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DINÁMICA DE VEHÍCULOS Ricardo Prado Gamez – Metalsa

ENERO 2016

Módulo 1

Módulo 2

Módulo 3

Módulo 4

Módulo 5

MECANICA DE LLANTAS Introducción Resistencia al Rodamiento Frenado Cornering Modelación Pacejka Entregable 1 MANIOBRABILIDAD EN ESTADO ESTABLE Angulo de Ackerman Sub-Viraje y Sobre Viraje Términos derivativos Velocidad crítica, característica, tangente y máxima Entregable 2 EXAMEN DE MEDIO TÉRMINO MANIOBRABILIDAD EN ESTADO TRANSITORIO Modelo de maniobrabilidad en espacio de estados Efecto de masas e inercias en la maniobrabilidad Entregable 3 ROLL Y TRANSFERENCIA DE CARGA Concepto de transferencia de Carga Ganancias y fuerzas verticales de reacci´no (Jacking Forces) Distribución de rigidez torsional Efecto de barra estabilizadora Entregable 4 COMFORT Umbrales de tolerancia y percepción Frecuencias naturales en manejo primario y secundario Modelo de cuarto de vehículo Modelo de medio vehículo Entregable 5 EXAMEN FINAL

PERIOD 1: Up to the early 1930's) Empirical Obervation about vehicle dynamic behaviour Concerns about wheel shimmy Ride comfort recognized as an important aspect of vehicle performance PERIOD 2: (1930-1952) Simple tyre mechanics/slip angle understood understeer/oversteer defined Steady concering understood Simple two degrees of freedom equations developed Ride experiments begun K2 rig and flat ride ideas proposed Independent front suspension introduced (1936) PERIOD III: (1952-Onwards) Understanding of tyre behaviour developed through rig results and modeling Three degree of freedom equations developed Analysis extended to include stability and directional response properties Ride predictions using random vibraiton theory initiated

Periods I, II ww1

Great Depresion

1905 1914 1918 1920 1929 1930 1931 1932 1933 1934

1939

ww1

1946 1952

1953 1954 1955 1956

Design philosophy focused on Engineering (not passengers) World War 1 (WW1) Rolls Royce started to manufacture cars in USA Rolls Royce concerned in ride Wall Street Crash Maurice Olley (From Rolls Royce) joined GM Cadillac as a chief engineer MO recognized problems on of shimmy, axle tramp, excessive vibration as a result of bad American Roads K2 Rig was built and Flat Ride Concept MO acknowledges importances of tyre and request Goodyear Force/Moment Tyre data GM acknowledges the importance of Independent Front Suspension (IFS) Chrysler showed interest on IFS Goodyear provides available Force/Moment Tyre Data to explain Understeer SAE Paper "Independent Suspension; its whys and wherefores" World War 2 (WW2) Vauxhaull impelmented IFS Paper "Road Manners of the modern car" Maurice Olley/Bob Schilling (Research and Development) First Corvette Chassis/Air Spring "We should do it" 25000 USD First contract with GM Research Lab Division Contract increased up to 125,000 USD Contract increased up to 250,000 USD Contract increased up to 500,000 USD IME Papers

K-Rig Concept

Source “Chassis Design” Principles & Analysis William Milliken

Front Axle Car Problems

“Road Maners from a Moden Carn” Maurice Olley

IME Papers

Maurice Olley

William Milliken

Leonard Segel

MAGIC NUMBERS INDEPENDENT FRONT SUSPENSION More Space for engine K-Rig Improvement Reduction (Elimination) of wheel shimmy Reduction of CG Styling Mass reduction Unsprung mass reduction  Higher frequency spread Lower Roll Center Small tread change  More Roll camber i.e less recovery Roll Center Constant  Excesive Roll (Bottoming)

Small tread change Roll Center Constant

+ Anti roll Bar

Lower Roll Center

Higher spring rates

Deteriorate comfort

Higher front track

+ More Roll camber (less recovery) Excesive Roll (Bottoming)

More oversteer behaviour

“No hay nada mas práctico que una buena teoría” Kurt Lewin

“El aprendizaje mas grande en la historia de la Ingeniería Mecánica es mediante errores, no ha lugar para prima donas” Maurice Olley

Mecánica de Llantas

X

Braking Y

Braking & cornering

ay

Braking

Acceleration

Acceleration & cornering

ax Acceleration

Ricardo Prado

The inertial forces should be in equilibrium of the tyre forces.

Ricardo Prado

The control of the vehicle, is all about understanding the twelve vectors between tyre patches and road.

Handling

Aceleración

Lat

Long Vert

Braking

Cruise

Objective: “To understand the the main principles underlying the handling and comfort of a vehicle.”

• • • • •

Tyres Load Transfer (Roll) Handling (Steady State) Handling (Maneouvers) Confort

Objective: “To understand the forces and moment generation at the tyres during car motion and handling.”

• • • • • • • •

Motivation Materials, Properties and type of Tyres Rolling Resistance Force Generation (Braking/Acceleration) Force Generation (Cornering) Relaxation Length Combined Cases. Modelling (Pacejka)

• Fx longitudinal force. • Fy lateral force. • Fz vertical force. • Mx overturning moment dirección de la rueda

• My rolling resistance moment. • Mz auto aligning moment.

Must support vehicle load Must absorb local surface irregularities

Must provide grip during brake/accel Must guide the vehicle during maneouverings (lateral grip) Must provide free motion on smooth surfaces -Solid in a perfectly straight guide way -Height of an axle must be constant

Must be durable to cyclic loads

Flexibility  Mechanical property restriction Developable Surface  Geometrical restriction

Solid tyre made to High geometrical precision Low modulus Highly elasticity Substantial deformation

Gas inflated tyre High geometrical precision Low modulus Highly elasticity Capable of substantial deformation

•No appreciable change of size upon inflation •Ability to envelop obstacles without sustaining damage •Ability to deform from a surface of double curvature to a plane surface •Enough rigidity to develop substantial forces

ángulo de los cables

Angle and orientation  Tyre deformation Highier  More deformation in lateral direction; i.e. Less Lat. Stiffness and Better comfort.

capas

(a) Bias Ply capas

ángulo de los cables

Small  Less deformation in lateral direction; i.e. more lateral stiffness but the road irregularities are taken by the mid layers. (b) Radial Ply

• Two force generation mechanisms • Adhesion • Hysteresis.

caucho

hysteresis

adhesion

Hysteresis F



Hysteresis

Reference Harty M & Blundel M. “The Multibody Systems Approach to Vehicle Dynamics”

Objective: “To understand the forces and moment generation at the Tyres during car motion and handling.”

• • • • • • • •

Motivation Materials, Propiedades y Types of Tyres Rolling Resistance Force Generation (Braking/Acceleration) Force Generation (Cornering) Relaxation Length Combined Cases. Modelling (Pacejka)

Rolling Resistance Moment (My)

Fr

x 

Fr Fz

Fz x

Reference Harty M & Blundel M. “The Multibody Systems Approach to Vehicle Dynamics”

Superficie

Rolling Resistance Coefficient ( x )

Gravel Highway Villages Rough Sand

0.02 0.008 -- 0.010 0.03 0.05 0.15 -- 0.30

Average values of rolling resistance coefficient for different roads and tyres

pn

pn

s pb

pb pn

coeficiente de resistencia al rodamiento

pn

Fr

Fr

s

s

ruedas bias-ply

Fr = fr Fz

fr =ruedas c0 + c1radial v2 ply

Velocidad del vehículo [km/h]

UP TO NOW…. •The equilibrium of inertial forces are at the tyre patches •Two types of Tyres Bias Ply

Several Cord Layers in angle

Either high lateral stiffness (handling) or soft long. Stiffness (confort)

Higher rolling resistance

Radial

Angled layers + radial layer

Higher flexibility to achieve handling and confort

Lower rolling resistance

•Forces generation mechanism: Adhesion & Hysteresis •Rolling resistance force is as a result of tyre hysteresis properties

Workshop: 1 Estimate the necessary power (hp) to keep in a driver controlled cruise speed of 40 and 100 km/h a 2 ½ ton pick up truck with bias ply and radial tyres. Use data below to estimate the rolling resistance coefficient. Note: Aerodynamic and grade effects ignored 0.022

f r  0.0169  0.19 106 v 2

0.021 0.02 0.019

Axis Title

0.018 0.017

f r  0.0136  0.4 107 v 2

0.016 0.015 0.014 0.013 0.012 0

20

40

60

Bias Ply

80 Km/hr

100

Radial

120

140

160

Workshop: 2 Use the SAE formula below to estimate the variation of the rolling resistance coefficient with respect to different payloads (Fz). 5.5  10 5  90 Fz 1100  0.0388Fz 2  K   5.1  fr   V  1000  p p  K’ takes value between 0.8 (llantas radiales) y 1.0 (ply) Fz (Newtons) P (Pa)  Newtons/m2 V (m/s).

Standing Waves (High Speeds)

Why is this important…?

Ref. SAE paper 2010-01-0763

Why is this important…?

Goodyear’s own test results show that its new EfficientGrip summer tire provides 15% lower rolling resistance. Tire manufacturer Goodyear introduced a new high-performance summer tire at the Geneva Motor Show. The tire is said to offer lower fuel consumption, long service life, and good wet braking performance. Goodyear’s own test results show that the new tire provides 15% lower rolling resistance, braking distances reduced by 3% in the wet, and 3% better wet handling characteristics. The company claims that the tire will also offer up to 25% greater mileage. The EfficientGrip tire incorporates Goodyear FuelSaving Technology, which consists of a number of developments that reduce the tire’s rolling resistance. These include a lightweight structure, reduced heat generation, advanced compound materials, and new manufacturing techniques. Goodyear reports weight reductions of around 10% for the tire compared with its predecessor. EfficientGrip features a lower polyester ply end and less material in the sidewall. Lower heat regeneration results from Goodyear’s patented CoolCushion Layer. This involves the use of a new thermoplastic ingredient used as a reinforcing agent, which is a partial substitute for carbon black. EfficientGrip compounds are produced using the latest generation of polymers. These feature improved energy dissipation characteristics, helping to reduce rolling resistance. The tread pattern includes a set of four wide grooves around the circumference. The shape and position of these are said to deliver better water dispersal characteristics and reduce the possibility of aquaplaning. To date, Renault has specified the new tire as original equipment for the Megane 3, and Mercedes-Benz has specified it for the E-Class. EfficientGrip is available in a range of sizes from 14- to 18-in diameter.

A new Bridgestone tire uses proprietary technologies to lessen the friction and heat buildup that can contribute to increased levels of rolling resistance, an enemy of fuel efficiency. Using Bridgestone's NanoPro-Tech, the Ecopia EP100—the first aftermarket product in North America's Ecopia tire line—controls the interaction between polymers, filler materials, and rubber chemicals used to manufacture the tire. "A tire that builds up a lot of heat increases rolling resistance, thereby increasing the amount of fuel that the vehicle consumes. When carbon particles rub against each other, that creates friction and heat, but NanoProTech keeps the carbon particles consistently spaced apart so there is not as much heat and friction generation," Kurt Berger, Manager of Consumer Products Sales Engineering for Bridgestone Americas Tire Operations, said at Ecopia EP100's unveiling at the 2009 Chicago Auto Show. The Ecopia EP100 is the first tire produced using NanoPro-Tech initiatives, but the new tire showcases other low rolling resistance attributes. "There are also interconnected tread blocks that prevent movement of the tread elements. The tread is essentially a continuous string of elements that are locked together, thereby minimizing movement, which contributes to rolling resistance. In addition, the interconnected tread blocks provide for enhanced wet performance," said Berger. Design elements of the Ecopia EP100 serve a role in helping to elicit better tire performance. For instance, high angle lateral grooves help prevent hydroplaning. Consistent surface contact via a special tread block design helps improve wet and dry handling and reduces irregular wear. And, 3-D cut circumferential ribs help reduce irregular wear as well as lessen road noise. The Ecopia EP100 is a summer replacement tire fitment available in H- and V-speed ratings and six different sizes, ranging from 14- to 16-in. "As Bridgestone's lowest rolling resistance tire to date, we expect the Ecopia EP100 to be a popular aftermarket choice for hybrid-electric and other fuelefficient vehicles," said Berger.

Yokohama debuted its new ADVAN ENV-R1 orange oil-infused racing tire at the Porsche GT3 Challenge at Sebring a few months back. At the time, the company promised to have new tires using the eco-friendly technology on the market for consumer use in short order. Apparently, that time is now. According to Dan King, Yokohama vice president of sales: The eco-focused dB Super E-spec mixes sustainable orange oil and natural rubber to drastically cut the use of petroleum, without compromising performance. It also helps consumers save money at the gas pump by improving fuel efficiency via a 20-percent reduction in rolling resistance. With these innovations, the dB Super E-spec could very well be the most technologically-advanced tire ever produced. At launch time, the new green orange tires will be available in four sizes. Not coincidentally, those sizes will fit popular hybrids like the Toyota Prius, Honda Civic Hybrid/Civic GX NGV, Toyota Camry Hybrid and Honda Accord Hybrid. Click past the break for the official press release.

Objective: “To understand the forces and moment generation at the Tyres during car motion and handling.”

• • • • • • • •

Motivation Materials, Propiedades y Types of Tyres Rolling Resistance Force Generation (Braking/Acceleration) Force Generation (Cornering) Relaxation Length Combined Cases. Modelling (Pacejka)

Braking

ixs: Slide during braking

vr: Rolling speed wb: Braking angular speed re: Effective radius

[0, +1]

SAE

[0, -1]

Acceleration

id: Acceleration slide

vr: Rolling speed wd: Tractive angular speed re: Effective radius

SAE

[0, >1] If wdre = 2vr  id = +1

However id can be up to > +1

Braking Curve

AB: 10-15% of slide (ixs). BC: Unstable region.

as ddx a

rodamiento libre

rueda bloqueada

OA: Linear region

Longitudinal Stiffness

Fx Cs  ixs

i xs  0

 1 .9

Dry tarmac

Wet tarmac .1.2 Snow

Slide .1.15

1

Hand Calculation Excersise: 1 Assume that a car is travelling at 25 mph with a tyre of 310 mm effective radius. After applying brake pedal, the wheel speed sensor indicates an actual tyre speed of 34 rad/sec (0.593 rpm). •Find the angular velocity of the tyre as a result of the travelling speed (wr) •Find the slip (ixs) •Find the braking force (use the grpah next slide)

Hand Calculation Excersise: 1 5000 4500 4000 3500

N

3000 2500 2000 1500 1000

500 0 0

20

40

60 ix (%)

80

100

120

Hand Calculation Excersise: 1 (cont) Assume that the wheel decelerates at a rate of 0.3g, obtain the force after 2 seconds after the application of brake pedal. Assume that the tyre contact point velocity is still the same (11.1736 m/s).

Repeat the same calculation but now assuming a higher deceleration of twice the above (0.6g)

Fuerzas Longitudinales en Llantas 16000

14000

12000

10000

N

Fz=2105N Fz=3995N

8000

Fz=6120N Fz=7950N

6000

Fz=10000N 4000

2000

0 0

0.2

0.4

0.6

%

0.8

1

Objective: “To understand the forces and moment generation at the Tyres during car motion and handling.”

• • • • • • • •

Motivation Materials, Propiedades y Types of Tyres Rolling Resistance Force Generation (Braking/Acceleration) Force Generation (Cornering) Relaxation Length Combined Cases. Modelling (Pacejka)

The cornering Forces depends mainly on the following parameters •Slipe angle (a) •Vertical Payload (Fz) •Road Conditions () •Longitudinal Loads (Fx) •Inflation Pressure •Temperature

Fy( a ,  , Fz , Fx , p ,T )

Fy( a ,  , Fz , Fx , p ,T )

Auto Aligning Moment Lateral force

Deformation shape Tyre deformation

sliding

Slip angle a Wheel Plane Forward Velocity V

Lateral (slip) = tan (a)

2000

Limit of adhesion

1800

Fy

1600

Adhesion Coefficient

1400

The curve depends on tyre adhesion properties and the road in the lateral direction

1200 1000 800

Fy( a ,  , Fz , Fx , p ,T )

600 400

200

a(deg)

0 0

1

2

3

4

5

6

Linear

7

8

K y

Fy Cornering Stiffness

Cornering Coefficient

Lateral Stiffness

Fy a

a 0

Fy( a ,  , Fz , Fx , p ,T )

Fy( a ,  , Fz , Fx , p ,T )

Fz

Fy( a ,  , Fz , Fx , p ,T )

Fy( a ,  , Fz , Fx , p ,T ) 800/820-15 Tyre (Fy vs Fz) 3000

2500

2000 Lbs

1 deg 2 deg

1500

3 deg 4 deg

1000

5 deg 6 deg

500

0

0

5

10 Lbs

15

20

*Obtained from Pacejka parameters

Fy( a ,  , Fz , Fx , p ,T ) P275/40 ZR17 Eagle ZR (Street Corvette)*

3000

Fy 2500

2000

Lbs

1 deg 2 deg

1500

3 deg 4 deg 5 deg

1000

6 deg

500

0 0

500

1000

1500

2000 2500 Lbs

3000

3500

4000

4500

Fz

*Obtained from Pacejka parameters

Auto Aligning Moment MZ = FYdp

T. Gillespie, Fundamentals of Vehicle Dynamics SAE Press, 1992, p 348.

The line of action of Fy lies behing the contact point, causing a Moment Mz which tends to provide stability towards the stable trim (Direction of travel). The relative distance between the line of action of Fy with respect to the contact point is called “Pneumatic Trail”

•

The first tyre-contact to surface is initially undeformed

•

The tyre roads but the point remains in contact with respect to the floor.

•

The point is deformed along the road with the tyre keeps rolling.

The force distribution is integrated along the length of the footprint to obtain a resultant force Fy

Fyx = Fy*Sin a Fy

Fyx Curve resistant (Drag Component) Reference Harty M & Blundel M. “The Multibody Systems Approach to Vehicle Dynamics”

Hand Calculation Excersise: 2 (Part 1) Characterize qualitatively the differences between GoodYear Tyre P275/40 ZR17 and Eagle and Goodyear Indy 27.0x14.5-15 Champ; considering: •Which one has higher adherence? •Which one has higher cornering stiffness? •Which one has higher auto aligning torque? Hand Calculation Excersise: 2 (Part 2) •Obtain the friction coefficient

•Obtain the cornering coefficient •Obtain the auto aligning torque coefficient

•Observe the trends and make your observations

9000

9000

8000

8000 7000

6000

6000

5000

5000 N

N

7000

4000

4000

3000

3000

2000

2000

1000

1000

0

0 0

2

1804 N

4 deg 4097 N

6258 N

6

8

8712 N

0

2

Fz: 4008 N

4 deg Fz: 6013 N

6

Fz: 8017 N

8

180

300

160

250 140 120

200

Nm

Nm

100 80

150

60

100 40 20

50

0 0

2

4 deg

Fz: 1804 N

Fz: 4097 N

Fz: 6258 N

Fz: 8712 N

6

8

0

0

1

2

3

4

5

deg Fz: 4008 N

Fz: 6013 N

Fz: 8017 N

6

7

Hand Calculation Excersise: 3 The Eagle ZR tire shown in the figure, Is used on a Corvette with a test weight of 3500 lbs, and having a 52/48 weight distribution. Use the cornering stiffnesses (i.e. a linearized tyre) and the actual curves themselves (a nonlinear tire) to calculate the cornering force at a slip angle of 1.5 deg on the front and rear tires. What is the percent of error between linearized and nonlinear models? Repeat the calculation at 4.0 deg. Interpolation between load curves will be necessary. For simplicity, ignore the weight transfer.

Hand Calculation Excersise: 3 9000

8000

7000

6000

5000 N

1804 N 4097 N

4000

6258 N 8712 N

3000

2000

1000

0 0

1

2

3

4 deg

5

6

7

Hand Calculation Excersise: 4 Consider a Corvette car with tyres P275/40ZR17 (see fig, next slilde). Obtain the total resistance force of the car if the kerb weight is 3000 lbs and all the tyres are operating with an slipe angle a = 3o and with a rolling resistance of fr = 0.022. Obtain the required power (hp) to drive this vehicle at 30 mph. Assume for simplicity that the weight is evenly distributed along all the roads and ignore any aerodynamic effect.

Hand Calculation Excersise 4 (cont) 9000

8000

7000

6000

5000 N

1804 N 4097 N 4000

6258 N 8712 N

3000

2000

1000

0 0

1

2

3

4 deg

5

6

7

Pneumatic and Mechanical Trail

Pneumatic and Mechanical Trail

Hand Calculation Excercise: 5 Consider a car with tyres P215/60-R15 Goodyear Eagle GT-S (see fig.), which in make a turns. Each front tyre is operating with a slip angle of a=3 deg. Due to the load transfer, the inside tyre has a normal force Fz of 900 lbs and the front outside of 1350 lbs. The mechanical trail is of 1.125 inches as a result of the caster angle inclination of the kingping axis. Obtain the total torque around the kingping axis considering also the effect of the pneumatic trail.

m = 1.125”

Hand Calculation Excersise: 5 (cont) 9000 8000

7000 6000

N

5000 4000 3000 2000 1000 0 0

1

2

3

4

5

deg Fz: 4008 N

Fz: 6013 N

Fz: 8017 N

6

7

Hand Calculation Excersise: 5 (cont) 9000 8000

7000 6000

N

5000 4000 3000 2000 1000 0 0

1

2

3

4

5

deg Fz: 4008 N

Fz: 6013 N

Fz: 8017 N

6

7

Objective: “To understand the forces and moment generation at the Tyres during car motion and handling.”

• • • • • • • •

Motivation Materials, Propiedades y Types of Tyres Rolling Resistance Force Generation (Braking/Acceleration) Force Generation (Cornering) Relaxation Length Combined Cases. Modelling (Pacejka)

Look up table Tyre testers

Fourier Series -Polynomial (high degrees) -Curve fitting (difficult) -Physical intuition (difficult) Special functions -Pacejka -Curve fitting (difficult) -Physical intuition (possible) -There are more than Pacejka Brake/Accel: (Fx vs i)

Cornering: (Fy vs a)

Auto aligning Moment: (Mz vs a)

Mathematical Model Adams Matlab Excell

Tyre Testers

Y  D * Sin( Bx )

Pacejka Model & Parameters

Y  D * sin( Arctg ( Bx )) Y  D * sin( C * Arctg ( Bx )) Y  D * sin( C * Arctg ( B ))  S v where :

Y: Fy, Fx, Mz x: Argument (a, i) C: Shape factor D: Peak Value BCD: Slope @ a=0 or i=0 E: Shape factor B: BCD/CD (Stiffness factor) Sv: Vertical offset Sh: Horizontal offset

Typical Starting Parameters

E * Arctg ( Bx ) C= 1.3  Cornering B E     ( 1  E )* x  S h  * Arctg B x  S h  C= 2.4  Auto Aligning Moment B

  ( 1  E )* x 

C = 1.65 Brake/Acceleration

Workshop 3 (excell) Find the Pacejka parameters of the P275/40 ZR17 Eagle tyre shown in the figure 9000

8000 7000 6000

N

5000 4000 3000 2000 1000 0 0

1

2

3

4

5

deg 1804 N

4097 N

6258 N

8712 N

6

7

Fy( a ,  , Fz , Fx , p ,T )

Fz

Cornering Forces, Auto Aligning Moment and Longitudinal Force are also dependent on vertical load (Fz) D  a1FZ 2  a2 FZ BCD  a3 sin( a4  tg 1( a5 FZ ))

BCD 

a3 sin( a4  tg 1( a5 FZ )) e a5 FZ

E  a6 FZ 2  a7 FZ  a8

Fz (kN) Cornering (Fy) Aligning Torque (MZ) Longitudinal Force (Fx)

Reference values from SAE Paper 870421

Fz (kN) 4 8

B

C

D

E

Sh

Sv

BCD

0.249 0.122

1.29 1.46

3850 6877

-0.678 -2.16

-0.049 0.125

-156 -240

1038 1017

Mz

4 8

0.244 0.137

2.78 2.51

-50.56 -193.3

-0.46 -3.21

-0.082 0.009

-11.7 -4.22

-30.45 -58.55

Fx

4 8

0.181 0.224

1.79 1.88

4436 7911

0.619 0.783

0.000 0.000

70.6 104

1224 2937

Fy

a1

a2

a3

a4

a5

a6

a7

a8

Fy

-24.1

1211

1178

1.82

0.208

0.000

-0.354

0.707

Mz

-4.72

-3.28

-1.96

-2.73

0.110

-0.070

0.643

-4.04

Fx

-23.3

1344

51.6

226

0.069

-0.006

0.056

0.486

"Tyre Modelling for Use in Vehicle Dynamics Studies," Bakker E, Nyborg L, Pacejka HB, SAE Paper No. 870421

Reference values from Mechanics of Pneumatic Tires; Wong

Fy

Mz

Fx

Fz (kN) 2 4 6 8 2 4 6 8 2 4 6 8

B

C

D

E

Sh

Sv

BCD

0.24 0.239 0.164 0.112 0.247 0.234 0.164 0.127 0.178 0.171 0.21 0.214

1.5 1.29 1.27 1.36 2.56 2.68 2.46 2.41 1.55 1.69 1.67 1.78

1936 3650 5237 6677 -15.53 -48.56 -112.5 -191.3 2193 4236 6090 7711

-0.132 -0.678 -1.61 -2.16 -3.92 -0.46 -2.04 -3.21 0.432 0.619 0.686 0.783

-0.28 -0.049 -0.126 0.125 -0.464 -0.082 -0.125 0.009 0.0 0.000 0.000 0.000

-118 -156 -181 -240 -12.5 -11.7 -6.0 -4.22 25.0 70.6 80.1 104

780.6 1038 1091 1017 -9.82 -30.45 -45.39 -58.55 605 1224 2136 2937

a1

a2

a3

a4

a5

a6

a7

a8

Fy

-22.1

1011

1078

1.82

0.208

0.000

-0.354

0.707

Mz

-2.72

-2.28

-1.86

-2.73

0.110

-0.070

0.643

-4.04

Fx

-21.3

1144

49.6

226

0.069

-0.006

0.056

0.486

“Mechanics of Pneumatic tires," Wong

Workshop 4 (excell) Use the excel file provided by the teacher to find the Pacejka parameters a1, a2, a3, a4 and a5 in order to adjust the four curves of Fy and Mz using the table below Fy

Mz

Fz (kN) 2 4 6 8 2 4 6 8

B

C

D

E

Sh

Sv

BCD

0.24 0.239 0.164 0.112 0.247 0.234 0.164 0.127

1.5 1.29 1.27 1.36 2.56 2.68 2.46 2.41

1936 3650 5237 6677 -15.53 -48.56 -112.5 -191.3

-0.132 -0.678 -1.61 -2.16 -3.92 -0.46 -2.04 -3.21

-0.28 -0.049 -0.126 0.125 -0.464 -0.082 -0.125 0.009

-118 -156 -181 -240 -12.5 -11.7 -6.0 -4.22

780.6 1038 1091 1017 -9.82 -30.45 -45.39 -58.55

Use the following parameters as starting point a1

a2

a3

a4

a5

Fy

-22.1

1011

1078

1.82

0.208

Mz

-2.72

-2.28

-1.86

-2.73

0.110