2 DOF Handling Model - Wheel...
Transcript of 2 DOF Handling Model - Wheel...
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• assumptions
• equations of motion
• steady cornering results
• stability/eigenvalue results
• frequency response results
• typical understeer/oversteer results
• Buick/Ferrari example
2 DOF Handling Model
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the moving axis system, A, showing the vehicle
velocity components
2 DOF Handling Model
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• simplest representation of vehicle handling which still
captures the key elements of behaviour
• road is level and flat
• vehicle body is rigid and suspension is ignored
• the steering system is assumed to be non-flexible: the
input is assumed to be a front wheel steer angle, but it
could equally be a handwheel angle
(= front wheel angle x steering ratio)
• aerodynamic forces are ignored
2 DOF Handling Model
assumptions
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• in general, three degrees of freedom are actually needed to define
the motion of a car on a horizontal surface:
• forward velocity
• lateral velocity
• yaw velocity
• however, forward velocity will be assumed constant and it
becomes, therefore, a parameter rather than a system variable
• problem formulated in velocities
• tyre forces are dependant on velocities
• nature of the vehicle motion i.e. velocity or lateral acceleration,
is more meaningful than absolute position on the surface
2 DOF Handling Model
assumptions
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• important omissions
• roll motion of sprung mass
• suspension effects, e.g. camber
• tyre load transfer
• steering system compliance
• however, some of these effects can nevertheless be
examined indirectly using the 2 dof model
2 DOF Handling Model
assumptions
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2 DOF Handling Model - Equations
Fyr
Fyf
m (v + ur) = Fyf + Fyr
I r = a Fyf - b Fyr
b
a
Distances
a = cg to
front axle
b = cg to
rear axle
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• dominant in controlling vehicle handling
• tyre side force depends primarily on slip angle
Tyre Force Properties
Resultant
motion
Slip
angle
Tyre
side force
• in the linear region, side force = cornering stiffness x slip angle
Fy = C
• lots of other secondary factors influence C - camber, load, tyre
pressure . . . .
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need to calculate front and rear slip angles in order
to calculate tyre lateral forces
2 DOF Handling Model
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the final equations of motion are then:
in matrix form
2 DOF Handling Model
m 0
0 I
v
r
Cf + Cr
U
aCf - bCr
U
aCf - bCr
U
a2Cf + b2Cr
U
mU +
+ v
r =
Cf
aCf f
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• the sign convention leads to the natural results of a
rightward steer leading to a rightward tyre force
• cornering stiffnesses have positive values (some sign
conventions lead to -ve values which seems unnatural for
a stiffness)
• in the 2 dof, “bicycle model” the C refers to the
cornering stiffness of the axle - i.e. twice that of the
individual tyres
• Fy actually acts perpendicular to the wheel, but since f is
assumed small, the cos f terms which should be in the
vehicle equations are assumed to be unity
2 DOF Handling Model notes to equations of motion
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• steady cornering
• fixed speed and steer angle
• consistent radius of turn
• important because it is a standard and relatively easy method of practical
testing
• stability
• straight running, no input, f = 0
• information about transient response to disturbances
• information about possible unstable conditions
• frequency response
• response to sinusoidal input applied at steering wheel for a range of
frequencies
• captures dynamic response of system to forcing inputs
2 DOF Handling Model - Results
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Front tyre side force
mass x lateral
acceleration
Rear tyre side force
Tyres operate at slip angles
Vehicle Cornering Typical steady cornering condition
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2 DOF Handling Model steady cornering results
• ceases to be a dynamic problem, since v and r are set to
zero
• model assumes small slip angles, so it is restricted to low
lateral accelerations (< 0.3g)
• equations manipulated to give (yaw rate output/steer
angle input):
in which l is the wheelbase, i.e. l =a+b, and the subscript
ss refers to steady state
U l Cf Cr
l2 Cf Cr + m U2(bCr - aCf)
rss
f
=
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2 DOF Handling Model
steady cornering
• lateral acceleration =
• yaw rate, r =
• path curvature, =
U2
R
U
R
1
R
r
U
R
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steady state equation re-written in terms of path
curvature:
where:
2 DOF Handling Model
1
l + K U2
ss
f
=
m (bCr - aCf)
l Cf Cr K =
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Understeer / Oversteer
K=0, Neutral steer
K<0, Oversteer
K<0, Oversteer
K>0, Understeer
Critical
speed
Forward speed, U
Steady state
curvature per
unit steer angle
Steady state turning responses predicted by the two DOF
vehicle
= 1
l + KU2 l = wheelbase
K = depends on stability margin
U = forward speed
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• K = 0 - NEUTRAL STEER
Path followed is same as that for a pure rolling vehicle, i.e. the Ackermann
condition
• K > 0 - UNDERSTEER
Always stable. More steering required than for the Ackermann vehicle.
Practically, vehicle is described as “running wide” as more lock than
anticipated must be applied
• K < 0 - OVERSTEER
Response increases with increasing speed. Critical speed at which response
becomes infinite is given by:
practically, vehicle wants to turn more than anticipated - rear end feels to be
swinging out. At speeds in excess of Ucrit - opposite lock solution
Steady Cornering Results
3 cases of interest (animation)
= l
-K Ucrit
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Understeer / Oversteer
Stability Margin = bCr - aCf
+ve Understeer
-ve Oversteer
a
b
Front cornering
stiffness = Cf
Rear cornering
stiffness = Cr
v r
2 degrees of freedom:-
- sideslip, v
- yaw rate, r
The analysis of the classic 2 degree of freedom vehicle identifies
the importance of stability margin:-
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• Understeer b Cr > a Cf
– car feels like it wants to run wide - not turning enough
– driver feels like he/she has to wind more lock on
– terminal understeer - can plough straight on irrespective
of how much steering is applied
Vehicle Handling - Subjective
Descriptions
Oversteer b Cr < a Cf
– car feels as if the rear end is breaking away
– driver actually has to reduce the steering input
– terminal oversteer - spin - leave the road backwards!
– not as bad as it sounds - vehicle is stable up to its
critical speed
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• critical terminology in order to discuss
vehicle handling
• understeer accepted as preferable – more natural feel as limit approached
– safer - slow down to regain control
• oversteer generally regarded as dangerous – less natural to reduce steering near limit
– spin may happen too quickly in limit conditions
– rally drivers may prefer it!
Understeer / Oversteer
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• K = Understeer parameter
• constant value from linear model
• in practice it is typically a non-linear function of lateral
acceleration
• practical results are normally obtained by driving at
increasing speed around a fixed radius (typically 33m)
circle
Steady Cornering Results
Ackermann
steer angle
Handwheel
steer angle,
deg
Lateral acceleration, g
120
90
60
30
0.2 0.4 0.6 0.8
K = slope of curve (deg/g)
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• effectively the free vibration response
• input i.e. steer angle is set to zero
• roots describe the natural frequency and damping of the
vehicle response
• practically, this characterises the transient response of the
vehicle lateral/yaw motion
• roots - which are typically complex numbers - can be
simply interpreted by plotting on the complex plane
2 DOF Handling Results
stability - same as eigenvalues or roots
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typical eigenvalues in the complex plane and their
relationship with natural frequency and damping ratio
Stability Results
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steady cornering results for an arbitrary vehicle:
Stability Results
Symbol Value
m
I
a
b
Cf
Cr
U
1
1.5
1.25
1.25
53
53
20
Parameter
Total mass
Total yaw inertia
CG to front axle
CG to rear axle
Front axle cornering stiffness
Rear axle cornering stiffness
Forward speed
Units
t
tm2
m
m
kN/rad
kN/rad
m/s
Units Understeer
m
m
kNm/rad
deg/g
m/s
1.15
1.35
+10.6
+0.85
-
Parameter
a
b
bCr - aCf
K
Ucrit
Neutral steer
1.25
1.25
0
0
Oversteer
1.35
1.15
-10.6
-0.85
41
modifications to the parameter set above:
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The eigenvalues of the basic vehicle as forward speed is
increased from 10m/s (smallest symbols) to 50m/s
(largest symbols)
Stability Results
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• sinusoidal input at steer wheels - linear system - results
in sinusoidal responses in yaw, sideslip and lateral
acceleration
• plot out gain and phase of outputs relative to input steer
angle - these are the frequency response functions
• describe the forced vibration characteristics of the system
- also known as the transfer function
Frequency Response Results
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Frequency Response Results
Yaw rate
response results
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Frequency Response Results
Lateral
acceleration
response results
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Frequency Response Results
Normalised yaw
rate and lateral
acceleration
gains
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• the two most meaningful outputs are:
• yaw rate - what the driver sees
• lateral acceleration - what the driver feels
• sideslip response is small and difficult to sense
• desirable properties
• flat gain - indicates consistent response
• gain does not roll off at low frequencies
• drivers cannot provide steering inputs beyond 2-3 Hz
• phase implies a time lag between input and output, which is
generally viewed as undesirable
Frequency Response Results
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2 DOF Handling Model - Results Summary
Understeer
• SS - less response than Ackermann vehicle
• response 0 as U
• always stable
• damped, oscillatory response
• good transient performance - responsive
• too much understeer - possibility of very light damping
at high speeds
• good straight running properties
8
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2 DOF Handling Model - Results Summary
Oversteer
• SS - more response than Ackermann vehicle
• stable below Ucrit • overdamped response
• poor transient performance
• damping increases with speed
• unstable above Ucrit • not as disastrous as it sounds!
• transition occurs gradually
• possible for driver to control an unstable system, providing rate of divergence is low
• poor straight running properties
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Linear handling case study
• 2 contrasting vehicles
– 1949 Buick
– Ferrari Monza
• simple 2 dof results
– steady state
– eigenvalues
– time history to a step input
– frequency responses
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Vehicle parameters
Parameter Symbol Units 1949 Buick Ferrari Monza
Mass m t 2.045 1.008
Yaw inertia I tm 2
5.428 1.031
CG to front axle a m 1.488 1.234
CG to rear axle b m 1.712 1.022
Front axle cornering stiffness C f kN/rad 77.85 117.44
Rear axle cornering stiffness C r kN/rad 76.51 144.93
Wheelbase a b + m 3.200 2.256
Stability margin bCr aCf - kNm/rad 15.15 3.20
Understeer gradient K deg/g 0.91 0.05
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Steady state cornering - Buick
• 0.3g turn at 20m/s, turn radius = 136m
lateral velocity, v -0.48 m/s
steer angle, f 1.62 deg
front slip angle, f -2.37 deg
rear slip angle, r -2.09 deg
front lateral force, Fyf 3.22 kN
rear lateral force, Fyr 2.80 kN
r
v
f f
r
Fyf
Fyr
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Steady state cornering - Ferrari
• 0.3g turn at 20m/s, turn radius = 136m
lateral velocity, v -0.07 m/s
steer angle, f 0.96 deg
front slip angle, f -0.66 deg
rear slip angle, r -0.64 deg
front lateral force, Fyf 1.34 kN
rear lateral force, Fyr 1.62 kN
r
v
f f
r
Fyf
Fyr
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Eigenvalues, damped natural frequencies and damping ratios
Buick Ferrari
Speed Eigenvalue w d , Hz z Eigenvalue w
d , Hz z
20 m/s -3.71 1.65 i 0.26 0.91 -14.5 0.91 i 0.14 1.00
30 m/s -2.48 1.66 i 0.26 0.83 -9.68 1.45 i 0.23 0.99
50 m/s -1.49 1.67 i 0.27 0.67 -5.81 1.65 i 0.26 0.96
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Root locus plots as speed increases from 10 m/s to 50 m/s
10 m/s
50 m/s
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Time histories
• yaw rate response of the Buick and Ferrari at 50 m/s, following a step steer input which results in a steady state lateral acceleration of 0.3g
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Frequency responses - yaw rate
• yaw rate responses for the Buick and Ferrari at two forward speeds
Ferrari
Buick
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Frequency responses - lateral acceleration
• lateral acceleration responses for the Buick and Ferrari at two forward speeds
Ferrari
Buick
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Buick vs Ferrari - Conclusions
• Buick
– high ratio of I:m
– low ratio of tyre force capacity (Cf, Cr) to vehicle mass
– marked understeer characteristic
– light damping at high speeds
• Ferrari
– high ratio of tyre force capacity (Cf, Cr) relative to vehicle mass/inertia
– very slight understeer - almost neutral steer characteristics
– rapid, well damped response to step steer input
– retains consistent gain properties up to much higher frequencies that the Buick