Goal: Simulate Ludicrous Mode P eak Acceleration Performance

Transcription

1 Tesla S P85D Ludicrous Mode Performance Simulation The Simulation can be run or modified with Mathcad 4/5. Free Trial at: Mathcad Simulation at: Goal: Simulate Ludicrous Mode P eak Acceleration Performance This paper shows a macro model for performance simulation of a Tesla S P85D Ludicrous Mode. The key parameters are peak motor torque, peak battery power (SOC), curb weiht, maximum tire traction, and some assumptions about the power loss/efficiency: hih efficiency induction motors (93%), and Inerter and power train (87%). Net System Efficiency, SysEff ~ 8%. From statements of Elon Musk, we would infer that tire traction is capable of rippin the Assume that with sufficient power, we can et a peak Motor Torque of 73 ft lb. For 5% SOC, battery max power is 74 hp, System Power of 586 hp. The model shows that for 5% SOC, the time from to 6 is.8 sec. This Analysis in done in the followin ten Sections. Section IV considers four different traction scenarios (Sec IV, p. 4). Results were calculated for, 9, 8, 7, 6, 5% State of Chare (SOC). See last raph p 5. NOTE: The Calculations & raphs in Sections III to VI are shown for % SOC. TABLE OF CONTENTS I. Introduction - Simple Analysis II. Macro Model Performance Discussion: -6 spec & the horsepower second (hp-s) III. Specifications & Enineerin Estimates: Ludicrous Mode Peak Acceleration IV. Four Tire Traction & Control Models: # Perfect Tire Grip, # Tires µ =., =., #3 Power Lited., #4 Fit to Dra Times Dyno Torque/Velocity Cure Shape V. Validation: The Model and Estimates of Parameters Aree with the Ludicrous Specs VI. Graphs of Model Results VII. Find the Sinle Chare Hihway Cruise Rane for a Gien Velocity and Final SOC VIII. Find Mileae Rane: Use Constant Velocity & Three Different EPA Driin Schedules IX. Extract Dra Times Dynamometer Torque Shape Cure X. Tire Friction/Grip (Tire Composition and Width) I. Introduction - Simple Analysis Exanin the Difference Between P85D Insane & Ludicrous Speed Uprade Insane Mode Specifications 3. seconds -6 Peak Acceleration: Front and Back Motor Spec: Power 69 hp Conentional Fuse: 3 amp battery litation Battery Peak Power: 3 A x = 695 hp Inerter and/or Alorithm sets max accel ~. Ludicrous Mode Specifications.8 seconds -6 with Ludicrous Speed Uprade Front and Back Motor Power Spec: 76 hp Electronic Fuse: 5 amp "effectie" battery uprade SOC Peak Power: 5 A x 36V = 74 hp Peak Motor Torque: 73 ft lb. New Inerter Alorithm to maxize Ludicrous acceleration. We assume that the implementation of the New Alorithm for Rear Motor Torque & Power Split in Ludicrous Mode, will proide the maximum tire rip acceleration,., and not optize Efficiency, that is, nize Power: = TS rear = 53/89 =.63 What is the nimum acceleration needed to meet the to 6 in.8 seconds? There are two factors inoled in this specification, elocity,, and time, t. For the sake of this Simple Analysis, let's assume that the acceleration is constant. Now acceleration defined as the rate of chane of elocity. In symbols, the constant acceleration, a. constant = chane of elocity/time. Then the acceleration needed to et to 6 in.8 seconds, or 6 /.8 s, is a constant.43 per sec. Thus a constant.43 /s in sufficient to meet our spec. In units of the earth's raitation, =.9 /s. We need a constant or "aerae" of.978. From Newtons nd Law we hae that F = m a. Then to accelerate a mass of 596 lb, we need a Force of 4984 lb or 4" tire radius and 9.73 Gear Ratio, a Motor Torque of 598 ft lbf. Then 73 ft lbf should be more than sufficient. Howeer, torque or acceleration is not constant. The 73 ft lb is peak torque. Torque falls off with ehicle speed or if the battery power is not sufficient to supply enouh power to keep it at its peak. Also, Ludicrous Mode uses a New Alorithm that proides complex Traction Control for max acceleration. Thus a more in-depth analysis is required. Amon other thins, it requires workin out what the torque falloff is with speed. We will find that with a net combined Inerter and Motor Power Efficiency of 8% and hih performance. tires (implied by Elon Musk), we can meet the Ludicrous Specs. This more in-depth analysis is what follows.

2 II. Macro Performance Model Discussion & Description of the Model Macro Model: Macro Models requires only lited knowlede of internal parameters. We treat the system as a Black Box. That is, we don't know the details of what's inside, just a few fundamental parameters. We are only interested in oerall performance. Inore the intricacies. Simple, but not too simple. May not know what is inside, but reardless, the laws of Physics still apply. We just need basic physical parameters such as: Vehicle mass (Mcurb), Coefficient of tire friction µ, and radius, Gear Ratio GR, max motor Torque & Power, battery power, and System Power Efficiency (Inerter, Gears, and Motors). The ehicle also has rotational enery from rotatin tires, motor rotor, and ear box. A factor km, which multiplies the mass, accounts for this added rotational mass. M curb = 4936 lbm, µ =. (equialently, max =.), 65/35ZR tires, tire radius=4 inches, GR = Then acceleration (a) is ien by: Newton's Second Law: a = Traction Force/m = Torque x GR/Mcurb See p 3 for Section on Traction Control. Then the Torque required to et to =., requires that Torque be at least: Torque_max_ = Weiht x km x. tire radius/gr = 68 ft lbf The present max Torque spec is 73 ft lbf. This is more than sufficient to ie.. What is the Estimated Motor Power needed to meet the - 6 in.8 s performance, P Spec? Ludicrous Mode Specs: The current has been increased from 3 to 5 amps (5 %), the total dual motor power increased from 68 hp to 76 hp (%), and the total torque from 77 to 73 ft lbf. Howeer, as we saw in section I, the System Power to the System Efficiency is lited to 649 hp There is a basic relationship between Torque, Motor RPM, and Motor Power: Assume that there is No Traction Control, this is tires can slip. Thus initially, full torque is applied to the wheels, until max motor power lits the torque. Refer to the below plot and exane the Power ersus time profile. The Power is ien by: Power = Torque x Anular Velocity, until Max Power is reached. This is shown in the raph below. Tire elocity, Pmax, to et to max motor power and Torque = The time to et to max motor power is t Pmax. The elocity at which this occurs is Pmax. There are paths to et to the max power: # Tires allowed to slip and # Tires do not slip (P. 3). Elon Musk said that the peak acceleration is.. Desinate the time to reach full power, t Pmax. Pm( ω) = T(ω) k π ω RPM What Max Power do we need to meet the - 6 in.8 s with No Traction Control? For a ehicle elocity,, the Vehicle Kinetic Enery, KE, required to et to 6, in units of horsepower seconds (hp-s), is ien by: A hp-s is the amount of enery one hp does in the time of one second. This enery unit more clearly reeals the needed hp oer time to to meet the -6 in.8 second spec. For the raph below, the aerae Motor power from the start at power to the peak of Power max is / Power max. If the time to et motor Power max is t Pmt, then the Enery is / Power max x t Pmax. After t Pmax the Motor power is constant. The Enery that oes into the Motor, E motor, in units of horsepower seconds, is shown below the raph. T max_ T max_ T max 4in GR 9.73 M curb 4936lbm k m.447 M curb k m = 68.3 ft lbf 73ft lbf Power max 76hp Velocity at Max Power Pmax Pmax = Power max T max GR 48.5 t Pmax.46s M ross M curb + 6lbm KE( ) M ross k m ( ) KE( 6) = 64.9 hp s. GR The most critical part of the model is the. seconds of Peak Power from the time the acceleration falls from. to the.8 second spec lit. The motor power cures belon to two different Traction Models: #: No Traction Control, Red Cure #: With Traction Control, Blue Cure t Pmt is the time to Peak Power =.46s t fall is the time for acceleration() to fall below. =.69 s. These times are in seconds. E motor 649 hp.46s + 649hp (.8.46 ) sec E motor = hp s This demonstrates that 649 hp is sufficient to meet the - 6 in.8 sec KE spec. What follows is a more detailed analysis.

3 III. Specifications & Enineerin Estimates: Ludicrous Mode Peak Acceleration System Efficiency: SysEff.8 % SOC Voltaes: V batt_ 398.4olt V batt_8 384olt V batt_5 36olt V batt V batt_ Results are Shown for % State of Chare. Battery and System SOC Power Batt V batt 5A = 8.39 hp Power System Power Batt SysEff = hp 85 kw-hr Battery De-acceleration Battery Enery Reeneration Factor: Reen.64 New Alorithm for Rear to Front Motor Torque & Power Split to maintain peak acceleration: = T Split = 53/59 ~.94Gear Ratio: Ludicrous Mode Max Power: Power max 649 hp RPM max 8 T Split.94 GR 9.73 Lit: Power System = hp Power max Power System Battery Enery: Enery bat 85 kw hr R phase.6ohm Max Dual Motor Torque: Power max = hp 6.3 From TeslaMotors.com/models 45/35R9 Tire Radius: r T tire Ultra Hih Performance Tire in T max GR max 73 ft lbf F Motor_Max 77 ft lbf Torque maxold μ. car max_ μ k τ sec Tire Coefficient of Friction, µ: Max. Quoted by Elon Musk Curb Weiht: M curb 4936lbm M ross M curb + 6lbm = 596 lbm Aerodynac Dra Coeff (TM): Cd. Aerae Wind Velocity: Vw Cross Wind Dra Coff: Cd cw.4 Effectie Cross Wind V: V cw max T max GR M ross k m Shape Correction Factor: SCF.85 Vehicle Frontal Dimensions: Af ( )in 77 in max =.9 Air Density, tire resistance: ρ.93 m Dra Frontal Area Ad Af SCF Ad =.7 m liter Tire Rollin Resist, Hys: RR tire. T hys sec Road Rollin Resistance: RR road.7 m Effectie Mass Coefficient: k m.447 EPA Rane Spec for P85 is 53 les.see pae 6 for EPA_RaneSim 5le IV. Tire Traction & Control Models: # Perfect Grip, # Tires slip, #3 No Slip, #4 Simple Step Model of Tire Traction ( Assume perfect weiht distribution per motor, i.e. same acceleration at each motor ) Dependin on road conditions, Tires do not hae perfect rip, they may slip. Vehicle acceleration, a eh is lited to the maximum tire traction (tire max_ ) =.. The tire rpm x GR = motor rpm, but because of slip, tire elocity can be reater than ehicle elocity. Therefore, ehicle acceleration and elocity are not directly proportional to rpm, that is, tires may slip: Case # No Traction Control, but no tire slip. Max motor power and torque are applied to tires. Perfect Tires that do not slip. Acceleration can exceed.. Case # - Ludicrous Mode Perfect Traction System and Hih Performance =. tires. Because of Traction Control and tire slip, effectie motor rpm can be reater than ehicle speed durin tire slip or Traction Control. Vehicle speed depends on tire coefficient of resistance, µ, which is equal to. for Ludicrous. This allows a max of.. For Case #, we assume Traction Control lits to.. Macro Model of Motor Dynacs : Velocity of Tire is Anular Velocity Symbol, Ω (units of radians/second) Anular Vel Power: Conert elocity to RPM Tire Velocity at Torque Fall: Tire Velocity to krpm: Road Resistance, Ft: Air Dra Force, Fa: Total Opposin Force, Fo: Ω( ω) πω n RPM/ Symbol, ω k RPM n Ω Pmax Ω Pmax Power max T max RPM Pmax RPM Pmax = RPM π VtoRPM( ) ( π RPM) ω Pfall RPM Pmax k = 4.78 RPM Tfall RPM Pmax π GR Tfall = t ( k π RPM) M ross sin( θ) VtokR t VtokR( 6 ) GR = 7.46 Ft T hys + ( RR tire + RR road ) cos( θ) + sin( θ) RPM pmax for Max Power: Note: For Dra and Road Resistance, Fa( ).5ρ Ad ( + Vw) Cd + Cd cw ( V cw ) approximate ehicle with tire. At Fo( ) Fa( ) + Ft( ) Fo( 6 ) = 39.4 lbf <6 Compared to Ftire, Fo is small. T PLt ( 55) = 6.99 ft lbf Power max Ω( ω k ) if ( ω k RPM, ) Torque/Force Falloff Cure: ω kmax 5.8 RPM T PLt ω k Tm is Torque of motor Fmot, Tractie Force from motor, not from slippin tires: T m ω k T m t ω Pfall, T PLt ω k T max P m ω k T m ( VtokR( t ) GR) F mot t GR T m ( t ) F PL t T m ( ω k ) k π ω k Power max ( t ) RPM

4 Sole for Velocity, Acceleration, and Distance ersus Time We are usin Mathcad 4, a Computer Math Proram, to do the Calculations: free-trial Case : Perfect Grip Tires at Maximum Motor Power, Coefficient of Tire Friction >. Newton's Third Law of Motion: F mot ( ) Fo( ) T max GR a ( ) a Tmax =.9 k m M ross M ross k m V V el ( t) root t sec dv, V a ( V ) time a ( ) d time a ( 6) =.58 s a ( ) el (.64) = 6.9 fall Velocity fall, a a ( fall ). root a ( V ) car max_, V = time a ( 6) =.58 s a t ( t) a el ( t) = a t =.7 Case -LudicrousMode: Hih Performance. Tires & Motor Drie Lited Accel <. / No Spin, but Max Power a acceleration is allowed by hih performance tires on dry road. a ( ) if a ( ) car max_, car max_, a ( ) car max_ =. V el ( t) root t sec dv, V time a ( ) d el (.8) = 6.87 a ( V ) a ( ) a ( fall ) =. t distance ( t) el ( t)τ dt a t ( t) a ( el ( t) ) a t ( ) =. time a ( 6) =.68 s distance (.5) =.5 le t fall time a fall = RPM at fall: R fall VtokR fall.69 s GR Case 3: Traction Control - Tire Force is Power Lited - No Tire Spin (This model not yet perfected) F _ F _ k m M ross car max_ = lbf T _ T 3 ω k GR P _ ω k T _ ω k if ω k R fall, T _, Power max Ω ω k k π RPM P _ ( 5.5) = hp ω 3Pmax 55 GR P 3 ( ω k ) T 3 ( ω k ) ( ω k k π RPM) F 3 ( ) T 3 ( VtokR( ) GR) Case 3: We end up ettin the same effectie peak torque, we just don't waste the power put into spinnin wheels. Tesla has a patent on how to split the power Case 4: Fit Torque to Dra-Times Dyno Torque Cure Shape Efficiency, Does Not Meet Specs a 4 () = P85D Acceleration/Torque, a () x T Shape (): a T Shape () was fitted to hae the same shape, as the 4 ( ) ifa ( ) T Shape Dra Times Torque/speed Cure - See Graph Below V el 4 ( t) root t sec car max_ a 4 ( V ) a 4t ( t) a 4 ( el ( t) ) time a4 ( ), car max_, a ( ) T Shape dv, V d a 4 ( ) el 4. ( t) el 4 ( t) Does not meet Specs time a4 ( 5) = 4.4 s Dyno Data by Dra Times for 5 Tesla S P85D Dyno Torque (Red) and Dyno Power (Green) DynoTorque Shape x (Blue), Extracted from Dyno Torque Model-S-Video s- 743.html V. Validation: The Model and the Estimate of System Parameters Meet the Ludicrous Specs time a ( 6 ) =.68 s distance (.46) =.4 le Calculated P9 EPA Rane: 3 Miles

5 VI. Graphs Compare with Torque & Power Cures at: Motor Torque el ( t) T m ( ω k ) ft lbf T 3 ω k ft lbf P 4 mt ( t) 3 P meh ( t) Li-Ion Cell Voltae (olts) Total Torque & hp s RPM/ ω Pfall ω k Motor RPM/ Velocity, Total hp/, Accel(s) s. sec.3 t Pmt t fall a t ( t) t Pmt.= t time (sec) a t ( t) P m ( ω k ) hp P 3 ω k hp el ( t) F mot. ( ) 6 spin lbf 5 F PL ( ) 4 lbf 3 time a. ( ) 7 6. el ( t) 5 Time -6 (Blue-secs) ersus Battery State of Chare, SOC% Time Meets Specs from to 5% SOC Time (Blue-secs) & Cell Voltae s. Battery Chare% Battery State of Chare% 4 Total Force, time(sec), hp s. () fall elocity () Velocity, Total hp/ Accel(s) s. sec Time to 6 (secs) t Pmt t fall. t time (sec) P ( ) k hp a ( ) a ( ) a t ( t) a t ( t) a 4t ( t) Panasonic Li-Ion P865 Approximate Cell Voltae Data: owthread.php/4469-p85d-69hp-s hould-hae-an-asterisk-*-next-to-it- Up-to-69HP/pae5?p=9487#post 9487 VII. Find the Sinle Chare. Hihway Cruise Rane for a Gien Velocity and Final SOC Driin Pattern/Profile: Assume we cruise at constant speed, but start, stop, and reen break four times per hour Drie Train Power Efficiency - Battery Loss for Commanded Vehicle Velocity and Final State of Chare, SOC f : SOC f is % at rechare. 4V HV battery idle power is Po. V battery ies Accessory Power. The Traction Inerter Efficiency - TInE, HV Power Electro nics at Idle Efficiency - IPEE, and Gear Power Efficiency - GPE are 9.5%, 95%, and 9%, respectiely. Brake Reen efficiency of kinetic enery is 64%. Then the number of starts per hour as a function of elocity, NS, NumStarts(, Po), is Chane in State of Chare = - SOC f TInE.95 IPEE.95 GPE.9 Reen.64 Power dissloss (, P o ) Fo( ) P o watt + Enery accel ( ) Power max time( ) hr TInE GPE IPEE

6 NSo, NS are iteratie conerin estimates of total NumStarts per chare 65 NS o ( ) NS, P o, SOC f ( +. ) Enery bat SOC f NS o ( ) Power dissloss, P o M ross 5 ( ) ( Reen) n floor NumStarts, P o, SOC f Enery bat SOC f NS, P o, SOC f Power dissloss, P o 5 M ross n ( ) ( Reen) Cruise_Rane, P o, SOC f Enery bat SOC f Reen M NumStarts(, P o, SOC f ) ross ( ) ( Reen) Power dissloss, P o Hihway Cruise Rane with Four Stops per Hour Estimate 4 Estimated Sinle Chare Hihway Mileae Cruise_Rane( 3,,.) = 3.55 Cruise_Rane( 4,,.) = Cruise_Rane( 5,,.) = 5.7 Cruise_Rane( 6,,.) = 6.8 Cruise_Rane( 7,,.) =.5 Cruise_Rane( 6,, ) = 5.4 Cruise Rane (les) Cruise_Rane(,, ) Cruise_Rane(,,.4) 35 3 Cruise_Rane(, 5,.5) 75 Cruise_Rane(,,.3) 5 5 Cruise_Rane(,,.4) 75 Cruise_Rane(,,.5) 5 5 Rane_ elocity () Opposin Force (Air Resistance, Tire, Road Resistance ) Power Loss Cruisin Power (kwatts) Power cruise, P o Power cruise (, 5) k watt Power dissloss (, P o ) Power cruise (, 5) k watt Power cruise (, 5) =.7 hp Cruise Power Cure elocity ()

7 VIII. Find Mileae Rane: Use 3 Different EPA Driin Schedules Alorithm to Calculate Rane, Rane(P,fHz), % Battery Dischare, Driin Profile Velocity/Time File, P and Samplin Rate, fhz Ebat E diss old Rane P, f Hz n N while rows( P) n n + t E diss P t Enery bat < kw hr mod( n, N) Enery bat = 85 kw hr P accel P accel f Hz k m M ross old sec if > old TInE GPE k m M ross old f Hz sec Reen sec Power dissloss (, ) + P accel E diss E diss + kw hr old Ebat n E diss f Hz otherwise If deceleratin, chare battery with Reen fraction of enery. Rane n m = ( P + P mod( m, N) mod( m+, N) ) sec f Hz Read US6 and FTP Dynamometer Drie Profile Files Refer to: The US6 cycle represents an 8. le (.8 km) route with an aerae speed of 48.4 les/h (77.9 km/h), maximum speed 8.3 les/h (9. km/h), and a duration of 596 seconds. Samplin can be either Hz or Hz The Federal Test Procedure (FTP) is composed of the UDDS followed by the first 55 seconds of the UDDS. It is often called the EPA75. Hz Samplin data is named FP and HY for the Hihway schedule. FTPF READPRN ("FedTestProc.txt" ) t FTPF FTP FTPF rows( FTP) = 875 UDDSF READPRN ("uddscol.txt" ) UDDS UDDSF rows( UDDS) = 37 HWYF READPRN ("hwycol.txt" ) HWY HWYF R hwy rows( HWY) FP READPRN ("FTPHz.TXT" ) FTPV submatrix( FP,, rows( FP),, cols( FP) ) HY READPRN ("HWYHz.TXT" ) HWYV submatrix( HY,, rows( HY),, cols( HY) ) US6F READPRN ("US6PROFILE.TXT" ) Time US6F US6 US6F n r.. rows( HY) HWY HWYV r r+ ceil, mod( r, )

8 Usin EPA Profiles and aboe Rane Proram, Calculate Tesla EV Rane for EPA Profiles Rane US6 Rane( US6, ) Rane FTP Rane( FTP, ) Rane HWY Rane( HWY, ) EPA 8 Cycle MPG Fuel Economy Least Squares Fit Reression for Rane MPG city MPG hwy Rane FTP Rane HWY MPG epa.55 MPG city +.45 MPG hwy Sinle Chare EPA Rane Calculations: Federal Test Procedure (FTP), Hihway, and US6 Model Validation: Published EPA Rane is 6 les Rane FTP = 9.44 Rane HWY = Rane US6 = 8. MPG city = 8.99 MPG hwy = MPG epa = 3 r rr r.. rows( FTP) Distance FTP rr.. rows( US6) Distance r r.rr US6 6 6 rr max( Distance) =.4 r = rr = max( Distance) =.4 max( Distance. ) = Plots of EPA Dynamometer Vehicle Testin Profiles Speed () US6 Distance US6 Drie Cycle: Speed and Distance les s time (seconds) Time Time (seconds) Distance (les) Petroleum S Th m A re Heatin V Speed () FTP Drie Cycle: Speed and Distance les s time (seconds) FTP Distance UDDS t Time (seconds) Distance (les)

9 X. Dra Times Tesla Torque and Power Dynamometer Model-S-Video s- 743.html Extract Points from Cures to cs data files Read Dyno Data cs data files T Dyn READPRN ("Tesla P85D DynoTorq Insane.cs" ) T Dyn READPRN ("Tesla P85D DynPowerR.cs" ) P Dyn P Dyn T Dyn 5.96 rows T Dyn P Dyn 5.96 = 7 rows( P Dyn ) = 4 Power Fit Torque to et Shape Cure Guess a b c d 7 e f 3 x Gien Torq T Dyn spd T Dyn Torq a ( spd) c ( spd d) e ( spd f) + x = At Minerr( a, c, d, e, f, x) Torque( s) At ( s) At s At ( ) At s At 3 ( 4) + At 5 Torque = 3.4 Guess a b c d e f 3 x h i Gien P wheel P Dyn spd P Dyn n.. 59 S n.5 Tx Torque S n n n P wheel a ( spd b) 3 c ( spd d) e ( spd f) + x ( spd h) + i = A Minerr( a, b, c, d, e, f, x, h, i) Pwr( s) A s A A s A A s A + A s A + A Pwr( 3) = ft lbf P Dyn_F T Dyn T Dyn P Dyn_F = 77.4 m x in 4 hp Normalize Torque Cure to Max = to extract shape only, TShape Torque Shape: NormT T Shape ( s) Torque( s) NormT T Shape. ( s) T Shape ( s) max( Tx) Torque (ft lbf) Multiple by to plot on same axis with Oriinal Torque Cure T Dyn T Shape. ( S) Dra Times Tesla S P85D Dyno T Dyn, S, PDyn, T Dyn Speed () 4 3 P Dyn At = A =

10 IX. Tire Friction (Composition and Width) Coefficient of Static Friction (µ) is the ratio of Tire Roa d Force to Vehicle Weiht. Values of µ for Conentional Ca On: Asphalt.7, Ca Grass.35. Top Fuel dra cas are ettin a coefficient of friction well oer 4.5. How is this possible? This material came from: See Mathcad/EVs/Tire Friction.doc Rubber enerates friction in three major ways: adhesion, deformation, and wear. Rubber in contact with a smooth surface (lass is often used in testin) enerates friction forces mainly by adhesion. When rubber is in contact with a rouh surface, another mechanism, deformation, comes into play. Moement of a rubber slider on a rouh surface results in the deformation of the rubber by hih points on the surface called irreularities or asperities. A load on the rubber slider causes the asperities to penetrate the rubber and the rubber drapes oer the asperities. The enery needed to moe the asperities in the rubber comes from the differential pressure across the asperities as shown in Fi. 3.4, where a rubber slider moes on an irreular surface at speed V. Tearin and Wear As deformation forces and slidin speeds o up, local stress can exceed the tensile strenth of the rubber, especially at an increase in local stress near the point of a sharp irreularity. Hih local stress can deform the internal structure of the rubber past the point of elastic recoery. When polymer bonds and crosslinks are stressed to failure the material can't recoer completely, and this can cause tearin. Tearin absorbs enery, resultin in additional friction forces in the contact surface. Wear is the ultimate result of tearin. Ftotal = Fadhesie + Fdefformation + Fwear Deformation Friction and Viscoelasticity Rubber is elastic and conforms to surface irreularities. But rubber is also iscoelastic; it doesn't rebound fully after deformation. Hysteresis Hysteresis, or enery loss, in rubber. where there is some slidin between the rubber and an irreular surface. If the rubber recoers slowly from the passin irreularity as in the hih-hysteresis rubber, it can't push on the downstream surfaces of the irreularities as hard as it pushes on the upstream surfaces. This pressure difference between the upstream and downstream faces of the irreularity results in friction forces een when the surfaces are lubricated. Wide Tires: It is true that wides commonly hae better traction. The main reason why this is so does not relate to contact patch, howeer, but to composition. Soft compound tires are required to be wider in order for the side-wa ll to support the weiht of the car. softes hae a larer coefficient of friction, therefore better traction. A narrow, soft tire would not be stron enouh, nor would it last ery lon. Wear in a tire is related to contact patch. Harder compound tires wear much lo ner, and can be narrower. They do, howeer hae a lower coefficient of friction, therefore less traction. Amon tires of the same type and composition, here is no appreciable difference in 'traction' with different widths. Wides, assun all other factors are equal, commonly hae stiffer side-walls and experience less roll. This ies better cornerin performance. Friction is proportional to the normal force of the asphalt actin upon the cas. This force is simply equal to the weiht which is distributed to each tire when the car is on leel round. Force can be stated as Pressure X Area. For a wide tire, the area is lare but the force per unit area is small and ice ersa. The force of friction is therefore the same whether the tire is wide or not. Howeer, asphalt is not a uniform surface. Een with steamrollers to flatten the asphalt, the surface is still somewhat irreular, especially oer the with of a tire. Dra racers can therefore increase the probability or likeliho od o f makin conta ct with the road by usin a wide In addition a secondary benefit is that the wide increased the support base a Friction force is independent of the apparent area o f co ntact. For hard materials, this is nearly correct. The true area of contact aries with the applied load. The apparent area does not. If you can imaine the contact zone from a croscopic iewpoint, only a tiny portion of the apparent area actually touches. This tiny area is the true area of contact. But this applies to hard materials. It does not apply to elastomers, such as rubber. Tire tread rubber compounds ary reatly from one application to another. Race car tire tread compounds can be ery soft, iscoelastic materials, while heay truck tread rubber can be quite hard. In eneral, soft rubber materials hae reater frictio n. With dra racin 'slicks,' the tire tread material literally sticks to the paement--and the rubber is sheared from the tire. Clearly, the reater the apparent contact area, the reater this shear force. Cleanliness is important to ettin the surfaces to 'stick.' This is one reason why dra racers hae a 'burn-out' before each race (another is to raise the tire tread surface temperature). Howeer, th ere is an other reason fo r wide tire tread s on some road an d track racin cars. They need tread olume to proide enouh wear life. Tires wear rapidly under racin conditions. Some lon races wear out seeral sets of tires. There are trade-offs with traction and tread life. That is why heay truck tire tread compounds do not hae as much friction as those used on passener cars. Howeer, truck tire tread compounds proide loner wear life and less heat build-up. Like many thins in this world, tire tread choices inole comproses.