MECA0492 : Vehicle dynamics

Pierre DuysinxResearch Center in Sustainable Automotive

Technologies of University of Liege

Academic Year 2017-2018

1

Bibliography

T. Gillespie. « Fundamentals of vehicle Dynamics », 1992, Society of Automotive Engineers (SAE)

W. Milliken & D. Milliken. « Race Car Vehicle Dynamics », 1995, Society of Automotive Engineers (SAE)

R. Bosch. « Automotive Handbook ». 5th edition. 2002. Society of Automotive Engineers (SAE)

J.Y. Wong. « Theory of Ground Vehicles ». John Wiley & sons. 1993 (2nd edition) 2001 (3rd edition).

M. Blundel & D. Harty. « The multibody Systems Approach to Vehicle Dynamics » 2004. Society of Automotive Engineers (SAE)

G. Genta. «Motor vehicle dynamics: Modelling and Simulation ». Series on Advances in Mathematics for Applied Sciences - Vol. 43. World Scientific. 1997.

2

INTRODUCTION TO HANDLING

3

Introduction to vehicle dynamics

Introduction to vehicle handling

Vehicle axes system

Tire mechanics & cornering properties of tires

Terminology and axis system

Lateral forces and sideslip angles

Aligning moment

Single track model

Low speed cornering

Ackerman theory

Ackerman-Jeantaud theory

4

Introduction to vehicle dynamics

High speed steady state cornering

Equilibrium equations of the vehicle

Gratzmüller equality

Compatibility equations

Steering angle as a function of the speed

Neutral, understeer and oversteer behaviour

Critical and characteristic speeds

Lateral acceleration gain and yaw speed gain

Drift angle of the vehicle

Static margin

Exercise

5

Introduction

In the past, but still nowadays, the understeer and oversteercharacter dominated the stability and controllability considerations

This is an important factor, but it is not the sole one…

In practice, one has to consider the whole system vehicle + driver

Driver = intelligence

Vehicle = plant system creating the manoeuvering forces

The behaviour of the closed-loop system is referred as the « handling », which can be roughly understood as the road holding

6

Introduction

Model of the system vehicle + driver

7

Introduction

However because of the difficulty to characterize the driver, it is usual to study the vehicle alone as an open loop system.

Open loop refers to the vehicle responses with respect to specific steering inputs. It is more precisely defined as the ‘directional response’ behaviour.

The most commonly used measure of open-loop response is the understeer gradient

The understeer gradient is a performance measure under steady-state conditions although it is also used to infer performance properties under non steady state conditions

8

AXES SYSTEM

9

Reference frames

Local reference frame oxyz attached to the vehicle body -SAE (Gillespie, fig. 1.4)

XY Z

O

Inertial coordinate system OXYZ

10

Reference frames

Inertial reference frame

X direction of initial displacement or reference direction

Y right side travel

Z towards downward vertical direction

Vehicle reference frame (SAE):

x along motion direction and vehicle symmetry plane

z pointing towards the center of the earth

y in the lateral direction on the right hand side of the driver towards the downward vertical direction

o, origin at the center of mass

11

Reference frames

z

x

y x

Système SAE

Système ISO

x z

yComparison of conventions of SAE and ISO/DIN reference frames

12

Local velocity vectors

Vehicle motion is often studied in car-body local systems

u : forward speed (+ if in front)

v : side speed (+ to the right)

w : vertical speed (+ downward)

p : rotation speed about a axis (roll speed)

q : rotation speed about y (pitch)

r : rotation speed about z (yaw)

13

Forces

Forces and moments are accounted positively when acting ontothe vehicle and in the positive direction with respect to the considered frame

Corollary

A positive Fx force propels the vehicle forward

The reaction force Rz of the ground onto the wheels is accounted negatively.

Because of the inconveniency of this definition, the SAEJ670e « Vehicle Dynamics Terminology » names as normal force a force acting downward while vertical forces are referring to upward forces

14

VEHICLE MODELING

15

The bicycle model

When the behaviours of the left and right hand wheels are not too different, one can model the vehicle as a single track vehicle known as the bicycle model or single track model.

The bicycle model proved to be able to account for numerous properties of the dynamic and stability behaviour of vehicle under various conditions.

x,u,p

y,v,qz,w,r

f

Velocity

r

Fyf

Tr

Tf

Fyr

Fxr

Fxf

L

b

c

t

16

The bicycle model

Geometrical data:

Wheel base: L

Distance from front axle to CG: b

Distance from rear axle to CG: c

Track: t

Tire variables

Sideslip angles of the front and rear tires: f and r

Steering angle (of front wheels)

Lateral forces developed under front and rear wheels respectively: Fyf and Fyr.

Longitudinal forces developed under front and rear wheel respectively: Fxf and Fxr.

x,u,p

y,v,qz,w,r

f

Velocity

r

Fyf

Tr

Tf

Fyr

Fxr

Fxf

L

b

c

t

17

The bicycle model

Assumptions of the bicycle model

Negligible lateral load transfer

Negligible longitudinal load transfer

Negligible significant roll and pitch motion

The tires remain in linear regime

Constant forward velocity V

Aerodynamics effects are negligible

Control in position (no matter about the control forces that are required)

No compliance effect of the suspensions and of the body

x,u,p

y,v,qz,w,r

f

Velocity

r

Fyf

Tr

Tf

Fyr

Fxr

Fxf

L

b

c

t

18

The bicycle model

Remarks on the meaning of the assumptions

Linear regime is valid if lateral acceleration<0.4 g

Linear behaviour of the tire

Roll behaviour is negligible

Lateral load transfer is negligible

Small steering and slip angles, etc.

Smooth floor to neglect the suspension motion

Position control of the command : one can exert a given value of the input variable (e.g. steering system) independently of the control forces

The sole input considered here is the steering, but one could also add the braking and the acceleration pedal.

19

The bicycle model

Assumptions :

Fixed: u = V = constant

No vertical motion: w=0

No roll p=0

No pitch q = 0

Bicycle model = 2 dof model :

r=wz, yaw speed

v, lateral velocity or b, side slip of the vehicle

Vehicle parameters:

m, mass,

Jzz inertia bout z axis

L, b, c wheel base and position of the CG

20

The bicycle model

x,u,p

y,v,qz,w,r

f

Velocity

f

Velocity

rr

Fyf

Fyr

Fyf

Tr

Tf

rv

u Vb

Fyr

Fxr Fxr

FxfFxf

L

b

c

h

m, J

21

LOW SPEED TURNING

22

Low speed turning

At low speed (parking manoeuvre for instance), the centrifugal accelerations are negligible and the tire have not to develop lateral forces

The turning is ruled by the rolling without (lateral) friction and without slip conditions

If the wheels experience no slippage, the instantaneous centres of rotation of the four wheels are coincident.

The CIR is located on the perpendicular lines to the tire plan from the contact point

In order that no tire experiences some scrub, the four perpendicular lines pass through the same point, that is the centre of the turn.

23

Ackerman-Jeantaud theory

24

Ackerman-Jeantaud condition

One can see that

This gives the Ackerman Jeantaud condition

Corollary e · i

25

Ackerman-Jeantaud condition

The Jeantaud condition is not always verified by the steering mechanisms in practice, as the four bar linkage mechanism

26

Jeantaud condition

The Jeantaud condition can be determined graphically, but the former drawing is very badly conditioned for a good precision

In practice one resorts to an alternative approach based on the following property

Point Q belongs to the line MF when the Jeantaud condition is fulfilled

The distance from Q to the line MF is a measure of the error from Jeantaud condition

27

Ackerman theory

rv

u Vb

L

R

centre du virage

vf

vr

RCG

28

Ackerman theory

The steering angle of the front wheels

The relation between the Ackerman steering angle and the Jeantaud steering angles 1 and 2

R=10 m, L= 2500 mm, t=1300 mm

1 = 15.090° 2= 13.305°

= 14.142°

(1+2)/2=14.197°

rv

u Vb

L

R

centre du virage

vf

vr

RCG

29

Ackerman theory

Curvature radius at the centre of mass

Relation between the curvature and the steering angle

Side slip b at the centre of mass

30

Ackerman theory

The off-tracking of the rear wheel set

rv

u Vb

L

R

centre du virage

vf

vr

RCG

31

HIGH SPEED STEADY STATE CORNERING

32

High speed steady state cornering

At high speed, it’s required that tires develop lateral forces to sustain the lateral accelerations.

The tire can develop forces if and only if they are subject to a side slip angle.

Because of the motion kinematics, the CIR is located at the intersection to the normal to the local velocity vectors under the tires.

The CIR, which was located at the rear axel for low speed turn, is now moved to a point in front.

33

High speed steady state cornering

f

f r

v f

v r

34

Dynamics equations of the vehicle motion

Newton-Euler equilibrium equation in the non inertial reference frame of the vehicle body

Model with 2 dof b & r

Equilibrium equations in Fy and Mz :

Operating forces

Tyre forces

Aerodynamic forces (can be neglected here)

f

Velocity

r

Fyf

Fyr

rv

u Vb

Fxr

Fxf

L

a

b M, J

e J x y = 0

e t J y z = 0

nt :

Fy = M ( _v + r u)

N = Jzz _r

35

Equilibrium equations of the vehicle

Equilibrium equations in lateral direction and rotation about z axis

Solutions

The lateral forces are in the same ratio as the vertical forces under the wheel sets

36

Behaviour equations of the tires

Cornering force for small slip angles

C ® =

@ F y

@ ®

® = 0

< 0

Gillespie, Fig. 6.237

Gratzmüller equality

Using the equilibrium and the behaviour condition, one gets

One yields the Gratzmüller equality

38

Compatibility equations

The velocity under the rear wheels are given by

The compatibility of the velocities yields the slip angle under the rear wheels

f

f r

v f

v r

39

Compatibility equations

The velocity under the front wheels are given by

The compatibility of the velocities yields the slip angle under the front wheels

f

f r

v f

v r

40

Steering angle

Steering angle as a function of the slip angles under front and rear wheels

The steering angle can also be expressed as a function of the velocity and the cornering stiffness of the wheels sets

41

Understeer gradient

The steering angle is expressed in terms of the centrifugal acceleration

So

With the understeer gradient K of the vehicle

42

Steering angle as a function of V

Gillespie. Fig. 6.5 Modification of the steering angle as a function of the speed

43

Neutralsteer, understeer and oversteer vehicles

If K=0, the vehicle is said to be neutralsteer:

The front and rear wheels sets have the same directional ability

If K>0, the vehicle is understeer :

Larger directional factor of the rear wheels

If K<0, the vehicle is oversteer:

Larger directional factor of the front wheels

44

Characteristic and critical speeds

For an understeer vehicle, the understeer level may be quantified by a parameter known as the characteristic speed. It is the speed that requires a steering angle that is twice the Ackerman angle (turn at V=0)

For an oversteer vehicle, there is a critical speed above which the vehicle will be unstable

45

Lateral acceleration and yaw speed gains

Lateral acceleration gain

Yaw speed gain

46

Lateral acceleration gain

Purpose of the steering system is to produce lateral acceleration

For neutral steer, lateral acceleration gain is constant

For understeer vehicle, K>0, the denominator >1 and the lateral acceleration is reduced with growing speed

For oversteer vehicle, K<0, the denominator is < 1 and becomes zero for the critical speed, which means that any pertubation produces an infinite lateral acceleration

47

Yaw velocity gain

The second raison for steering is to change the heading angle by developing a yaw velocity

For neutral vehicles, the yaw velocity is proportional to the steering angle

For understeer vehicles, the yaw gain angle is lower than proportional. It is maximum for the characteristic speed.

For oversteer vehicles, the yaw rate becomes infinite for the critical speed and the vehicles becomes uncontrollable at critical speed.

48

Yaw velocity gain

Gillespie. Fig. 6.6 Yaw rate as a function of the steering angle

49

Sideslip angle

Definition (reminder)

Value

Value as a function of the speed V

Becomes zero for the speed

independent of R !

50

Sideslip angle

Gillespie. Fig. 6.7 Sideslip angle for a low speed turn

Gillespie. Fig. 6.8 Sideslip angle for a high speed turn

This is true whatever the vehicle is understeer or oversteer

b > 0 b < 0

51

Static margin

The static margin provides another (equivalent) measure of the steady-state behaviour

Gillespie. Fig 6.9 Neutral steer linee>0 if it is located in front of the vehicle centre of gravity

52

Static margin

Suppose the vehicle is in straight line motion (=0)

Let a perturbation force F applied at a distance e from the CG (e>0 if in front of the CG)

Let’s write the equilibrium

The static margin is the point such that the lateral forces do not produce any steady-state yaw velocity

That is:

53

Static margin

It comes

So the static margin writes

A vehicle is

Neutral steer if e = 0

Under steer if e<0 (behind the CG)

Over steer if e>0 (in front of the CG)

54

Static margin

Gillespie. Fig. 6.10 Maurice Olley’s definition of understeer and over steer

55

Exercise

Let a vehicle A with the following characteristics:

Wheel base L=2,522m

Position of CG w.r.t. front axle b=0,562m

Mass=1431 kg

Tires: 205/55 R16 (see Figure)

Radius of the turn R=110 m at speed V=80 kph

Let a vehicle B with the following characteristics :

Wheel base L=2,605m

Position of CG w.r.t. front axle b=1,146m

Mass=1510 kg

Tires: 205/55 R16 (see Figure)

Radius of the turn R=110 m at speed V=80 kph

56

Exercise

Rigidité de dérive (dérive <=2°) :

0

200

400

600

800

1000

1200

1400

1600

1800

0 1000 2000 3000 4000 5000 6000

Charge normale (N)

Rig

idit

é (

N/°

)

175/70 R13

185/70 R13

195/60 R14

165 R13

205/55 R16

57

Exercise

Compute:

The d’Ackerman angle (in °)

The cornering stiffness (N/°) of front and rear wheels and axles

The sideslip angles under front and rear tires (in °)

The side slip of the vehicle at CG (in °)

The steering angle at front wheels (in °)

The understeer gradient (in °/g)

Depending on the case: the characteristic or the critical speed (in kph)

The lateral acceleration gain (in g/°)

The yaw speed velocity gain (in s-1)

The vehicle static margin (%)

58

Exercise 1

Data

mb 562,0 2,522 0,562 1,960c L b m

2228,0L

b0,7772

c

L

kgm 1431

NL

cmgW

f8714,10909 3127,6909r

bW mg N

L

smkphV /2222,2280

²/4893,4²

smR

Va

y mR 110

59

Exercise 1

Ackerman angle

Tire cornering stiffness of front wheels

Tire cornering stiffness of rear wheels

3134,10229,0)0229,0arctan()arctan( radR

L

NL

cmgW

f8714,10909

NFz

93,5454

deg/1550)1( NCf

3110 / deg 177616,98 /fC N N rad

3127,6909r

bW mg N

L

1563,84zF N

(1) 500 / degrC N

1000 / deg 57295,8 /rC N N rad 60

Exercise 1

Side slip angles under the front tires

NR

Vm

L

cFyf 879,54992

²

²/4893,4²

smR

Va

y Nma

y1883,6424

yfffFC

3110 / degfC N

radC

F

f

yf

f0281,06106,1

61

Exercise 1

Side slip angles under the rear tires

Side slip angle at CG

²1431,3092yf

b VF m N

L R

r r yrC F 1000 / degrC N

1,4313 0,0250yr

r

r

Frad

C

rr

R

c

V

crb

1,9600,0250 0,0072 0,4105

110radb

62

Exercise 1

Steering angle at front wheels

Understeer gradient

R

V

C

Lmb

C

Lmc

R

L

rf

²//

f r

L

R

1,3134 1,6106 1,4313 1,4927

/ / 1112,1732 318,8268

3100 1000f r

mc L mb LK

C C

2/0399,0 msK ' * 0,3918 deg/K K g g

63

Exercise 1

Understeer gradient: check!

Characteristic speed

r

r

f

f

gC

W

gC

WK

R

VK

R

L ²3,57

4925,14893,43134,1 K

R

L2

K

LV

carac

²//49639,6deg/0399,0 2 smradEmsK

2,52260,1793 / 216,64

6,9639 4carac

LV m s kph

K E

64

Exercise 1

Lateral acceleration gain

Yaw speed gain

deg/3066,04927,1

81,9/4893,4g

aG y

ay

/ 22,222 /1107,7543 deg/ / deg

1,4927r

r V RG s

65

Exercise 1

Neutral maneuver point

Static margin

10003100

1000.9600,13100.562,0

rf

rf

CC

cCbCe

me 0531,0

%11,2L

e

66