VeD Chapter3 Lecture - uni-due.de · Lecture: Vehicle Dynamics Tutor: T. Wey Date: ... contribution...

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



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



3Chapter 3

1. Vehicle Dynamics Basics

• Longitudinal Dynamics

• Lateral Dynamics

• Vertical Dynamics

z

y

x

vertical

lateral

longitudinal

yaw

pitch

roll



4Chapter 3


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



5Chapter 3



Equation of motion

• Body

• Front axle

• Rear axle

Source: Willumeit



6Chapter 3



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



7Chapter 3



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



8Chapter 3




• Rolling resistance coefficient

– fR≈0.0014 [-]

• In general, fR is a

function of vehiclespeed

– Measured data

Source: Willumeit



9Chapter 3




• Rolling resistance coefficient and tire pressure

– Measured data

Source: Willumeit



10Chapter 3



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



11Chapter 3



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



12Chapter 3



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



13Chapter 3


Lateral Dynamics

• Vehicle model

– Side slip angle

– Yaw rate

– Longitudinal and lateralforces at each tire

– Steering wheelangle (different forleft and right frontwheel)



14Chapter 3


Lateral Dynamics

• Single track model

– Rolling degree of freedom neglected

– Tire non-linearities neglected

– Restricted to small angles and distances (for linearity)



15Chapter 3


Lateral Dynamics

• Single track model

– Mean wheel loads at frontand rear axle

– Slip angles at frontand rear wheels



16Chapter 3


Lateral Dynamics

• Single track model - Variables



17Chapter 3


Lateral Dynamics

Single track model

• Equations of motion



18Chapter 3


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



19Chapter 3


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



20Chapter 3


Lateral Dynamics

Yaw eigen frequency: example for typical parameter values

• 2nd ordersystembehavior

Source: Willumeit

slip



21Chapter 3


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



22Chapter 3


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.



23Chapter 3


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−



24Chapter 3


Lateral Dynamics

Typical behavior of side side slip angle over lateral acceleration for steady-state cornering

Source: Willumeit



25Chapter 3


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



26Chapter 3


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



27Chapter 3


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



28Chapter 3


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



29Chapter 3


Lateral Dynamics

Test maneuver: vehicle behavior during Power-off in a turn

• Load change atrear axle

• Increase in yawspeed

Source: ATZ



30Chapter 3


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



31Chapter 3


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



32Chapter 3


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



33Chapter 3


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



34Chapter 3


Vertical Dynamics


• 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



35Chapter 3


Vertical Dynamics


• 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



36Chapter 3


Vertical Dynamics

Simple 3-dimensional vertical dynamics model

• Height, pitch, roll modeling

Source: Willumeit



37Chapter 3


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



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


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



39Chapter 3


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



40Chapter 3


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



41Chapter 3


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



42Chapter 3


Vertical Dynamics

Road surfacecharacteristics z0(t)

• Typical PSDs for roadsurfaces

•Tarmac•Cement•Macadam medium•Cobblestone medium•Dirt road



43Chapter 3


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



44Chapter 3


Vertical Dynamics

Seat rail vertical acceleration

Source: ATZ



45Chapter 3


Vertical Dynamics

Seat rail vertical acceleration: PSD (Power Spectrum Density)

Source: ATZ



46Chapter 3


Suspension

Front axle: McPherson

Transverse Link

• Long. forces: rotationaround front bush

• Lat. forces: radiallystiff front lower controlarm bush



47Chapter 3


Suspension

Rear axle: Multilinksuspension

Rear axle: SLA suspension

• Maximum loadcompartment:SLA selected for wagon

• Lat. forces:radially stiff front lowercontrol arm bush



48Chapter 3


Suspension

McPherson front axle

Source: Ford



49Chapter 3


Suspension

McPherson front axle: Roll center height

Source: Ford



50Chapter 3


Suspension

Trailing arm SLA rear axle

Source: Ford



51Chapter 3


Suspension

Trailing arm SLA rear axle:Bump steer

Source: Ford



52Chapter 3


Suspension

Multilink rear axle: Roll center height

Source: Ford



53Chapter 3


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



54Chapter 3


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)



55Chapter 3


Spring and Shock

McPherson strut

• Coil spring

• Damper inside spring

• Distributed mass

• Stiffness c=f(y)



56Chapter 3


Spring and shock

Hydropneumatic springCitroën

• “Air spring”

• Force transferby oil

• Height controlby oilvolume change

• Damping withnitrogenreservoir

• Spring elementsconnected to arms

Cylinder

Nitrogen

Damper



57Chapter 3


Spring and shock

Hydro pneumatic spring

• Function



58Chapter 3


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)



59Chapter 3


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



60Chapter 3


Spring and Shock

Damper dynamics

• Measurements

• Bode diagram ideal vs. Maxwell model behavior

Measured

Maxwell

Ideal



61Chapter 3


Spring and Shock

Damper realization

• Monotube andtwintube

Dynamic Damper Properties

• Hysteresis

– Different forces injounce and rebound

• Friction forces

• Cavitation

• Reduced dampingat higherfrequencies

Twintube Monotube



62Chapter 3


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



63Chapter 3


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



64Chapter 3


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



65Chapter 3


Steering

Driver demands

• Reasonable SteeringWheel Torque (SWT)

• Sensitivity

• Auto Return

• Passive Safety

Vehicle demands

• Steeringkinematics

• Package

• Cost

Source: RWTH Aachen, Kraftfahrzeugtechnik



66Chapter 3


Steering

Steering rack realization

• Different gearmechanisms

• Gear rackshowsadvantages

Source: RWTH Aachen, KraftfahrzeugtechnikSource: RWTH Aachen, KraftfahrzeugtechnikSource: RWTH Aachen, Kraftfahrzeugtechnik



67Chapter 3


Steering

Steering rack realization

• Tie-rods connectedvia ball joints

• Attachment pointson the outside (a)or in the middle (b)part


Tie-rods



68Chapter 3


Steering

Power Assist

• Hydraulic assist

• Open-centervalve withtorsion bar

• Constantflow pump




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



70Chapter 3


• 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



71Chapter 3


• 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

Date post:	30-Jul-2018
Category:	Documents
Upload:	lammien
View:	226 times
Download:	6 times

VeD Chapter3 Lecture - uni-due.de · Lecture: Vehicle Dynamics Tutor: T. Wey Date: ... contribution...

Documents