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Lateral dynamics: steadyLateral dynamics: steady--state state corneringcornering
Basics of Vehicle DynamicsBasics of Vehicle Dynamics
corneringcornering
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Steady-state cornering
Introduction: steering with and w/o tyre side slip
Instantaneous centre changes place due to tyre side slip.
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Steady-state cornering
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)
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Steady-state cornering
Introduction: high speed cornering
Tyre side forces introduce side slipModifying motion direction
Migration of instantaneous centre changes curvature radius and therefore cornering “sharpness”.
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Steady-state cornering
Introduction: “bicycle model”
“Effective wheel” front
“Effective wheel” rear
CM CM
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Steady-state cornering
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
+
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Steady-state cornering
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
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Steady-state cornering
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
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Steady-state cornering
Cornering at high speed
• 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
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Steady-state cornering
Cornering at high speed
• Three types os steering behavior
autozine.org
pussiriot.wordpress.com
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Steady-state cornering
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
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Steady-state cornering
Basic analysis of steering behavior
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
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Steady-state cornering
Basic analysis of steering behavior
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
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Steady-state cornering
Basic analysis of steering behavior
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)
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Steady-state cornering
Basic analysis of steering behavior
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
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Steady-state cornering
• Cornering at high speed: impact of load distribution –“steering by gas-pedal”
Basic analysis of steering behavior
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
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Lateral dynamics: transient Lateral dynamics: transient maneuversmaneuvers
Basics of Vehicle DynamicsBasics of Vehicle Dynamics
maneuversmaneuvers
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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
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Transient maneuvers
Equations of motion
• Kinematic parameters and forces
)β(ρRv C
f
C
fv l
r
C
rv l
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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...
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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
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Transient maneuvers
• Harmonic input
• Steering ramp
• Steering pulse
Some typical transient maneuvers
• Braking in the curve
• “Scandinavian flick”
• Sudden side wind gust
• etc.
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Transient maneuvers
Steering ramp
Steering ramp @ 1 sec
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Transient maneuvers
Braking in the curve
15 sec:Steering ramp
30 sec:Brake on
10 sec:Throttle off
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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
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Transient maneuvers
Sudden side wind with steering
Sudden wind gust @ 2 sec
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Transient maneuvers
“Scandinavian flick”
Caranddriver.com
“Scandinavian flick” video
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Vehicle vibrations and vertical Vehicle vibrations and vertical dynamicsdynamics
Basics of Vehicle DynamicsBasics of Vehicle Dynamics
dynamicsdynamics
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Vehicle vibrations
Basic modelling approaches
scielo.br intechopen.com
sharetechnote.com
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Vehicle vibrations
Topics of interest
From: Thomas D. Gillespie: Fundamentals of Vehicle Dynamics, Society of Automotive Engineers, 1992, ISBN 1560911999
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Vehicle vibrations
Equations for quarter-car model
From: Thomas D. Gillespie: Fundamentals of Vehicle Dynamics, Society of Automotive Engineers, 1992, ISBN 1560911999
Natural frequency
Damped frequency
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Vehicle vibrations
Equations for quarter-car model
From: Thomas D. Gillespie: Fundamentals of Vehicle Dynamics, Society of Automotive Engineers, 1992, ISBN 1560911999
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Vehicle vibrations
Main I/O relations
From: Thomas D. Gillespie: Fundamentals of Vehicle Dynamics, Society of Automotive Engineers, 1992, ISBN 1560911999
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Vehicle vibrations
Main I/O relations
Body natural frequency: 1.5 HzWheel natural frequency: 1015 Hz
From: Thomas D. Gillespie: Fundamentals of Vehicle Dynamics, Society of Automotive Engineers, 1992, ISBN 1560911999
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Vehicle vibrations
Impact of stiffness
From: Thomas D. Gillespie: Fundamentals of Vehicle Dynamics, Society of Automotive Engineers, 1992, ISBN 1560911999
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Vehicle vibrations
Impact of damping
From: Thomas D. Gillespie: Fundamentals of Vehicle Dynamics, Society of Automotive Engineers, 1992, ISBN 1560911999
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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)
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Further readingFurther reading
Basics of Vehicle DynamicsBasics of Vehicle Dynamics
Further readingFurther reading
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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!