basics of vehicle dynamics lateral dynamics: steady-state...

Lateral dynamics: steadyLateral dynamics: steady--state state corneringcornering

Basics of Vehicle DynamicsBasics of Vehicle Dynamics

corneringcornering

Steady-state cornering

Introduction: steering with and w/o tyre side slip

Instantaneous centre changes place due to tyre side slip.


Introduction: kinematic (“Ackermann”) steering

Cornering at low speed, negligible side forces (and therefore side slip)

Steering mechanism configuration named after Rudolph Ackermann(patent agent), invented by Georg Lankensperger XIX century (Germany)


Introduction: high speed cornering

Tyre side forces introduce side slipModifying motion direction

Migration of instantaneous centre changes curvature radius and therefore cornering “sharpness”.


Introduction: “bicycle model”

“Effective wheel” front

“Effective wheel” rear

CM CM


Cornering at low speed

C

AA

R

)θ2/sin(β)sin(θ

π

a OCA:

KRsinβ

b OCB:

C

AAA

R

cosθsinβcosθcosβsinθ

a Acosθ

a

CA

Rsinβcosβtgθ

a

CRsinβ

b

CA

Rcosβtgθ

l

+


Cornering at low speed

Further we assume:Further we assume:

•• RRCC >> >> ll

CA

Rcosβtgθ

l

•• angles angles AA, , are small are small tgtgAA AA ((radians!radians!), ), coscos 11

CA

Rθ

l

Wheelbase of the vehicleWheelbase of the vehicle

CM’s trajectory curvature radiusCM’s trajectory curvature radius

Required steering angle


Cornering at high speed

C

ff

R

))δ(θ/sin(β))δsin((θ

2

a

C

fr

R

)δ2/sin(β)sin(δ

b+

sin (/2– x) = cosxSmall angles: sinxx, cosx1

rfC

δδR

θ l



• Three types os steering behavior

rfC

δδR

θ l

CA

Rθ

l To achieve the same RC, driver

has to turn the steering wheel:C

f > r

f = r

f < r

UNDERSTEER

NEUTRAL STEER

OVERSTEER

has to turn the steering wheel:

MORE

EQUAL

LESS



• Three types os steering behavior

autozine.org

pussiriot.wordpress.com


Basic analysis of steering behavior

• Assumption: small angles, large curve radius

• Constant velocity v, constant constant RC

CM

CMFyf

Fyf Fyr

Fyr

vfvr

FC

FC

RC

C

2

CR

vmF "centrifugal force“

(D'Alembert's principle of inertial forces)

f

r



CMFyf Fyr

FC

C

2

YZYPR

vmFF

aFYP = bFYZa b

a + b = l -- wheelbasewheelbase

C

2

yr

C

2

yf

R

vmaF

R

vmbF

l

l

Further we apply:

Wb

Wf l

Wa

Wr l

W

Wb fl

W

Wa rl

C

2r

yr

C

2f

yf

R

vm

W

WF

R

vm

W

WF

Finaly:

Important:Important:W

W

F

Ff

C

yf W

W

F

Fr

C

yr f

r

yf

yr

W

W

F

F



For moderate Fy (small ): Fy = c

2r

C

2f

yf

vmWF

R

vm

W

WF

Fyf = cff

Fyr = crr

Linear approximation:FY = c

C

ryr

RWF Fyr = crr

gRC

vW

gRC

vW

Cr

2

rr

Cf

2

ff

δ

δ

δ

δ

rfC

δδR

θ l

Required steering angle

gRC

vW

gRC

vW

Rθ

Cr

2r

Cf

2f

C

δδ

l



C

2

r

r

f

f

C Rg

v

C

W

C

W

R

δδ

lθ

K – UNDERSTEER GRADIENTg

aK

R

y

C

l

θr

r

f

f

C

W

C

WK

δδ

Solution for most simple model – comprises most fundamental factors

Further factors affecting steering behavior:

Wheel alignment and elastokinematics of the wheel suspension

Distribution of vertical loads amongst wheels at the same axle

Non-linear tyre behavior

Presence of driving/braking torque (i.e. longitudinal tyre force)



1

1,2

1,4

1,6

1,8

2

g

aK

R

y

C

l

θ

K > 0 - UNDERSTEER

Req

uir

ed s

teer

ing

angl

e o

f th

e r

oad

wh

eel

0

0,2

0,4

0,6

0,8

1

0 20 40 60 80 100 120

Velocity (km/h)

K = 0 - NEUTRAL

K < 0 - OVERSTEER

Req

uir

ed s

teer

ing

angl

e o

f th

e r

oad

wh

eel

Oversteer vehicle is unstable by nature above critical speed!

Drift video


• Cornering at high speed: impact of load distribution –“steering by gas-pedal”


r

r

f

f

C

W

C

WK

δδ

WW

Braking/accelerating: redistribution of WBraking/accelerating: redistribution of Wff / W/ Wrr

Tyre load (WTyre load (WTT) influences c) influences c (larger (larger WWTT larger larger cc))

Influence is degressive!Influence is degressive!

Deccelerating (braking) in the curve Deccelerating (braking) in the curve car behavior changes towards more oversteercar behavior changes towards more oversteer

Aeccelerating in the curve Aeccelerating in the curve car behavior changes towards more understeercar behavior changes towards more understeer

WWrrWWff

“Lift-off oversteer” video

Lateral dynamics: transient Lateral dynamics: transient maneuversmaneuvers


maneuversmaneuvers

Transient maneuvers

Equations of motion

iC FΣam

(1,2)

iFCCz ΣMJ (3)

• General equations of planar motion

NTCIN amamamFD'Alembert's principle:

0FΣamam0FΣFΣ iCNCTINi

CyCxCNTCC aaaaa

Vector equation to scalar (axes: T/N or x/y)

ρ

va

2

CN vdt

dvaCT

Transient maneuvers

Equations of motion

• Kinematic parameters and forces

)β(ρRv C

f

C

fv l

r

C

rv l

Transient maneuvers

Equations of motion

0FFsinθFcosθFsinβρ

vmcosβvm Wxryfxf

2

(1,2)

0FFcosθFsinθFcosβρ

vmsinβvm Ayyryfxf

2

x:

y:ρ

(3) 0FlFcosθFsinθFJ AyAyrryfxffCz ll

FY = FY()

= (t)

FX, v(t) – assumptions...

Transient maneuvers

• Special case for constant velocity:

• d/dt and d/dt are directly related

• One ODE remains in the form of:

Equations of motion

θkθkβCβBβA 21

0βCβBβA

With homogenwous part:

Vehicle can undergo damped angular oscillationsVehicle can undergo damped angular oscillations

Transient maneuvers

• Harmonic input

• Steering ramp

• Steering pulse

Some typical transient maneuvers

• Braking in the curve

• “Scandinavian flick”

• Sudden side wind gust

• etc.

Transient maneuvers

Steering ramp

Steering ramp @ 1 sec

Transient maneuvers

Braking in the curve

15 sec:Steering ramp

30 sec:Brake on

10 sec:Throttle off

Transient maneuvers

Braking in the curve – impact of wheel lock

2vm

Front wheel lock UNSTEERABLE, STABLE VEHICLE

Rear wheel lockSTEERABLE, UNSTABLE VEHICLE!

couple

CR

vm VEHICLE

More favorable situation for untrained driver!

(Higher probability of avoiding the accident)

Lateral forces at both axles – STEERABLE AND STABLE VEHICLE

Transient maneuvers

Sudden side wind with steering

Sudden wind gust @ 2 sec

Transient maneuvers

“Scandinavian flick”

Caranddriver.com

“Scandinavian flick” video

Vehicle vibrations and vertical Vehicle vibrations and vertical dynamicsdynamics


dynamicsdynamics

Vehicle vibrations

Basic modelling approaches

scielo.br intechopen.com

sharetechnote.com

Vehicle vibrations

Topics of interest

From: Thomas D. Gillespie: Fundamentals of Vehicle Dynamics, Society of Automotive Engineers, 1992, ISBN 1560911999

Vehicle vibrations

Equations for quarter-car model


Natural frequency

Damped frequency

Vehicle vibrations

Equations for quarter-car model


Vehicle vibrations

Main I/O relations


Vehicle vibrations

Main I/O relations

Body natural frequency: 1.5 HzWheel natural frequency: 1015 Hz


Vehicle vibrations

Impact of stiffness


Vehicle vibrations

Impact of damping


Vehicle vibrations

Impact of suspension and tyre elasticity on dynamic axle load gain

EVERYTHING RIGID

RIGID WHEEL

Lecture notes FHTW Berlin

RIGID SUSPENSION (e.g. tractor)

EVERYTHING ELASTIC(road vehicle)

Further readingFurther reading


Further readingFurther reading

Further reading

• Thomas D. Gillespie: Fundamentals of Vehicle Dynamics, Society of Automotive Engineers, 1992, ISBN 1560911999

• Georg Rill: Road Vehicle Dynamics - Fundamentals and Modeling, CRC Press, 2012, ISBN 978-1-4398-3898-3

• Masato Abe: Vehicle Handling Dynamics - Theory and Application, Butterworth-Heinemann, 2nd Edition 2015, ISBN 9780081003909

If this was interesting for you...

Heinemann, 2nd Edition 2015, ISBN 9780081003909

• William F. Milliken, Douglas L. Milliken: Race Car Vehicle Dynamics, SAE International, 1994, ISBN of 978-1-56091-526-3

• Mitschke M., Wallentowitz H.: Dynamik der Kraftfahrzeuge, VDI-Buch 2004, 2014

• Hans Pacejka: Tire and Vehicle Dynamics, Butterworth-Heinemann, 3rd Edition 2012, ISBN 9780080970165

•• ...and plenty more!...and plenty more!

THE ENDTHE ENDTHANKS FOR YOUR ATTENTIONTHANKS FOR YOUR ATTENTION

[email protected]@UNS.AC.RS

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