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