AV_VEHDYN_2012 Lecture Slides and Calcs

Introduction to Vehicle Dynamics

Pierre DuysinxLTAS Automotive Engineering

University of LiegeAcademic Year 2011-2012

Bibliography

T. Gillespie. « Fundamentals of vehicle Dynamics », 1992, Society of Automotive Engineers (SAE)W. Milliken & D. Milliken. « Race Car Vehicle Dynamics », 1995, Society of Automotive Engineers (SAE)R. Bosch. « Automotive Handbook ». 5th edition. 2002. Society of Automotive Engineers (SAE)J.Y. Wong. « Theory of Ground Vehicles ». John Wiley & sons. 1993 (2nd edition) 2001 (3rd edition).M. Blundel & D. Harty. « The multibody Systems Approach to Vehicle Dynamics » 2004. Society of Automotive Engineers (SAE)G. Genta. «Motor vehicle dynamics: Modelling and Simulation ». Series on Advances in Mathematics for Applied Sciences - Vol. 43. World Scientific. 1997.

INTRODUCTION TO HANDLING

Introduction to vehicle dynamics

Introduction to vehicle handlingVehicle axes system

Tire mechanics & cornering properties of tiresTerminology and axis systemLateral forces and sideslip anglesAligning moment

Bicycle modelLow speed cornering

Ackerman theoryAckerman-Jeantaud theory

Introduction to vehicle dynamics

High speed steady state cornering Equilibrium equations of the vehicle

Gratzmüller equalityCompatibility equationsSteering angle as a function of the speedNeural, understeer and oversteer behaviourCritical and characteristic speedsLateral acceleration gain and yaw speed gainDrift angle of the vehicleStatic margin

Exercise

Introduction

In the past, but still nowadays, the understeer and oversteercharacter dominated the stability and controllability considerationsThis is an important factor, but it is not the single one…

In practice, one has to consider the whole system vehicle + driver

Driver = intelligenceVehicle = plant system able to create the manoeuvre forces

The behaviour of the closed-loop system is referred as the « handling », which can be roughly understood as the road holding

Introduction

Conducteur

Vision

Système de directionTransmissionSystème freinage

SuspensionLiaisonPneus

AutomobileCaisse

Pédaleaccél.

Anglevolant.

Pédalefrein Couple

auxroues

Anglesyst.direct.

Forces etmoments

Environnement

Bruits, vibrationsBruit, vibrations

Forces d’inertie

Model of the system vehicle + driver

Introduction

However because of the difficulty to characterize the driver, it is usual to characterize the vehicle alone as an open loop system.

Open loop refers to the vehicle responses to specific steering inputs. It is more precisely defined as ‘directional responses’behaviour.

The most commonly used measure of open-loop response is the understeer gradient

The understeer gradient is a performance measure under steady-state conditions although it is also used to infer performance properties under non steady state conditions

AXES SYSTEM

Reference frames

Local reference frame oxyzattached to the vehicle body -SAE (Gillespie, fig. 1.4)

XY Z

O

Inertial coordinate system OXYZ

Reference frames

Inertial reference frameX direction of initial displacement or reference directionY right side travelZ towards downward vertical direction

Vehicle reference frame (SAE):x along motion direction and vehicle symmetry planey in the lateral direction on the right hand side of the driver towards the downward vertical directiono, origin at the center of mass

Reference frames

z

x

y x

Système SAE

Système ISO

x z

yComparison of conventions of SAE and ISO/DIN reference frames

Local velocity vectors

Vehicle motion is often studied in car-body local systems

u forward speed (+ if in front)v side speed (+ to the right)w vertical speed (+ downward)p rotation speed about a axis (roll speed)q rotation speed about y (pitch)r rotation speed about z (yaw)

Forces

Forces and moments are accounted positively when acting ontothe vehicle and the positive direction with respect to the considered frame

Corollary A positive Fx force is propelling the vehicle forwardThe reaction force of the ground onto the wheels is accounted negatively.

Because of the inconveniency of this definition, the SAEJ670e « Vehicle Dynamics Terminology » is naming as normal force a force acting downward while vertical forces are referring to upward forces

TIRE MECHANICS: CORNERING PROPERTIES

Terminology and axis system (SAE)

Wheel plane : central plane of the tire normal to the axis of rotationWheel centre: intersection of the spin axis with the wheel planeCentre of Tire Contact: intersection of the wheel plane and the projection of the spin axis onto the road planeLoaded radius: distance from the centre of tire contact to the wheel centre in the wheel plane


Longitudinal force Fx :component of the force acting on the tire by the road in the plane of the road and parallel to the intersection of the wheel plane with the roadLateral force Fy : component of the force acting on the tire by the road in the plane of the road and normal to the intersection of the wheel plane with the road planeNormal force Fz : component of the force acting on the tire by the road which is normal to the plane of the road


Overturning moment Mx : moment acting on the tire by the road in the plane of the road and parallel to the intersection of the wheel plane with the road planeRolling Resistance Moment My :moment acting on the tire by the road in the plane of the road and normal to the intersection of the wheel plane with the roadAligning Moment Mz : moment acting on the tire by the road which is normal to the road plane


Slip Angle (α): angle between the direction of the wheel heading and the direction of travel. A positive sideslip corresponds to a tire moving to the right while rolling in the forward directionCamber Angle (γ): angle between the wheel plane and the vertical. A positive camber corresponds to the top of the tire leaned outward from the vehicle

LATERAL FORCES

When a lateral force is applied to the tire, one observes that the contact patch is deformed and the tire develops a lateral force that is opposed to the applied force.When rolling, applying a lateral force to the tire, the tire develops a lateral force opposed to the applied force and the tire moves forward with an angle α with respect to the heading direction, called slip angle.The relation between the lateral force that is developed and thesideslip angle is a fundamental matter for the study the vehicledynamics and its stability

ORIGIN OF LATERAL FORCES

The origin of the tire deformation is the applied lateral force

Gillespie, Fig 10.10Wong Fig 1.22

LATERAL FORCES

Milliken, Fig. 2.1

Milliken, Fig. 2.2

Chevrolet R&D experiment

LATERAL FORCES

The name of sideslip is misleading since there is no global slip of the tire with respect to the ground (except in a very limited part at the back of the contact patch).The sideslip of the tire is due to the flexible character of the rubber tire that allows keeping a heading while having a lateral motionThe lateral force may be a cause or a consequence of the sideslip.

Lateral forces (gust) sideslip reaction forces under the tireSteering the wheel sideslip lateral forces to turn

Analogy with walking a snow slope

LATERAL FORCES

Gough machine (1950)(Dunlop Research Center)

Milliken, Fig. 2.4

Records:• lateral displacement (a,b)• the force of a slice (c)• the reaction force on the tire groove

LATERAL FORCES

Side displacements

Lateral speedA>B : V sinα

B>C: local slip

Genta Fig 2.24

Distribution of contact pressure σz: centre of

pressure is in the front part of the contact patch rolling

resistance

Distribution of τzy = lateral shear of the tire: Triangular distribution.Its resultant force is

located in the back of the contact patch

aligning moment

LATERAL FORCES

From experimental observations, one notices that:The resultant of the lateral force is located in the rear part of the pneumatic contact patch. The distance between the resultant position and the centre of the contact pact is the pneumatic trail tThe lateral forces produces also an aligning moment

The local slippage of tire on the road is limited to a small zone at the back of the contact patch. Its span depends on the sideslip angle.The characteristics of the lateral forces are related, in the small sideslip angles, to the lateral displacements due to the rollingprocess but they are mostly independent of the speed.

z yM F t=

LATERAL FORCES

Genta Fig 2.23 : Contact zone in presence of sideslipa/ Contact zone and trajectory of a point in the treadb/ Contact zones et slippage zone for different sideslip angles

LATERAL FORCES

3 parts in the curves of Fy as a function of the sideslip angleLinear part for small sideslip angles (< 3°)In the frictional part (>7°) most of the contact patch experiences dry friction. Over the peak of the curves, the lateral force can fall rapidly or remain smoothTransition part: end of linear regime (~3°) to the peak value of the curve (~7°)On wet roads, the peak value is reduced and drops rapidly

P215/60 R15 GoodYear Eagle GT-S(shaved for racing) 31 psiVertical load 1800 lb

Milliken. Fig. 2.7

LATERAL FORCESP215/60 R15 Goodyear Eagle GT-S

(saved for racing) 31 psi

Milliken. Fig. 2.8 et Fig. 2.9

µ=Fy/Fz

LATERAL FORCES

One introduces the friction coefficient as the ratio of the lateral force and of the applied vertical load

µ = Fy/ Fz

The peak of the lateral force is reduced with the vertical load: this is the load sensitivity phenomenon. The modification of the lateral force with the vertical load can be important with old bias tires. This affects the total lateral force capability of the wheel sets when experiencing lateral load transfer.

Milliken. Fig. 2.10

Load sensitivity

Heisler: Fig 8.42 et 8.43 : Sensitivity of the lateral force with respect to the load transfer

Cornering stiffness

In its linear part (small sideslip angles) the lateral force curve can be approximated by its Taylor first order expansion :

Fy = -Cα αCα is called as the cornering stiffness.The cornering stiffness is negative

Gillespie Fig. 6.2

Cornering stiffness

The cornering stiffness depends on several parameters:The type of the tire, its sizes and width, The inflating pressureThe vertical load

The order of magnitude of the cornering stiffness is about 50000 N/radAs the lateral force is quite sensitive to the vertical load, one generally prefers to use the cornering coefficient that is defined as the ratio of the cornering stiffness by the vertical load:

CCα = Cα / Fz

The order of magnitude of CCα is in the range 0.1 to 0.2 N/N degre-1

Cornering stiffness

Gillespie: Fig 10.14 : cornering coefficient for different populations of tires

Cornering stiffness

Gillespie: Fig 10.15 : Load sensitivity of the cornering stiffness and cornering coefficient

ALIGNING MOMENT AND PNEUMATIC TRAIL

The aligning moment reflects the tendency of the tire to rotate about its vertical axis to self align with the motion direction

For small and medium sideslip angles, the tire tends to align itself with the velocity vector direction

Origin of the aligning moment: Triangular distribution of the local shear forces in the contact patch with a resultant located behind the contact centre

The pneumatic trail is the distance between the lateral force resultant and the contact centre

Trail = Aligning moment / Lateral force

Aligning moment and pneumatic trail

Linear part - small sideslip angles (<3°):

The largest shear stresses work to reduce the sideslip

Non-linear part – medium and large sideslip angles

Maximum of the curves about 3 to 4°When the rear of the contact patch is invaded with local friction, the aligning moment is reducedAt the maximum of the lateral force, the aligning moment goes to zero and may even become negative for larger sideslip angles α > 7° à 10°

Milliken. Fig. 2.11


The aligning moment can also be reinforced with a mechanical trail coming from the wheel and the steering geometryOptimum combination of both trails :

Too small mechanical trail, the vehicle can be inclined to reduce its turn radius with the saturation of lateral forceToo large mechanical trail, no feeling of the maximum of the lateral force curve

Milliken. Fig. 2.12


Lateral force and aligning moment for a tire 175/70 R 13 82SFrom Reimpell et al.


Trail = Aligning Moment / Lateral Force

Important lateral forces result in low aligning moment and a small pneumatic trail.

For small sideslip angles, only the profile is deformed, which gives rise to a resultant far to the back.For important side slip angles, the sidewall works much more and the resultant gets closer to the centre of the contact patch

Pneumatic trail for a tire 175/70 R 13 82S from Reimpell et al.

VEHICLE MODELING

The bicycle model

When the behaviours of the left and right hand wheels are not too different, one can model the vehicle as a single track vehicle known as the bicycle model.

The bicycle model proved to be able to account for numerous properties of the dynamic and stability behaviour of vehicle under various conditions.

x,u,p

y,v,qz,w,r

δ αf

Velocity

αr

Fyf

Tr

Tf

Fyr

Fxr

Fxf

L

b

c

t

The bicycle model

Assumptions of the bicycle modelNegligible lateral load transferNegligible longitudinal load transferNegligible significant roll and pitch motionThe tires remain in linear regimeConstant forward velocity VAerodynamics effects are negligible Control in position (no matter about the control forces that are required)No compliance effect of the suspensions and of the body

x,u,p

y,v,qz,w,r

δ αf

Velocity

αr

Fyf

Tr

Tf

Fyr

Fxr

Fxf

L

b

c

t

The bicycle model

Remarks on the meaning of the assumptionsLinear regime is valid if lateral acceleration<0.4 g

Linear behaviour of the tireRoll behaviour is negligibleLateral load transfer is negligible

Small steering and slip angles, etc. Smooth floor to neglect the suspension motionPosition control of the command : one can exert a given value ofthe input variable (e.g. steering system) independently of the control forcesThe sole input considered here is the steering, but one could also add the braking and the acceleration pedal.

The bicycle model

Assumptions :Fixed: u = V = constantNo vertical motion: w=0No roll p=0No pitch q = 0

Bicycle model = 2 dof model :r, yaw speedv, lateral velocity or β, side slip of the vehicle

Vehicle parameters:m, mass, Jzz inertia bout z axisL, b, c wheel base and position of the CG

The bicycle model

x,u,p

y,v,qz,w,r

δ αf

Velocity

δ αf

Velocity

αrαr

Fyf

Fyr

Fyf

Tr

Tf

rv

u Vβ

Fyr

Fxr Fxr

FxfFxf

L

b

c

h

m, J

LOW SPEED TURNING

Low speed turning

At low speed (parking manoeuvre for instance), the centrifugal accelerations are negligible and the tire have not to develop lateral forcesThe turning is ruled by the rolling without friction and withoutslip conditionsIf the wheel have no slippage, the instantaneous centres of rotation of the four wheels are coincident.The CIR is located on the perpendicular lines to the tire plan from the contact point In order that no tire experiences some scrub, the four perpendicular lines pass through the same point, that is the centre of the turn.

Ackerman-Jeantaud theory

Ackerman-Jeantaud condition

One can see that

This gives the Ackerman Jeantaud condition

Corollary δe · δi

t a n ± i = L = ( R ¡ t = 2 )

t a n ± e = L = ( R + t = 2 )

c o t ± e ¡ c o t ± i =

t

L

Ackerman-Jeantaud condition

The Jeantaud condition is not always verified by the steering mechanisms in practice, as the four bar linkage mechanism

s i n ( ° ¡ ± 2) + s i n ( ° + ± 1) = s µ l 1

l 2¡ 2 s i n °

¶ 2

¡ ( c o s ( ° ¡ ± 2 ) ¡ c o s ( ° + ± 1) ) 2

Jeantaud condition

The Jeantaud condition can be determined graphically, but the former drawing is very badly conditioned for a good precisionIn practice one resorts to an alternative approach based on the following property Point Q belongs to the line MF when the Jeantaud condition is fulfilledThe distance from Q to the line MF is a measure of the error from Jeanteaud condition

Ackerman theory

r v

u VβL

R

δ

δcentre du virage∆

vf

vrRCG

Ackerman theory

The steering angle of the front wheels

The relation between the Ackerman steering angle δ and the Jeantaud steering angles δ1 and δ2

R=10 m, L= 2500 mm, t=1300 mmδ1 = 15.090° δ2= 13.305°

δ = 14.142°(δ1+δ2)/2=14.197°

t a n ± =

L

R

c o t ( ± ) =

R

L

=

c o t ± e + c o t ± i

2

r v

u VβL

R

δ


vf

vrRCG

Ackerman theory

Curvature radius at the centre of mass

Relation between the curvature and the steering angle

Side slip β at the centre of mass

1 = R

±

=

1

L

¯

±

=

c

L

Ackerman theory

The off-tracking of the rear wheel set

r v

u VβL

R

δ


vf

vrRCG

HIGH SPEED STEADY STATE CORNERING

High speed steady state cornering

At high speed, its required that tires develop lateral forces to sustain the lateral accelerations.The tire can develop forces if and only if they are subject to a side slip angle.Because of the motion kinematics, the CIR is located at the intersection to the normal to the local velocity vectors under the tires.The CIR, which was located at the rear axel for low speed turn, is now moving to a point in front.

High speed steady state cornering

δ−αf

αfαrδ

vf

vr

Dynamics equations of the vehicle motionNewton-Euler equilibrium equation in the non inertial reference frame of the vehicle body

Model with 2 dof β & r

Equilibrium equations in Fy and Mz :

Operating forcesTyre forcesAerodynamic forces (can be neglected here)

δ αf

Velocity

αr

Fyf

Fyr

rv

u Vβ

Fxr

Fxf

L

a

b M, J

J x y = 0

J y z = 0

X¡ ! F =

d

d t ( M ¡ ! V ) = M

_ ¡ ! V + M

¡ ! ! £ ¡ ! V X ¡ !

M = d

d t ( J ¡ ! ! ) =

d ( J ¡ ! ! )

d t + ¡ ! ! £ ( J

¡ ! ! )

¡ !

V = [ u v 0 ] T

¡ ! ! = [ 0 0 r ] T

Fy = M( _v+ ru)

N = Jzz _r

Equilibrium equations of the vehicle

Equilibrium equations in lateral direction and rotation about yaw axis

Solutions

The lateral forces are in the same ratio as the vertical forces under the wheel sets

F y f + F y r = m

V 2

R

F y f b ¡ F y r c = 0

F y f =

c

L

m

V 2

R

F y r =

b

L

m

V 2

R

Behaviour equations of the tires

Cornering force for small slip angles

F y = C ® ®

C ® =

@ F y

@ ®

¯ ¯ ¯ ¯

® = 0

< 0

Gillespie, Fig. 6.2

Gratzmüller equality

Using the equilibrium and the behaviour condition, one gets

One yields the Gratzmüller equality

F y f = C ® f ® f =

c

L

m

V 2

R

F y r = C ® r ® r =

b

L

m

V 2

R

® f

® r

=

c C ® r

b C ® f

Compatibility equations

The velocity under the rear wheels are given by

The compatibility of the velocities yields the slip angle under the rear wheels

t a n ® r =

¡ v r

u r

=

¡ v + c r

V

V = r R

® r = ¡ ̄ +

c

R

u r = u ' V

v r = v ¡ c r

δ−αf

αfαrδ

vf

vr

Compatibility equations

The velocity under the front wheels are given by

The compatibility of the velocities yields the slip angle under the front wheels

u f = u ' V

v f = v + b r

V = r R

t a n ( ± ¡ ® f ) =

v f

u f

=

v + b r

V

± ¡ ® f = ¯ +

b

R

δ−αf

αfαrδ

vf

vr

Steering angle

Steering angle as a function of the slip angles under front and rear wheels

The steering angle can also be expressed as a function of the velocity and the cornering stiffness of the wheels sets

± =

L

R

+ ® f ¡ ® r

± =

L

R

+ (

m c

C ® f L

¡

m b

C ® r L

)

V 2

R

± =

L

R

+ (

W f

C ® f

¡

W r

C ® r

)

V 2

g R

Understeer gradient

The steering angle is expressed in term of the centrifugal acceleration

So

With the understeer gradient K of the vehicle

± =

L

R

+ K

V 2

R

K =

m c

C ® f L

¡

m b

C ® r L

± =

L

R

+ (

m c

C ® f L

¡

m b

C ® r L

)

V 2

R

Steering angle as a function of V

Gillespie. Fig. 6.5 Modification of the steering angle as a function of the speed

Neutralsteer, understeer and oversteer vehicles

If K=0, the vehicle is said to be neutralsteer:

The front and rear wheels sets have the same directional ability

If K>0, the vehicle is understeer :

Larger directional factor of the rear wheels

If K<0, the vehicle is oversteer:

Larger directional factor of the front wheels

K = 0 , c C ® r = b C ® f

K > 0 , c C ® r > b C ® f

K < 0 , c C ® r < b C ® f

Characteristic and critical speeds

For an understeer vehicle, the understeer level may be quantified by a parameter known as the characteristic speed. It is the speed that requires a steering angle that is twice the Ackerman angle (turn at V=0)

For an oversteer vehicle, there is a critical speed above which the vehicle will be unstable

± = 2 L = R

V c a r a c t ¶ e r i s t i q u e =

r

L

K

± = 0

V c r i t i q u e =

s

L

j K j

Lateral acceleration and yaw speed gains

Lateral acceleration gain

Yaw speed gain

± =

L

R

+ K a y

a y

±

=

V 2

L

1 + K V 2

L

r =

V

R

r

±

=

V

L

1 + K V 2

L


Purpose of the steering system is to produce lateral acceleration

For neutral steer, lateral acceleration gain is constantFor understeer vehicle, K>0, the denominator >1 and the lateral acceleration is always reducedFor oversteer vehicle, K<0, the denominator is < 1 and becomes zero for the critical speed, which means that any pertubation produces an infinite lateral acceleration

± =

L

R

+ K a y

a y

±

=

V 2

L

1 + K V 2

L

Yaw velocity gain

The second raison for steering is to change the heading angle by developing a yaw velocity

For neutral vehicles, the yaw velocity is proportional to the steering angleFor understeer vehicles, the yaw gain angle is lower than proportional. It is maximum for the characteristic speed.For oversteer vehicles, the yaw rates becomes infinite for the critical speed and the vehicles becomes uncontrollable at critical speed.

r =

V

R

r

±

=

V

L

1 + K V 2

L

Yaw velocity gain

Gillespie. Fig. 6.6 Yaw rate as a function of the steering angle

r

±

=

V

L

1 + K V 2

L

Sideslip angle

Definition (reminder)

Value

Value as a function of the speed V

Becomes zero for the speed

independent of R !

¯ =

v c g

u c g

¯ =

c r

V

¡ ® r = ± ¡ ® f ¡

b r

V

V ̄ = 0 =

r

c g

C ® r

W r

¯ =

c

R

¡

W r

C ® r

V 2

g R

Sideslip angle

Gillespie. Fig. 6.7 Sideslip angle for a low speed turn

Gillespie. Fig. 6.8 Sideslip angle for a high speed turn

This is true whatever the vehicle is understeer or oversteer

β > 0 β < 0

Static margin

The static margin provides another (equivalent) measure of the steady-state behaviour

Gillespie. Fig 6.9 Neutral steer linee>0 if it is located in front of the vehicle centre of gravity

Static margin

Suppose the vehicle is in straight line motion (δ=0)Let a perturbation force F applied at a distance e from the CG (e>0 if in front of the CG)Let’s write the equilibrium

The static margin is the point such that the lateral forces do not produce any steady-state yaw velocityThat is:

F y f + F y r = F

F y f b ¡ F y r c = F e

± = 0

r = 0 R = 1

± =

L

R

+ ® f ¡ ® r , ® f = ® r

F y f ( b ¡ e ) ¡ F y r ( c + e ) = 0

Static margin

It comes

So the static margin writes

A vehicle isNeutral steer if e = 0Under steer if e<0 (behind the CG)Over steer if e>0 (in front of the CG)

e =

b C ® f ¡ c C ® r

C ® f + C ® r

C ® f ( b ¡ e ) ¡ C ® r ( c + e ) = 0

Static margin

Gillespie. Fig. 6.10 Maurice Olley’sdefinition of understeer and over steer

Exercise

Let a vehicle A with the following characteristics:Wheel base L=2,522mPosition of CG w.r.t. front axle b=0,562mMass=1431 kgTires: 205/55 R16 (see Figure)Radius of the turn R=110 m at speed V=80 kph

Let a vehicle B with the following characteristics :Wheel base L=2,605mPosition of CG w.r.t. front axle b=1,146mMass=1510 kgTires: 205/55 R16 (see Figure)Radius of the turn R=110 m at speed V=80 kph

Exercise

Rigidité de dérive (dérive <=2°) :

0

200

400

600

800

1000

1200

1400

1600

1800

0 1000 2000 3000 4000 5000 6000

Charge normale (N)

Rig

idité

(N/°)

175/70 R13

185/70 R13

195/60 R14

165 R13

205/55 R16

Exercise

Compute: The d’Ackerman angle (in °)The cornering stiffness (N/°) of front and rear wheels and axlesThe sideslip angles under front and rear tires (in °)The side slip of the vehicle at CG (in °)The steering angle at front wheels (in °)The understeer gradient (in °/g)Depending on the case: the characteristic or the critical speed (in kph)The lateral acceleration gain (in g/°)The yaw speed velocity gain (in s-1)The vehicle static margin (%)

Exercise 1

Data

mb 562,0= 2,522 0,562 1,960c L b m= − = − =

2228,0=Lb

0,7772cL

=

kgm 1431=

NLcmgWf 8714,10909== 3127,6909r

bW mg NL

= =

smkphV /2222,2280 ==

²/4893,4² smR

Vay == mR 110=

Exercise 1

Ackerman angle

Tire cornering stiffness of front wheels

Tire cornering stiffness of rear wheels

°==== 3134,10229,0)0229,0arctan()arctan( radRLδ

NLcmgWf 8714,10909==

NFz 93,5454=

deg/1550)1( NC f =α

3110 / deg 177616,98 /fC N N radα = =

3127,6909rbW mg NL

= =

1563,84zF N=

(1) 500 / degrC Nα =

1000 / deg 57295,8 /rC N N radα = =

Exercise 1

Side slip angles under the front tires

NR

VmLcFyf 879,54992²

==

²/4893,4² smR

Vay == Nmay 1883,6424=

yfff FC =αα3110 / degfC Nα =

radCF

f

yf

f 0281,06106,1 =°==α

α

Exercise 1

Side slip angles under the rear tires

Side slip angle at CG

² 1431,3092yfb VF m NL R

= =

r r yrC Fα α = 1000 / degrC Nα =

1,4313 0,0250yrr

r

Frad

Cα

α = = ° =

rr Rc

Vcr ααβ −=−=

1,960 0,0250 0,0072 0,4105110

radβ = − = − = − °

Exercise 1

Steering angle at front wheels

Understeer gradient

RV

CLmb

CLmc

RL

rf

²//⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

αα

δ f rLR

δ α α= + −

1,3134 1,6106 1,4313 1,4927δ = ° + ° − ° = °

/ / 1112,1732 318,82683100 1000f r

mc L mb LKC Cα α

⎛ ⎞= − = −⎜ ⎟⎜ ⎟

⎝ ⎠2/0399,0 −°= msK ' * 0,3918 deg/K K g g= =

Exercise 1

Understeer gradient: check!

Characteristic speed

r

r

f

f

gCW

gCW

Kαα

−= RVK

RL ²3,57 +=δ

4925,14893,43134,1 =+= Kδ

RL2=δ

KLVcarac =

²//49639,6deg/0399,0 2 smradEmsK −== −

2,522 60,1793 / 216,646,9639 4carac

LV m s kphK E

= = = =−

Exercise 1


Yaw speed gain

deg/3066,04927,1

81,9/4893,4 ga

G y

ay =°

==δ

/ 22,222 /110 7,7543 deg/ / deg1,4927r

r V RG sδ δ

= = = =

Exercise 1

Neutral maneuver point

Static margin

100031001000.9600,13100.562,0

+−

=+−

=rf

rf

CCcCbC

eαα

αα

me 0531,0−=

%11,2=Le

Documents

AV_VEHDYN_2012 Lecture Slides and Calcs