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Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:21
Professur fürSteuerung, Regelungund Systemdynamik
1
Content
Overview
0. Basics on Signal Analysis
1. System Theory
2. Vehicle Dynamics Modeling
3. Active Chassis Control Systems
4. Signals & Systems
5. Statistical System Analysis
6. Filtering
7. Modeling, Simulation with Matlab/Simulink
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:21
Professur fürSteuerung, Regelungund Systemdynamik
2Chapter 3
Vehicle Dynamics Modeling1. Vehicle Dynamics Basics
– Longitudinal Dynamics
– Lateral Dynamics
– Vertical Dynamics
2. Chassis Components
– Suspension
– Spring and Shock
– Anti-roll-bar
– Steering
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:21
Professur fürSteuerung, Regelungund Systemdynamik
3Chapter 3
1. Vehicle Dynamics Basics
• Longitudinal Dynamics
• Lateral Dynamics
• Vertical Dynamics
z
y
x
vertical
lateral
longitudinal
yaw
pitch
roll
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:22
Professur fürSteuerung, Regelungund Systemdynamik
4Chapter 3
1. Vehicle Dynamics Basics
Longitudinal Dynamics
• 1-dimensional (plane) model
• Supply side
– Powertrain
– Drive line
• Demand side / Drivingresistances
– Rolling resistance
– Air drag
– Climbing resistance
– Vehicle inertia
– Braking
Source: Willumeit
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:22
Professur fürSteuerung, Regelungund Systemdynamik
5Chapter 3
1. Vehicle Dynamics Basics
Longitudinal Dynamics
Equation of motion
• Body
• Front axle
• Rear axle
Source: Willumeit
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:23
Professur fürSteuerung, Regelungund Systemdynamik
6Chapter 3
1. Vehicle Dynamics Basics
Longitudinal Dynamics
Driving torque
• Vehicle inertia
• Climbing resistance
• Air drag
• Rolling resistance: contribution by tire, track, skew
FGxgVH x
rdyn s
FG sinSt
F WL
F WR
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:23
Professur fürSteuerung, Regelungund Systemdynamik
7Chapter 3
1. Vehicle Dynamics Basics
Longitudinal Dynamics
Rolling resistance: Tire
• Pressure distribution non-symmetric due to hysteresis⇒ MP~e
• Rolling resistance
• Directly proportional towheel load
F WR=−F U=M P
s
F WR= f R F P=es
F P
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:24
Professur fürSteuerung, Regelungund Systemdynamik
8Chapter 3
1. Vehicle Dynamics Basics
Longitudinal Dynamics
Rolling resistance: Tire
• Rolling resistance coefficient
– fR≈0.0014 [-]
• In general, fR is a
function of vehiclespeed
– Measured data
Source: Willumeit
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:24
Professur fürSteuerung, Regelungund Systemdynamik
9Chapter 3
1. Vehicle Dynamics Basics
Longitudinal Dynamics
Rolling resistance: Tire
• Rolling resistance coefficient and tire pressure
– Measured data
Source: Willumeit
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:24
Professur fürSteuerung, Regelungund Systemdynamik
10Chapter 3
1. Vehicle Dynamics Basics
Longitudinal Dynamics
Rolling resistance: Skew
• Wheel plane and direction of movement differ by slip angle
• Causation for slip angle
– Geometrical alignment offront wheels
– Lateral charge of wheel
– Curve driving (centripetal)
– Cambered road
– Crosswind
• Rolling resistance at skew
Source: WillumeitF WR=FWR cosF sin
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:25
Professur fürSteuerung, Regelungund Systemdynamik
11Chapter 3
1. Vehicle Dynamics Basics
Longitudinal Dynamics
Rolling resistance: Skew
• For small (<5°) we have linear relation between and lateral force
• This yields
• For large slip anglenon-linear relation
Source: Willumeit
F =k
F WR=FWRk 2
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:25
Professur fürSteuerung, Regelungund Systemdynamik
12Chapter 3
2. Chassis Components
Longitudinal Dynamics
Braking: pitch axis
• Braking induces longitudinal, vertical and pitch oscillations
• For identical brake torques front and rear no vertical movement of COG
• For COG position identical to pitch axis no pitching oscillation
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:26
Professur fürSteuerung, Regelungund Systemdynamik
13Chapter 3
1. Vehicle Dynamics Basics
Lateral Dynamics
• Vehicle model
– Side slip angle
– Yaw rate
– Longitudinal and lateralforces at each tire
– Steering wheelangle (different forleft and right frontwheel)
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:26
Professur fürSteuerung, Regelungund Systemdynamik
14Chapter 3
1. Vehicle Dynamics Basics
Lateral Dynamics
• Single track model
– Rolling degree of freedom neglected
– Tire non-linearities neglected
– Restricted to small angles and distances (for linearity)
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:29
Professur fürSteuerung, Regelungund Systemdynamik
15Chapter 3
1. Vehicle Dynamics Basics
Lateral Dynamics
• Single track model
– Mean wheel loads at frontand rear axle
– Slip angles at frontand rear wheels
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:31
Professur fürSteuerung, Regelungund Systemdynamik
16Chapter 3
1. Vehicle Dynamics Basics
Lateral Dynamics
• Single track model - Variables
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:35
Professur fürSteuerung, Regelungund Systemdynamik
17Chapter 3
1. Vehicle Dynamics Basics
Lateral Dynamics
Single track model
• Equations of motion
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:37
Professur fürSteuerung, Regelungund Systemdynamik
18Chapter 3
1. Vehicle Dynamics Basics
Lateral Dynamics
• Single track model as AS :Substitution v=r −
Singletrack
t COG x , y ,
F x , i
Steering wheelangle
Longitudinal forces Center of gravitymotion
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:37
Professur fürSteuerung, Regelungund Systemdynamik
19Chapter 3
1. Vehicle Dynamics Basics
Lateral Dynamics
Yaw eigen frequency
• Implicit stiffness and dampingrate by tires
• 2nd order model
• Yaw and side slip havesame eigen frequency
Source: Willumeit
Damping
Eigen freq.
∆k: side force delta front
to rear
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:38
Professur fürSteuerung, Regelungund Systemdynamik
20Chapter 3
1. Vehicle Dynamics Basics
Lateral Dynamics
Yaw eigen frequency: example for typical parameter values
• 2nd ordersystembehavior
Source: Willumeit
slip
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:38
Professur fürSteuerung, Regelungund Systemdynamik
21Chapter 3
1. Vehicle Dynamics Basics
Lateral Dynamics
Steady-state Cornering
• Constant steering angle or constant curve radius
• Seldom in reality (Autobahn entry). Most of curves are build as clothoid (bending proportional arc length)
• Very useful as 'artificial' manoeuvre for analyzinglateral dynamics – skid-pad is standard for automotivecompanies
• Important data are steering angle and the resulting sideslip angle
– Understeer, Oversteer, Neutral steer
• Steerable rear axle would allow to compensate side slip angle completely
clothoid
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:39
Professur fürSteuerung, Regelungund Systemdynamik
22Chapter 3
1. Vehicle Dynamics Basics
Lateral Dynamics
Understeer
• In technical terms understeer is the condition when the slip angles of the front tires are greater than the slip angles of the rear tires. In other words the front wheel angle is greater than the angle normally required for the turn. The slip angle is the angle of the tire in relation to it's direction of travel.
Oversteer
• Oversteer is the reverse of understeer, that is where the slip angles of the rear tires are greater than the front tires. In cases of oversteer the rear tires tend to describe a larger cornering radius than the front tires.
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:39
Professur fürSteuerung, Regelungund Systemdynamik
23Chapter 3
1. Vehicle Dynamics Basics
Lateral Dynamics
Steady-state cornering
• This results in a direct relation between side slip angle and steering angle (cf. slide 17, assume sin =, cos =1-, sin =, cos =1-)
• At lateral acceleration equal to zero yields a negative side slip angle (assumption: front wheel steering only)
=0 ; =0 ; =0 ; F x ,F=0
=F a
F aF b
F a=F x ,RF x ,F1−−F y ,FF b=−F y ,R−F x , F−F y ,F1−
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:39
Professur fürSteuerung, Regelungund Systemdynamik
24Chapter 3
1. Vehicle Dynamics Basics
Lateral Dynamics
Typical behavior of side side slip angle over lateral acceleration for steady-state cornering
Source: Willumeit
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:39
Professur fürSteuerung, Regelungund Systemdynamik
25Chapter 3
1. Vehicle Dynamics Basics
Lateral Dynamics
Ackermann vehicle
• Assumption: no slip angles
Instantaneous center of rotation
l Fl R
COG
vF
vR
v
r
ICR
A
A
A
A
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:40
Professur fürSteuerung, Regelungund Systemdynamik
26Chapter 3
1. Vehicle Dynamics Basics
Lateral Dynamics
General case with slip angles
F y ,F=k FF
F y ,R=kRR
ki: Side force
coefficent
l Fl R
COG
vF
vR
r
ICR
F
R
v
vR
R
=F−RA
=R−A
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:40
Professur fürSteuerung, Regelungund Systemdynamik
27Chapter 3
1. Vehicle Dynamics Basics
Lateral Dynamics
• Steering wheel angle vs. lateral acceleration
Measurement: Ford MondeoCalculation
Single-track vehicle
Ackermann vehicle
aq / g
δ / °
1 F x ,FF
F y ,F =m v2
r lRlFlR
F
F y ,F−
l FlFlR
R
F y ,R A
«1 for rear wheel drive»1 for front wheel drive
Source: ATZ
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:40
Professur fürSteuerung, Regelungund Systemdynamik
28Chapter 3
1. Vehicle Dynamics Basics
Lateral Dynamics
Understeer, oversteer and neutral in steady-state cornering
• In real driving conditionsnon-linear behaviorof tire side forces
• Radius const.
• Under-/Oversteerdefinition resultsfrom gradientd/daq
Source: Willumeit
SingleTrack-SWA
Ackermann-SWA
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:40
Professur fürSteuerung, Regelungund Systemdynamik
29Chapter 3
1. Vehicle Dynamics Basics
Lateral Dynamics
Test maneuver: vehicle behavior during Power-off in a turn
• Load change atrear axle
• Increase in yawspeed
Source: ATZ
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:41
Professur fürSteuerung, Regelungund Systemdynamik
30Chapter 3
1. Vehicle Dynamics Basics
Lateral Dynamics
Side forcecoefficient
Source: Willumeit
F y ,F=k FF
F y ,R=kRR
k F/R~F z , F/R
Fz
Fy
Wheel loaddifference
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:41
Professur fürSteuerung, Regelungund Systemdynamik
31Chapter 3
1. Vehicle Dynamics Basics
Lateral Dynamics
Kamm'scher Circle
• Sum of longitudinal forces and lateral forces cannot exceed a certain value max Fp
– FP: wheel load
– max: maximum friction
coefficient
• If longitudinal forces are fullyutilized, no side forces can begenerated and vice versa
F x
F y
max F p
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:41
Professur fürSteuerung, Regelungund Systemdynamik
32Chapter 3
1. Vehicle Dynamics Basics
Lateral Dynamics
Steering Wheel Torque: beside steering wheel angle (SWA) the torque is of interest as a design parameter
• Side force at frontwheels
• Power assist
– Hydraulic(cf. Chapter 4)
– Electrical(cf. Chapter 4)
Measurement: Ford Mondeo
Source: ATZ
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:42
Professur fürSteuerung, Regelungund Systemdynamik
33Chapter 3
1. Vehicle Dynamics Basics
Vertical Dynamics
Quarter Vehicle Model (QVM)
• No pitch, no roll modeling
• Mass m1 equal unsprung
masses (wheel, tire, arms,brake,...)
• Mass m2 equal ¼ body mass
• Spring (linear)
• Damper (linear)
• No tire damping
m2
m1
c2
d
c1
z0
z1
z2
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:42
Professur fürSteuerung, Regelungund Systemdynamik
34Chapter 3
1. Vehicle Dynamics Basics
Vertical Dynamics
Quarter Vehicle Model (QVM)
• Equations of motion
Possible output variables
– z1(wheel movement)
– z2 (body movement)
[m1 00 m2][ z1
z2][d −d−d d ][ z1
z2][c1c2 −c2
−c 2 c2 ][z1
z2]=[P 1
0 ]with P 1=c1z0
Mass matrix Damping matrix Stiffness matrix
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:43
Professur fürSteuerung, Regelungund Systemdynamik
35Chapter 3
1. Vehicle Dynamics Basics
Vertical Dynamics
Quarter Vehicle Model (QVM)
• State space description
• Eigen values
– Body eigen frequency (¼ car)
– Wheel eigen frequency
x=[−dm1
dm1
−c1c 2
m1
c 2
m1
dm2
−dm2
c2
m2−c2
m2
1 0 0 00 1 0 0
]x[c1
m1
000]u
y=[0 0 0 1]x
x4=z2 ; x2= z2x3=z1 ; x1= z1
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:43
Professur fürSteuerung, Regelungund Systemdynamik
36Chapter 3
1. Vehicle Dynamics Basics
Vertical Dynamics
Simple 3-dimensional vertical dynamics model
• Height, pitch, roll modeling
Source: Willumeit
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:44
Professur fürSteuerung, Regelungund Systemdynamik
37Chapter 3
1. Vehicle Dynamics Basics
Vertical Dynamics
Simple 3-dimensional vertical dynamics model
• Body mass distribution
• No body elasticity
– Valid only belowbending eigenfrequency of body
• Point P: part of body
– Accelerations at PP'
P
Source: Willumeit
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:44
Professur fürSteuerung, Regelungund Systemdynamik
38
Vertical Dynamics
3D vertical dynamics model: block diagram
z0l,r(t): Road input (partially correlated)
H1(s): Quarter vehicle model
H2z
(s): Pitch model between upper point QVM and P'
H3(s): Roll model between left side body and P
H4(s): Roll model between right side body and P
Chapter 3
1. Vehicle Dynamics Basics
H1(s)
z0l t H
2z(s) H
3(s)
u t
H1(s)
z0r t H
2z(s) H
4(s)
v t
z2Pt
Source: Willumeit
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:45
Professur fürSteuerung, Regelungund Systemdynamik
39Chapter 3
1. Vehicle Dynamics Basics
Vertical Dynamics
3D vertical dynamics model: linear transfer functions
H1(s): Quarter vehicle model
H2z
(s): Pitch model between upper point QVM and P'
H 2z j=11e− j l 'l
Time-delay from front to rear axle =1v
P: position in x-direction
Wheel base
QVM from state spacedescription (slide 34)
H 1 j=cT [ I j−A]−1b
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:45
Professur fürSteuerung, Regelungund Systemdynamik
40Chapter 3
1. Vehicle Dynamics Basics
Vertical Dynamics
3D vertical dynamics model: linear transfer functions
H3(s): Roll model between left side body and P (static)
H4(s): Roll model between right side body and P (static)
H 3 j=1 s 's
H 4 j=− s 's
P: position in y-direction
Track width
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:45
Professur fürSteuerung, Regelungund Systemdynamik
41Chapter 3
1. Vehicle Dynamics Basics
Vertical Dynamics
Quarter Vehicle Model PSD (chapter 1, slide 35f)
• Road surface characteristics z0(t) are stationary with normal
distribution
• Vehicle speed is const.
• Vehicle model behaves linear
H(s)z0t z t
P zz =∣H j∣2 P z0z0
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:45
Professur fürSteuerung, Regelungund Systemdynamik
42Chapter 3
1. Vehicle Dynamics Basics
Vertical Dynamics
Road surfacecharacteristics z0(t)
• Typical PSDs for roadsurfaces
•Tarmac•Cement•Macadam medium•Cobblestone medium•Dirt road
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:46
Professur fürSteuerung, Regelungund Systemdynamik
43Chapter 3
1. Vehicle Dynamics Basics
Vertical Dynamics
Quarter Vehicle Model PSD
• Position & acceleration PSD
H(jω)
|H(jω)|2
PSD acceleration body
PSD acceleration road
P ¨z0 ¨z0 =4 P z0z0
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:46
Professur fürSteuerung, Regelungund Systemdynamik
44Chapter 3
1. Vehicle Dynamics Basics
Vertical Dynamics
Seat rail vertical acceleration
Source: ATZ
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:47
Professur fürSteuerung, Regelungund Systemdynamik
45Chapter 3
1. Vehicle Dynamics Basics
Vertical Dynamics
Seat rail vertical acceleration: PSD (Power Spectrum Density)
Source: ATZ
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:47
Professur fürSteuerung, Regelungund Systemdynamik
46Chapter 3
2. Chassis Components
Suspension
Front axle: McPherson
Transverse Link
• Long. forces: rotationaround front bush
• Lat. forces: radiallystiff front lower controlarm bush
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:48
Professur fürSteuerung, Regelungund Systemdynamik
47Chapter 3
2. Chassis Components
Suspension
Rear axle: Multilinksuspension
Rear axle: SLA suspension
• Maximum loadcompartment:SLA selected for wagon
• Lat. forces:radially stiff front lowercontrol arm bush
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:48
Professur fürSteuerung, Regelungund Systemdynamik
48Chapter 3
2. Chassis Components
Suspension
McPherson front axle
Source: Ford
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:49
Professur fürSteuerung, Regelungund Systemdynamik
49Chapter 3
2. Chassis Components
Suspension
McPherson front axle: Roll center height
Source: Ford
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:49
Professur fürSteuerung, Regelungund Systemdynamik
50Chapter 3
2. Chassis Components
Suspension
Trailing arm SLA rear axle
Source: Ford
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:50
Professur fürSteuerung, Regelungund Systemdynamik
51Chapter 3
2. Chassis Components
Suspension
Trailing arm SLA rear axle:Bump steer
Source: Ford
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:51
Professur fürSteuerung, Regelungund Systemdynamik
52Chapter 3
2. Chassis Components
Suspension
Multilink rear axle: Roll center height
Source: Ford
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:51
Professur fürSteuerung, Regelungund Systemdynamik
53Chapter 3
2. Chassis Components
Spring and shock
Perfect coil spring modeling
• No spring mass
• Constant stiffness c
• 2nd order modely1y2
m2 m1
F1(t)c
G1s=Y 1s s2
F 1 s=
s2 cm2
m1s2 cm1
cm2
G2 s=Y 2 s s2
F 1 s=
cm2
m1s2 cm1
cm2
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:51
Professur fürSteuerung, Regelungund Systemdynamik
54Chapter 3
2. Chassis Components
Spring and shock
Bode diagram of perfect coil spring
• Above 40Hz is the real spring behavior not covered by perfect model: resonances
• Possible solution:FEM modeling
-150
-100
-50
0
50
100
150
Magn
itu
de (
dB
)
10-1
100
101
-360
-180
0
180
Phase
(d
eg
)
Bode Diagram
Frequency (rad/sec)
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:51
Professur fürSteuerung, Regelungund Systemdynamik
55Chapter 3
2. Chassis Components
Spring and Shock
McPherson strut
• Coil spring
• Damper inside spring
• Distributed mass
• Stiffness c=f(y)
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:52
Professur fürSteuerung, Regelungund Systemdynamik
56Chapter 3
2. Chassis Components
Spring and shock
Hydropneumatic springCitroën
• “Air spring”
• Force transferby oil
• Height controlby oilvolume change
• Damping withnitrogenreservoir
• Spring elementsconnected to arms
Cylinder
Nitrogen
Damper
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:52
Professur fürSteuerung, Regelungund Systemdynamik
57Chapter 3
2. Chassis Components
Spring and shock
Hydro pneumatic spring
• Function
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:53
Professur fürSteuerung, Regelungund Systemdynamik
58Chapter 3
2. Chassis Components
Spring and Shock
Damper
• Hydraulic principle
• Ideal damper modeling
• Optimized modeling(Maxwell model)
– Additional spring
F D= y
y2
m2
m1
F1(t)
cy1
d
y2
y1
d
m2
m1
F1(t)
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:54
Professur fürSteuerung, Regelungund Systemdynamik
59Chapter 3
2. Chassis Components
Spring and Shock
Damper dynamics: transfer functions
– Ideal model
– Maxwell model
G1 s=Y 1s s2
F 1s=
m2 sdm1m2 sd m1m2
G2s=Y 2s s2
F1 s= d
m1 m2 sd m1m2
G1 s=Y 1s s2
F 1s =d
m2
cs2m2 sd
dm! m2
cs2m1m2 sd m1m2
G1 s=Y 1s s2
F 1s= d
dm! m2
cs2m1m2 sd m1m2
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:54
Professur fürSteuerung, Regelungund Systemdynamik
60Chapter 3
2. Chassis Components
Spring and Shock
Damper dynamics
• Measurements
• Bode diagram ideal vs. Maxwell model behavior
Measured
Maxwell
Ideal
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:55
Professur fürSteuerung, Regelungund Systemdynamik
61Chapter 3
2. Chassis Components
Spring and Shock
Damper realization
• Monotube andtwintube
Dynamic Damper Properties
• Hysteresis
– Different forces injounce and rebound
• Friction forces
• Cavitation
• Reduced dampingat higherfrequencies
Twintube Monotube
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:55
Professur fürSteuerung, Regelungund Systemdynamik
62Chapter 3
2. Chassis Components
Spring and Shock
Monotube properties
• Compression: displaced oil compensated by gas reservoir
• Damping work by piston valve
• Prevent cavitation: pre-charge pressure 30 ~ 40 bars in passive state
Twintube properties
• Compression: displaced oil pressed in outer tube through base valve
• Damping work by base valve
• Package: diameter
Source: ZF Sachs
Twintube
Monotube
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:56
Professur fürSteuerung, Regelungund Systemdynamik
63Chapter 3
2. Chassis Components
Anti-roll-bar (ARB)
Spring connection between wheels of one axle
• Effect only during inverse z-movement of wheels
• Increased roll stiffness of vehicle
• Reduced roll angle and roll speed
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:56
Professur fürSteuerung, Regelungund Systemdynamik
64Chapter 3
2. Chassis Components
Anti-roll-bar (ARB)
Effect on side forces
• Increasing wheel load difference due to ARB
• Side force depends non-linear onwheel load
• ARB as tuning parameter for understeer resp. oversteer
Higher ARB stiffness at front axle: understeer tendencyHigher ARB stiffness at rear axle: oversteer tendency
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:57
Professur fürSteuerung, Regelungund Systemdynamik
65Chapter 3
2. Chassis Components
Steering
Driver demands
• Reasonable SteeringWheel Torque (SWT)
• Sensitivity
• Auto Return
• Passive Safety
Vehicle demands
• Steeringkinematics
• Package
• Cost
Source: RWTH Aachen, Kraftfahrzeugtechnik
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:57
Professur fürSteuerung, Regelungund Systemdynamik
66Chapter 3
2. Chassis Components
Steering
Steering rack realization
• Different gearmechanisms
• Gear rackshowsadvantages
Source: RWTH Aachen, KraftfahrzeugtechnikSource: RWTH Aachen, KraftfahrzeugtechnikSource: RWTH Aachen, Kraftfahrzeugtechnik
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:58
Professur fürSteuerung, Regelungund Systemdynamik
67Chapter 3
2. Chassis Components
Steering
Steering rack realization
• Tie-rods connectedvia ball joints
• Attachment pointson the outside (a)or in the middle (b)part
Source: RWTH Aachen, Kraftfahrzeugtechnik
Tie-rods
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:59
Professur fürSteuerung, Regelungund Systemdynamik
68Chapter 3
2. Chassis Components
Steering
Power Assist
• Hydraulic assist
• Open-centervalve withtorsion bar
• Constantflow pump
Source: RWTH Aachen, Kraftfahrzeugtechnik
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:07:59
Professur fürSteuerung, Regelungund Systemdynamik
69Chapter 3
Matlab/Simulink Exercise
• Single Track Model – Simple Simulink model– Equation of Motion - assume steering wheel angle is equal to zero
– Tire forces – function of wheel slipand driving situation
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:08:00
Professur fürSteuerung, Regelungund Systemdynamik
70Chapter 3
Matlab/Simulink Exercise
• Single Track Model – Simple Simulink model– Simulink modeling of 2nd order differential equation
– Implement Simulink representation of single track model based on the given equations
Lecture: Vehicle DynamicsTutor: T. WeyDate: 17.01.09, 11:08:00
Professur fürSteuerung, Regelungund Systemdynamik
71Chapter 3
Matlab/Simulink Exercise
• Vertical Dynamics
– A QVM as described in fig. 1 should be modelled in Simulink. Parameters are• c1 = 150kN/m
• c2 = 15000N/m
• m1 = 30kg
• m2 = 300kg
• d = 0kg/s
– Show for z0(t)=1(t) and z0(t)=sin(t) the body motion
– In general, the eigen frequency of body and wheelare easily calculated by
Mathematically this is not correct. Evaluate graphically the difference between the solution above and the approximation.
– What happens to the eigen frequency for d = 1400kg/s?
Body= c2
m2; Wheel= c1c2
m1
fig. 1