, a sudy by Long showed ha applying he value of 6% o he popular linear decay acceleraion model produced erroneous resuls when compared o field daa (). Using he linear decay model, Long recommends using an alpha value of m/s and a bea value of. o depic average acceleraions of passenger cars. Oher values for various vehicles are recommended by Long in he lieraure (). Many oher models have been developed o predic vehicle acceleraion paerns based on empirical relaionships generaed from field daa. However, hese models mus been updaed ofen as he capabiliies of vehicles change, speed limis on our highways change, and he demographics of drivers on he road change. The goal of his paper is o develop a disribuion of facors o represen he acceleraion paerns of he driving public and herefore creae an acceleraion model ha more accuraely represens acual condiions.. Maximum Model Before creaing a model for ypical acceleraion, we mus sar wih a model for maximum acceleraion. The Rakha model is a vehicle dynamics model ha has been used o predic he maximum speed and acceleraion profiles of a variey of passenger vehicles and rucks based on he forces acing on he vehicle. Using readily available inpu parameers abou he vehicle and roadway, he Rakha model has been shown o 79

89 Chaper : Typical Acceleraion replicae daa colleced from field research in he domains of speed versus ime, speed versus disance, acceleraion versus ime, acceleraion versus disance, and acceleraion versus speed. A horough descripion of he Rakha model can be found in Chaper. The goal of his chaper is o develop an appropriae modificaion o he Rakha model ha will enable i o predic ypical driver behavior for a variey of driver ypes.. Field Tess The firs sep in he research effor was he collecion of a new se of field daa. Exising daa ses did no conain informaion abou he individual drivers and vehicles ha was necessary for he model. Therefore, field ess were performed as described in he following secion... Tes Procedure Tesing was performed in he spring and summer of a he Smar Road research faciliy in Blacksburg, Virginia. The es involved acceleraing a vehicle from a sop over a disance of approximaely fee ( m) a a driver s normal acceleraion rae. The es was performed on a relaively fla and sraigh srech of roadway conrolled by a sop sign and subjec o very ligh raffic volumes ha would no inerfere wih each run. The goal was o simulae he ypical acceleraion profile of a lead vehicle a a sop line. Tweny differen drivers voluneered for he es, and hey are described in he following secion. The es vehicle used in he es was a 999 Ford Crown Vicoria, which had been equipped wih a GPS uni ha colleced speed daa for he vehicle every second. The same vehicle was used for each driver, in order o serve as he conrol for he experimen. The drivers were each allowed o drive he vehicle for a while o ge familiar wih he car before saring he esing. The drivers were aware ha hey were being esed, and were old o accelerae a heir normal rae unil he end of he es secion. To accoun for variabiliy wihin he driver, up o weny-five runs were conduced. From hese runs, speed, ime, and disance measuremens were recorded. The resuls are presened laer in his repor... Tes Drivers The es drivers were random voluneers chosen o reflec he driving populaion on he road oday. Tweny drivers overall were chosen, including eleven men and nine women. The drivers ranged in age from o. Table - shows a breakdown of licensed drivers on he road for he year 999, lised by age and gender, according o federal highway adminisraion saisics. Noe ha he age group sudied in his research projec reflecs over half of he driver populaion on he road. Furher research needs o be done on eenage drivers and elderly drivers, which was beyond he scope of his paper. Over fify percen of drivers naionally are male, and herefore % of drivers sudied for his research were male. Table - shows a breakdown of he drivers used in he sudy. 8

90 Chaper : Typical Acceleraion Table -: Naional Driver Saisics, Source FHWA DISTRIBUTION OF LICENSED DRIVERS 999 BY SEX AND PERCENTAGE IN EACH AGE GROUP AND RELATION TO POPULATION OCTOBER TABLE DL- MALE DRIVERS FEMALE DRIVERS TOTAL DRIVERS AGE PERCENT DRIVERS AS PERCENT DRIVERS AS PERCENT DRIVERS AS NUMBER OF TOTAL PERCENT OF NUMBER OF TOTAL PERCENT OF NUMBER OF TOTAL PERCENT OF DRIVERS AGE GROUP / DRIVERS AGE GROUP / DRIVERS AGE GROUP / UNDER 6 7,..,76.., , ,6.8 7.,8, ,,7. 7.9,,. 8.,, ,,67. 7.,, ,767,. 7. 9,6, ,7, ,, (9 AND UNDER),969,6..9,6,78.. 9,6,..7,9, ,8, ,, ,6, ,9,.6 8.,,.6 8.,6,.7 88.,6,.7 88.,6,.7 88.,87, ,,7.6 9.,8,.7 9.,6, ,9,7.7 9.,9, (-) 7,9, ,89, ,9, ,67, ,67, ,77, ,6, ,79,. 9. 8,9, ,6,9. 9.8,6,6. 9.,, ,76,89. 9.,,. 9.,7, ,8, ,88, ,66, ,96, ,6, ,, ,, ,7, ,8, ,88, ,8, ,66, ,6, ,67, ,, ,6,8.9 9.,86, ,, ,777, ,,77. 7.,789, ,8, ,78,.9 8.,, AND OVER 889, ,,8..,89,6.. TOTAL 9,66,. 7. 9,, ,7,. 7. / These percenages are compued using populaion esimaes of he Bureau of he Census. Under-6 age group is compared o and -year-old populaion esimaes; he oher age brackes coincide wih hose from he Bureau of he Census. Table -: Tes Driver Characerisics Driver # Age Gender Male Male 7 Male 9 Male Male 6 Male 7 Male 8 Male 9 Male 8 Male Male Female Female Female Female 6 Female 7 Female 8 Female 9 Female Female 8

91 Chaper : Typical Acceleraion. Resuls The Rakha model for predicing he maximum acceleraion behavior of vehicles had been validaed for he es vehicle, he 999 Ford Crown Vicoria, in he previous chaper. Figure - shows he srong correlaion beween he Rakha model and field daa colleced represening he maximum acceleraion of he Crown Vicoria. Noe ha his daa was colleced on a roadway wih a subsanial uphill grade. Because he Rakha model uses grade as an inpu parameer, i was recalibraed for a grade of % as shown in Figure - o accoun for he level grade used in his sudy. This model was used as a saring poin and compared o he field resuls of each of he weny drivers. Daa was colleced for he drivers, and plos of speed versus ime, speed versus disance, acceleraion versus ime, acceleraion versus disance, and acceleraion versus speed were generaed. Then, a reducion facor was inroduced ino he model for each driver unil he model presened he bes fi o he field daa. The driver facor was muliplied by he prediced maximum acceleraion o generae he acual acceleraion a any ime. Therefore, a driver facor of would represen he maximum capabiliy of he vehicle as shown in Figure -, while a driver facor of.7 would signify ha he driver acceleraed a 7% of he maximum acceleraion possible, for example. The maximum acceleraion behavior exhibied by he Crown Vicoria and by oher vehicles can be classified as saring a a maximum value a low speeds and gradually diminishing o lesser values as he speed of he vehicle increases, as demonsraed in he plo of acceleraion versus speed in Figure -. While i is clear ha he ypical acceleraion behavior of drivers will reflec some value below he maximum vehicle capabiliy, here is no guaranee ha his lesser acceleraion behavior will follow he same profile shape as he maximum behavior for every driver. Three ypes of driver behavior can herefore be observed. The firs driver ype can be classified as a Sandard Acceleraor. Drivers in his caegory will end o follow he shape of he maximum acceleraion profile, wih he model simply shifed by some consan reducion facor hroughou he run. A second driver classificaion is he Hard Acceleraor. These drivers end o accelerae from a sop a a high iniial rae ha is close o he maximum value and hen decrease heir acceleraion rae as hey approach heir desired speed. A hird group of drivers can be classified as Gradual Acceleraors. These drivers are primarily concerned wih passenger comfor, and accelerae a a relaively consan and moderae rae hroughou he period of acceleraion, despie he abiliy of he vehicle o accelerae more aggressively a low speeds. Figure - shows a sample es run for a driver in each of he hree caegories. Noe ha he final speed reached by each of he drivers is he same a he end of he es secion. Therefore, hese hree drivers would have approximaely he same driver reducion facor. However, Figure - clearly demonsraes hree differen approaches o obaining he desired speed. As will be discussed in he following secion, he majoriy of he drivers observed in his research effor refleced he Sandard Acceleraor behavior. Because he Sandard Acceleraors follow he same rend as he maximum acceleraion paern, inroducing a consan driver reducion facor o he model creaes a good fi o he field daa from heir runs. Therefore, a consan reducion facor was applied o he maximum model o ge 8

92 Chaper : Typical Acceleraion he bes fi for each driver. The following secion describes he breakdown for each of he drivers, grouped by heir classificaion, and shows he fi of he model o each se of field daa Acceleraion (m/s ) 7 - Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -: Crown Vic Maximum Acceleraion Daa on Smar Road wih Rakha Model 8

93 Chaper : Typical Acceleraion Proposed Model Proposed Model Proposed Model Proposed Model Acceleraion (m/s ) Acceleraion (m/s ) 7 Acceleraion (m/s ) Proposed Model Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -: Rakha Model for Crown Vic Maximum Acceleraion on Level Roadway Gradual Driver Sandard Driver Hard Driver Figure -: Example Profiles of Three Differen Driver Types 8

94 Chaper : Typical Acceleraion.. Driver Facors and Classificaion a. Sandard Acceleraors The larges group of drivers observed in he sudy fell ino he caegory of Sandard Acceleraors. The defining characerisic for his acceleraion behavior is ha hey follow he paern of maximum acceleraion, only a a lower rae. Therefore, he acceleraion profiles for hese drivers are similar o he maximum acceleraion profile ha can be achieved by he vehicle, excep ha hey are shifed by some consan reducion facor. Consequenly, he daa from hese drivers shows he sronges correlaion o he model. Figure - shows he effec of applying he driver facor o he acceleraion versus speed plo. Noe ha he facor does no simply reduce he maximum acceleraion rae by a cerain percenage a a given speed. This is because he effec of he driver facor is cumulaive hroughou he acceleraion, and herefore he plo is shifed owards he origin raher han jus owards he x-axis. This enables he modified Rakha model o achieve a beer fi o acual daa han oher models like he linear decay model, as will be discussed laer in his repor. Foureen of he weny drivers (7%) fell ino he sandard acceleraor caegory. Differen driver facors were applied o he model for each driver o form he bes fi o he field daa. Larger driver facors indicae more aggressive driving. A good fi o he field daa was achieved for each Sandard Driver in all of he speed profiles. The rend in he acceleraion plos also maches he model predicion for all of he drivers in his caegory. Please see Figures - hrough -8 for all plos of he differen Sandard Acceleraors.... Facor. Facor.8 Facor.7 Facor.6 Facor. Facor. Acceleraion (m/s ) Figure -: Example Effec of Driver Facors on Acceleraion Profile 8

95 Chaper : Typical Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -: Driver # Daa wih Rakha Model, Reducion Facor =.6 86

96 Chaper : Typical Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -6: Driver # Daa wih Rakha Model, Reducion Facor =.78 87

97 Chaper : Typical Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -7: Driver # Daa wih Rakha Model, Reducion Facor =.6 88

98 Chaper : Typical Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -8: Driver #7 Daa wih Rakha Model, Reducion Facor =.7 89

99 Chaper : Typical Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -9: Driver #8 Daa wih Rakha Model, Reducion Facor =.6 9

100 Chaper : Typical Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -: Driver #9 Daa wih Rakha Model, Reducion Facor =.6 9

101 Chaper : Typical Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -: Driver # Daa wih Rakha Model, Reducion Facor =. 9

102 Chaper : Typical Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -: Driver # Daa wih Rakha Model, Reducion Facor =.6 9

103 Chaper : Typical Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -: Driver # Daa wih Rakha Model, Reducion Facor =.8 9

104 Chaper : Typical Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -: Driver # Daa wih Rakha Model, Reducion Facor =.8 9

105 Chaper : Typical Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -: Driver #6 Daa wih Rakha Model, Reducion Facor =. 96

106 Chaper : Typical Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -6: Driver #7 Daa wih Rakha Model, Reducion Facor =. 97

107 Chaper : Typical Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -7: Driver #8 Daa wih Rakha Model, Reducion Facor =. 98

108 Chaper : Typical Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -8: Driver #9 Daa wih Rakha Model, Reducion Facor =.7 99

109 Chaper : Typical Acceleraion b. Gradual Acceleraors Three individuals esed exhibied gradual acceleraion behavior. These drivers ended o accelerae a a lower rae han he Sandard Acceleraors a he beginning of he run. Therefore, here are several indicaors of Gradual Acceleraor behavior ha can be observed in he plos of Figures -9 hrough -. For he Gradual Acceleraors, he model ends o overesimae speeds owards he beginning of he run. The model also overesimaes acceleraions a low speeds for his driver ype. Finally, he slopes of acceleraion versus ime plos for hese drivers appear o be less severe han he model predics. Overall, however, he model predics reasonable speed profiles, even for drivers in his caegory. Only in he mos exreme case of gradual acceleraion behavior (Driver #, Figure -) does he model provide somewha inaccurae resuls. c. Hard Acceleraors Three individuals exhibied hard acceleraion behavior. These drivers accelerae very aggressively iniially, when he acceleraion capabiliy of he vehicle is greaes, bu rapidly decrease heir acceleraion once higher speeds are achieved. Figures - hrough - show he plos of he drivers in his caegory. The speed versus disance plos show ha he model overesimaes speeds owards he end of he run for hese drivers because he model is fi o he beginning of he run when he acceleraion is more aggressive. The acceleraion versus speed plos also show ha he model underesimaes acceleraion slighly a low speeds. The model also underesimaes speed for hese drivers in he speed versus ime plo. Despie hese minor errors, he model sill provides a reasonable approximaion of he speed profiles for hese drivers... Driver Summary A wide range of driver facors was observed in his sudy, from a highly aggressive value of.8 o a very non-aggressive value of.. The observed speed profiles are shown in Figure -. The disribuion of driver facors is analyzed in he following secion. Three driver ypes were observed Gradual Acceleraor, Sandard Acceleraor, and Hard Acceleraor. Sandard Acceleraors accouned for of he drivers, while only examples of Gradual Acceleraors and Hard Acceleraors were observed. The assumpion of applying a consan reducion facor o he maximum acceleraion model hroughou he run creaes a good fi for he Sandard Acceleraors. Since his group represens he vas majoriy of driver ypes, his assumpion is valid. In fac, he consan reducion facor also enabled a reasonable fi for mos of he Gradual and Hard Acceleraors. Only he exreme examples of hese driver classificaions resuled in daa inconsisen wih he model predicions. I would be unreasonable o develop a model for hese exreme siuaions, and we can herefore conclude ha he model effecively represens he range of ypical driver behavior.

110 Chaper : Typical Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -9: Driver # Daa wih Rakha Model, Reducion Facor =.6

111 Chaper : Typical Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -: Driver # Daa wih Rakha Model, Reducion Facor =.7

112 Chaper : Typical Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -: Driver # Daa wih Rakha Model, Reducion Facor =.

113 Chaper : Typical Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -: Driver # Daa wih Rakha Model, Reducion Facor =.6

114 Chaper : Typical Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -: Driver # Daa wih Rakha Model, Reducion Facor =.6

115 Chaper : Typical Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -: Driver #6 Daa wih Rakha Model, Reducion Facor =.6 6

116 Chaper : Typical Acceleraion Maximum acceleraion 8 6 Field observed acceleraions Figure -: Observed Typical Speed Profiles Versus Maximum Profile.. Disribuion This research has shown ha a model for ypical acceleraion behavior can be generaed for any driver by applying a cerain reducion facor o he maximum acceleraion Rakha model. Obviously, every driver on he road canno be esed o obain his or her facor. However, a saisical disribuion of driver facors can be generaed based on he drivers sudied in his paper. Wih ha in mind, he reducion facors for each of he weny drivers in his sudy were grouped ino bins and ploed as shown in Figure -6. Based on Figure -6 and common sense, a reasonable assumpion can be made ha he driver facors could be fi o a normal saisical disribuion. This would mean ha he majoriy of drivers would exhibi some average driver facor, and he probabiliy of observing differen facors would reduce as values moved away from he average. A normal disribuion was herefore fi o he daa as shown in Figure -7. Figure -8 shows a good fi beween he cumulaive normal disribuion and he one observed during he field ess. The besfi normal disribuion of driver facors had a mean of.6 and a sandard deviaion of.8. The acual mean value observed for he drivers was.6. 7

117 Chaper : Typical Acceleraion 7 Number of Observaions Driver Facor Figure -6: Disribuion of Observed Driver Facors Likelihood Driver Facor Figure -7: Normal Disribuion Fi o Observed Driver Facors Cumulaive Probabiliy Observed Daa. Normal Dis Driver Facor Figure -8: Cumulaive Normal Disribuion Fi o Daa 8

118 Chaper : Typical Acceleraion To es he validiy of applying he normal disribuion o he se of observed facors, a chi-squared es was performed. The facors were grouped ino hree bins less han., beween. and.6, and greaer han.6. This insured ha each bin would have a leas five observaions, as required for he chi-squared es. Table - shows he number of observed and expeced observaions for each bin, as well as he deviaion. A chi-squared value of.8 was calculaed from he daa. This value is well below he criical value of.99. Therefore, his daa se shows a srong correlaion o he normal disribuion a he % confidence limi. Furher evidence of he srong correlaion beween he normal disribuion and he es daa is shown in Figure -9. This figure shows he daa poins from all runs for each of he weny drivers. The speed profile for he mean driver facor as well as for he % and 9% confidence limis are also depiced on he graph. Several observaions can be made from his graph. The firs observaion is ha a large percenage of he runs are grouped near he mean, which should be expeced. Also, he plo shows ha daa poins from one driver are below he % confidence limi and daa poins from one driver are above he 9% confidence limi. This should be expeced because his represens % ( of ) of he es subjecs a eiher end of he specrum. Table -: Chi Squared Tes Calculaion Class Observed Expeced Deviaion d squared d/e < > Chi Squ Mean and 9% Confidence Limis Figure -9: Driver Run Daa wih Disribuion Mean and Seleced Perceniles 9

119 Chaper : Typical Acceleraion.. Age and Gender Variabiliy In addiion o developing a disribuion of driver facors, ess were done o see if age or gender affeced he observed driver facors. As migh have been expeced, he mean driver facor for men was higher han for women (.6 o.69) and he mean driver facor for ages -9 was higher han for ages and above (.69 o.6). An analysis of variance es was performed on he daa o deermine if hese means were saisically differen. Table - shows he compued values for he gender comparison, while Table - shows he values for he age comparison. An analysis of variance was also performed for boh facors simulaneously, as shown in Table -6. The resuls of hese ess showed ha here was no saisical difference in he means a he % confidence limi ha could be aribued o age or gender. Therefore, we can conclude ha he normal disribuion wih a mean of.6 and a sandard deviaion of.8, as deermined earlier in his repor, does no need o be modified for driver populaions of differen ages or genders. Table -: ANOVA Table, Male versus Female Anova: Single Facor SUMMARY Groups Coun Sum Average Variance Female Male ANOVA Source of Variaion SS df MS F P-value F cri Beween Groups Wihin Groups Toal Table -: ANOVA Table, Young versus Old Anova: Single Facor SUMMARY Groups Coun Sum Average Variance Over ANOVA Source of Variaion SS df MS F P-value F cri Beween Groups Wihin Groups Toal.997 9

120 Chaper : Typical Acceleraion Table -6: ANOVA Table, Boh Facors Terms: Sex Age Sex:Age Residuals Sum of Squares.6.6 E Deg. Of Freedom 6 Residual sandard error:.776 Df Sum of Sq Mean Sq F Value Pr(F) Sex Age Sex:Age..9E Residuals Linear Decay Model Comparison The model presened in his paper has been shown o provide a good fi for each driver in he domains of speed versus ime, speed versus disance, acceleraion versus ime, acceleraion versus disance, and acceleraion versus speed. For comparison, he linear decay model recommended in he lieraure (Long, ) was applied o he daa ses colleced for his sudy o deermine heir fi. The resuls of his effor are discussed in he following paragraphs. The firs ask was o calibrae he linear decay model o fi he maximum acceleraion profile of he Ford Crown Vicoria. This was accomplished by seing he maximum acceleraion a a value of. m/s and by seing he maximum speed o 6 km/h. This resuled in alpha and bea values of. and.6, respecively. Using hese values, he linear decay model showed an excellen fi o he maximum acceleraion field daa colleced in a prior research effor. The nex sep was o apply he driver facors deermined in his repor o he linear decay model for each driver and compare he resuls o he ypical acceleraion field daa. One example of a gradual acceleraion driver, a sandard acceleraion driver, and a hard acceleraion driver were chosen for illusraion purposes in his repor. Figure - shows he fi of he linear decay model o a driver classified as a gradual acceleraor. The model demonsraes a good fi in all domains for his driver, and even capures he slope of he acceleraion versus speed plo beer han he model developed in his paper for his paricular driver. This demonsraes ha using a reducion facor along wih he linear decay model is effecive in modeling drivers ha exhibi gradual acceleraion endencies. However, only % of he drivers sudied in his es exhibied his behavior.

121 Chaper : Typical Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -: Sample Gradual Acceleraion Driver versus Linear Decay Model

122 Chaper : Typical Acceleraion Figure - shows he plos of a sandard acceleraing driver versus he linear decay model. The speed versus disance plo shows a reasonable fi, which indicaes ha he driver facor used was appropriae. However, he linear decay model seems o underesimae he driver acceleraion for he firs half of he run. The model does no seem o capure he appropriae slope in he acceleraion versus speed domain for he sandard acceleraors. This is a major limiaion of he linear decay model, because he majoriy of drivers in his sudy 7% - were classified in his group. Therefore, he linear decay model would no accuraely describe he acceleraion behavior of mos of he drivers on he road. This is consisen wih he findings of Long (), who found ha he slope of he acceleraion versus speed plo was oo fla when a reducion facor was applied o he linear decay model o predic ypical acceleraion behavior. Figure - demonsraes ha he fi of he linear decay model o individuals classified as hard acceleraors is even worse. In addiion o he slope being erroneous in he acceleraion versus speed plo, he speed profiles prediced by he linear decay model are also flawed for he hard acceleraors. We can conclude ha using a reducion facor wih he linear decay model o predic ypical acceleraion raes generaes a profile ha is consisen wih he gradual acceleraor driver ype. Because only a small percenage of drivers exhibi his behavior, i is no recommended o use he model in his fashion o predic ypical behavior. The Rakha model provides a good fi for all driver ypes, paricularly hose in he sandard acceleraor caegory. Since his group makes up he larges percenage of drivers, he Rakha model is more effecive a predicing ypical acceleraion behavior using driver reducion facors han he linear decay model..6 Imporance of Modeling a Range of Typical Behavior This paper has shown ha ypical driver behavior varies grealy from driver o driver. I has also shown ha a good fi o field daa can be generaed by applying various driver facors o he maximum acceleraion capabiliies of vehicles prediced by he Rakha model. A normal disribuion of driver facors was deermined wih a mean of.6 and a sandard deviaion of.8 ha represens he driver populaion on he road. However, a quesion remains; is i really necessary o use he driver disribuion in simulaion models, or is i sufficien o use he mean value for all drivers? To answer his quesion, a simple es was conduced. The Rakha model was used o predic he speed profiles of a simulaed vehicle driven by five differen drivers wih reducion facors ranging from. o.8. The. facor represened a driver wih a low aggressiveness, while he.8 facor represened a highly aggressive driver. The driver facor of.6 represens average driver aggressiveness. The speed profile for each driver was deermined over an 8-meer disance on a road wih a speed limi of mph (88. km/h). Each driver was simulaed o accelerae from a sop o he speed limi, and hen coninue raveling a he speed limi unil he end of he 8-meer secion. Figure - shows he corresponding speed profiles in his simulaion for he differen driver ypes.

123 Chaper : Typical Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -: Sample Sandard Acceleraion Driver versus Linear Decay Model

124 Chaper : Typical Acceleraion Acceleraion (m/s ) Acceleraion (m/s ) Acceleraion (m/s ) Figure a: Speed versus disance Figure b: Speed vesus ime Figure c: Acceleraion versus disance Figure d: Acceleraion versus ime Figure e: Acceleraion versus speed Figure -: Sample Hard Acceleraion Driver versus Linear Decay Model

125 Chaper : Typical Acceleraion Figure -: Sample Speed Profiles for Differen Driver Facors The simulaion was performed o illusrae he difference in vehicle behavior beween an aggressive driver, an average driver, and a passive driver. The firs hing o noe is he amoun of ime required o obain he speed limi. The average driver akes. seconds o accelerae up o a speed of mph, and accomplishes his over a disance of abou meers. A highly aggressive driver can reach he same speed in only. seconds in a disance of 9 meers, while a non-aggressive driver requires. seconds and 8 meers o reach mph. These differences in acual driver performance are significan and could change he resuls of a simulaion model, especially when accumulaed hroughou he nework over ime. If only average drivers are modeled in he simulaion, acual roadway condiions will no be observed. This could resul in he model generaing inappropriae signal imings, for example. In addiion o vehicle behavior varying grealy from driver o driver, oher facors relaed o he acceleraion profile of he vehicles, including fuel consumpion and emissions, can vary as well. To es he significance, he five vehicle profiles for he 8-meer simulaion were inpu ino he VT-Micro model. This model can deermine he fuel consumpion and emissions for any speed profile (Rakha and Ahn, ). The resuls are shown in Table -7. The able indicaes ha modeling all vehicles using he average driver behavior may overesimae fuel consumpion, as he fuel consumpion is lower for 6

126 Chaper : Typical Acceleraion boh very aggressive drivers and very non-aggressive drivers. The emissions produced by he vehicle also vary grealy depending on driver aggressiveness. A highly aggressive driver emis nearly double he amoun of carbon monoxide as a ypical driver. This furher demonsraes he need o model a disribuion of drivers in he simulaion. Table -7: Fuel Consumpion and Emissions Raes for Various Driver Types Uni: fuel (grams), HC, CO, NOx (grams) Fuel Consumpion HC CO NOx Facor Facor Facor Facor Facor Variaion from Mean Fuel Consumpion HC CO NOx Facor. -% % -% -% Facor. % % -% -% Facor.6 % % % % Facor.7 -% % % -% Facor.8 -% 8% 9% -%.7 Conclusions and Recommendaions This paper has presened a modificaion o he Rakha vehicle dynamics model for predicing he acceleraion behavior of vehicles. The modificaion involves applying a reducion facor o he value of maximum acceleraion o accoun for he ypical behavior of a given driver. Field daa was colleced for weny drivers operaing he same es vehicle. The drivers included men and women and ranged from age o. Driver facors were observed ha fi a normal disribuion wih a mean of.6 and a sandard deviaion of.8. A consan driver reducion facor hroughou he run was shown o generae a good fi o he field daa for mos of he drivers. Males and younger drivers ended o be more aggressive in his sudy, bu he saisical means of hese groups did no vary significanly when an analysis of variance was applied. The disribuion of driver reducion facors can be used wih he Rakha model and incorporaed ino simulaion models o reflec he populaion of drivers on he roadway more accuraely. The following recommendaions for fuure research acion are proposed o enhance he capabiliies of he updaed acceleraion model:. Tess should be performed wih eenage drivers and older drivers. As menioned in he repor, drivers ouside he range of ages sudied in his research accoun for nearly half he drivers on he road. These groups need o be included in fuure daa collecion o enhance he accuracy of he model. I is of paricular imporance o sudy hese wo groups because eenagers generally represen an aggressive group and older drivers 7

127 Chaper : Typical Acceleraion generally reflec more conservaive driving behavior. Therefore, he saisical means of hese wo groups may be differen from he means for he - age group.. Tesing should be performed wih oher vehicles. I is unclear from his esing wheher he driver facor would change based on he capabiliy of he vehicle. For example, would he same driver accelerae more rapidly in a vehicle wih more horsepower, or would he acceleraion profile for a given driver be similar regardless of vehicle?. Tesing should be done on a vehicle wih manual ransmission. The es vehicle used in his sudy was auomaic ransmission and herefore may have limied he abiliy of he drivers o accelerae a heir normal levels.. Tesing should be performed in real on-road driving condiions. The poin of his research was o compare driver behavior o maximum vehicle capabiliy in an unconsrained esing environmen. Research needs o be done o deermine how he ypical acceleraion rae varies in normal driving condiions, where car following is presen. 8

128 Chaper 6: Conclusions Chaper Six: Summary, Conclusions, and Recommendaions 6. Summary The paper firs consruced a daabase of unconsrained maximum vehicle acceleraion daa for hireen ligh duy vehicles and rucks. In addiion, he paper validaed a vehicle dynamics model for predicing maximum ligh duy vehicle acceleraion raes. The model was compared o four of sae-of-ar acceleraion models using he developed daa se. Advanages of he vehicle dynamics model were presened, including is abiliy o accuraely predic vehicle behavior wih readily available inpu parameers and is flexibiliy in esimaing acceleraion raes of boh large and small vehicles on varied errain. This paper hen presened a modificaion o he vehicle dynamics model in order o predic he ypical acceleraion behavior of drivers. The modificaion involved applying a reducion facor o he value of maximum acceleraion o accoun for he ypical behavior of a given driver. Field daa was colleced for weny drivers operaing he same es vehicle. The drivers included men and women and ranged from age o. Driver facors were observed ha fi a normal disribuion wih a mean of.6 and a sandard deviaion of.8. A consan driver reducion facor hroughou he run was shown o generae a good fi o he field daa for mos of he drivers. Males and younger drivers ended o be more aggressive in his sudy, bu he saisical means of hese groups did no vary significanly when an analysis of variance was applied. The disribuion of driver reducion facors can be used wih he Rakha model and incorporaed ino simulaion models o reflec he populaion of drivers on he roadway more accuraely. 6. Conclusions The vehicle dynamics model proposed originally by Rakha e al. () o predic he maximum acceleraion behavior of rucks can be applied successfully o predic he maximum acceleraion behavior of a variey of passenger vehicles, ranging from small subcompac cars, o larger vans, spor uiliy vehicles, and pickup rucks. While a variable power modificaion o he model is necessary o accoun for he build-up of power hrough gear shifing in rucks, his modificaion is no necessary when modeling cars because hey have fewer gears and generally use auomaic ransmission. Therefore, he more-basic consan power model can be applied successfully when simulaing passenger vehicles. This vehicle dynamics model was shown o ouperform oher similar acceleraion models recommended in he lieraure. The model is able o generae a good fi o field daa in all possible speed profile domains, including speed versus ime, speed versus disance, acceleraion versus ime, acceleraion versus disance, and acceleraion versus speed. Several limiaions of exising models were found, including heir reliance on field daa, 9

129 Chaper 6: Conclusions and heir endency o only provide a reasonable fi o field daa for he domain in which hey were calibraed. Based on a comparison sudy, hree primary advanages of he Rakha dynamics model were observed:. Good Fi in all Domains: The vehicle dynamics model offers a good fi o field daa in each of he domains, namely speed versus disance, speed versus ime, acceleraion versus speed, acceleraion versus ime, and acceleraion versus disance. Alernaively, he oher models, especially he empirical models, are designed o fi a single domain, and herefore do no fi well over all domains.. Model Flexibiliy: The vehicle dynamics model is able o adap o differen vehicle ypes, differen pavemen condiions, and ravel over differen errain.. Simple Calibraion: Anoher advanage of he vehicle dynamics model is ha i does no require field daa ses o calibrae inpu parameers. Specifically, inpu parameers can be obained from vehicle manuals, car magazines, and/or he World Wide Web. Alernaively, he oher models require field daa for he calibraion of inpu parameers. The Rakha model can also be applied o ypical driver behavior by inroducing a reducion facor o he maximum acceleraion model for each driver. A differen facor can be chosen for each driver ha will enable he model o provide a good fi o field daa in each domain. The disribuion of observed driver facors fi a normal disribuion wih a mean of.6 and a sandard deviaion of.8. This disribuion can be used o model he driver populaion in simulaion siuaions. The Rakha model was shown o be superior o similar ypical acceleraion models because of is abiliy o properly predic he acceleraion profiles of he majoriy of drivers. Three driver ypes were idenified, including Sandard Acceleraors, Gradual Acceleraors, and Hard Acceleraors. The Rakha model was mos successful in modeling he behavior of Sandard Acceleraors, which comprised 7% of hose esed. 6. Recommendaions The following recommendaions for fuure acion are suggesed o furher enhance he capabiliies of he models presened in his paper for predicing maximum and ypical acceleraion levels:. Tesing should be performed in real on-road driving condiions. The poin of his research was o observe driver behavior and maximum vehicle capabiliy in an unconsrained esing environmen. Research needs o be done o deermine how he ypical acceleraion rae varies in normal driving condiions, where car following is presen.

130 Chaper 6: Conclusions. Daa are also required on ypical vehicle acceleraion during criical evens, including merging wih freeway raffic a an on-ramp and acceleraing o overake a vehicle. For all hese scenarios vehicle acceleraion models should be developed o capure driver behavior in he non-seady sae car-following mode of operaion. To expand he capabiliies of he ypical acceleraion model presened in Chaper, he following acions are recommended:. Tess should be performed wih eenage drivers and older drivers. As menioned in he repor, drivers ouside he range of ages sudied in his research accoun for nearly half he drivers on he road. These groups need o be included in fuure daa collecion o enhance he accuracy of he model. I is of paricular imporance o sudy hese wo groups because eenagers generally represen an aggressive group and older drivers generally reflec more conservaive driving behavior. Therefore, he saisical means of hese wo groups may be differen from he means for he - age group.. Tesing should be performed wih oher vehicles. I is unclear from his esing wheher he driver facor would change based on he capabiliy of he vehicle. For example, would he same driver accelerae more rapidly in a vehicle wih more horsepower, or would he acceleraion profile for a given driver be similar regardless of vehicle?. Tesing should be done on a vehicle wih manual ransmission. The es vehicle used in his sudy was auomaic ransmission and herefore may have limied he abiliy of he drivers o accelerae a heir normal levels.

131 References References. Akcelik, R. and Biggs, D.C. (987) Acceleraion Profile Models for Vehicles in Road Traffic. Transporaion Science, Vol., No... Archilla, A.R., and De Cieza, A.O.F. (999). Truck Performance on Argeninean Highways. Transporaion Research Record, Transporaion Research Board, Washingon, D.C.. Bham, G.H. and Benekohal, R.F. () Developmen, Evaluaion, and Comparison of Acceleraion Models, Paper No. -767, CD-ROM, 8 s Annual Meeing of he Transporaion Research Board, Washingon, D.C.. Dockery, A. (966) Acceleraions of Queue Leaders from Sop Lines. Traffic Engineering and Conrol. Vol. 8, No... Drew, D.R. (968) Traffic Flow Theory and Conrol. McGraw-Hill. 6. Fich, J.W. (99). Moor Truck Engineering Handbook. Sociey of Auomoive Engineers, h Ediion. 7. Lee, C., and T. Rioux (977), The TEXAS Model for Inersecion Traffic Developmen Research Repor. Universiy of Texas a Ausin. 8. Long, G. () Acceleraion Characerisics of Saring Vehicles. Transporaion Research Record, No Louzenheiser, D.W. (98) Speed-Change Raes of Passenger Vehicles. HRB Proc., Vol. 8.. Mannering, F.L. and Kilareski, W.P. (99). Principles of Highway Engineering and Traffic Analysis. John Wiley & Sons.. A Policy on Geomeric Design of Highways and Srees (99). AASHTO, Washingon, D.C.. A Policy on Geomeric Design of Rural Highways (9). AASHTO, Washingon, D.C.. Rakha H. and K. Ahn (), The INTEGRATION Framework for Esimaing Mobile Source Emissions, Submied for publicaion in he Journal of Advanced Transporaion.. Rakha H. and B. Crowher (), A Comparison of he Greenshields, Pipes, and Van Aerde Car-Following and Traffic Sream Models, Acceped for publicaion in he Transporaion Research Record.. Rakha H., Lucic I., Demarchi S., Sei J., and Van Aerde M. () Vehicle Dynamics Model for Predicing Maximum Truck Acceleraion Levels. Journal of Transporaion Engineering, Vol. 7, No.. 6. Rakha H., and Lucic I. () Variable Power Vehicle Dynamics Model for Esimaing Maximum Truck Acceleraion Levels, Acceped for publicaion in he Journal of Transporaion Engineering.

132 References 7. Searle, J. (999) Equaions for Speed, Time and Disance for Vehicles Under Maximum Acceleraion. Advances in Safey Technology, Sociey of Auomoive Engineers Special Publicaions, No.. 8. Sociey of Auomoive Engineers (SAE). (996). Commercial Truck and Bus SAE Recommended Procedure for Vehicle Performance Predicion and Charing. SAE Procedure J88. Warrendael, PA. 9. Vara, M.S. and Husher, S.E. (), Vehicle Impac Response Analysis Through he Use of Acceleromeer Daa. Acciden Reconsrucion: Analysis, Simulaion, and Visualizaion, SP-9, Sociey of Auomoive Engineers, Inc., Warrendale, PA.. Waanada, T., e al. (987). Descripion of he HDM-III Model. The Highway Design and Mainenance Sandard Model, Vol., Johns Hopkins Universiy, Balimore.

133 Via Via Ma Snare was born in Phillipsburg, New Jersey o Parker and Sherryl Snare. He laer moved o Eason, Pennsylvania wih his family and wen on o graduae from Eason Area High School as he saluaorian of he Class of 997. Ma aended Virginia Tech afer high school and he received a Bachelor of Science degree in Civil Engineering in. Ma eleced o say a Virginia Tech for his graduae work, and he received he Charles Edward Via fellowship o pursue his graduae sudies. A Virginia Tech, Ma became he presiden of he universiy chaper of he Insiue of Transporaion Engineers and also served as a eaching assisan. Ma will be married on Sepember, o his fiancée Miriam, and he couple will be relocaing o Cockeysville, Maryland where Ma will work as a raffic engineer wih he consuling firm of Rummel, Klepper, and Kahl in Balimore.