av_vehdyn_2012 lecture slides and calcs
DESCRIPTION
Vehicle dynamics lectureTRANSCRIPT
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
Terminology and axis system (SAE)
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
Terminology and axis system (SAE)
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
Terminology and axis system (SAE)
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
Aligning moment and pneumatic trail
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
Aligning moment and pneumatic trail
Lateral force and aligning moment for a tire 175/70 R 13 82SFrom Reimpell et al.
Aligning moment and pneumatic trail
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
δcentre du virage∆
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
δcentre du virage∆
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
Lateral acceleration gain
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
Lateral acceleration gain
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