MECA0525 : Vehicle dynamics

Pierre DuysinxResearch Center in Sustainable Automotive

Technologies of University of Liege

Academic Year 2020-2021

1

Lesson 1: Steady State Cornering

2

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.

3

INTRODUCTION TO HANDLING

4

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

5

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

6

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 closed loop system vehicle + driver

◼ Driver = intelligence

◼ Vehicle = plant system creating the manoeuvring forces

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

7

Introduction

Model of the system vehicle + driver

8

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

9

AXES SYSTEM

10

Reference frames

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

XY Z

O

Inertial coordinate system OXYZ

11

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

12

Reference frames

z

x

y x

Système SAE

Système ISO

x z

yComparison of conventions of SAE and ISO/DIN reference frames

13

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 x axis (roll speed)

◼ q : rotation speed about y (pitch)

◼ r : rotation speed about z (yaw)

14

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

15

VEHICLE MODELING

16

The bicycle model

◼ When the behaviours of the left and right hand wheels are not that much 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

17

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

18

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

19

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 ground 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.

20

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 about z axis

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

21

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

22

LOW SPEED TURNING

23

Low speed turning

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

◼ The turning is ruled by the rolling conditions 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 have to pass through the same point, that is the centre of the turn.

24

Ackerman-Jeantaud theory

25

Ackerman-Jeantaud condition

◼ One can see that

◼ This gives the Ackerman Jeantaud condition

◼ Corollary

26

Ackerman-Jeantaud condition

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

27

Genta Fig. 5.2

Jeantaud condition

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

◼ Actually, 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

28

Wong Fig. 5.2

Wong Fig. 5.4

Ackerman theory

29

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°

30

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

31

Ackerman theory

◼ The off-tracking of the rear wheel set

32

HIGH SPEED STEADY STATE CORNERING

33

High speed steady state cornering

◼ At high speed, the tires have todevelop 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 kinematics of the motion, the IC is located at the intersection of the normal lines to the local velocity vectors under the tires.

◼ The IC, which was located at the rear axel for low-speed turn, is now moving to a point in front.

34

High speed steady state cornering

− f

f r

v f

v r

35

Sideslip angles have been assumed to be on left side of the wheel. We consider the modulus of .

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)

e J x y = 0

e t J y z = 0

36

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

◼ Inertia tensor

37

e J x y = 0

e t J y z = 0

Dynamics equations of the vehicle motion

◼ It comes

◼ And finally

◼ The only nontrivial equations are

38

Dynamics equations of the vehicle motion

◼ Circular motion

◼ wz: rotation speed about vertical axis

◼ V tangent velocity

◼ R radius of the turn

◼ Steady state

◼ Equation of motion

39

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

40

Equilibrium equations of the vehicle

◼ Solving

◼ Can be made by using Horner rule

41

Behaviour equations of the tires

◼ Cornering force for small slip angles

Gillespie, Fig. 6.242

Gratzmüller equality

◼ Using the equilibrium and the behaviour condition, one gets

◼ One yields the Gratzmüller equality

43

Compatibility equations

◼ Compatibility equation consists in evaluating the side slip angles in terms of the velocities

44

Because of assumption r<0!

Because of steering action !

Compatibility equations

◼ Evaluation of velocities under front and rear axles thanks to the Poisson transport equation

45

Compatibility equations

◼ The velocity under the rear wheels are given by

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

46

Compatibility equations

◼ The velocity under the front wheels are given by

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

47

Steering angle

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

◼ This gives relation between the steering angles and the

48

Correction due to side slipAckerman angle

Steering angle

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

◼ Let’s insert the expression of the side slip angles in terms of lateral forces and cornering stiffness

49

Steering angle

◼ The expression of the steering angle as a function of the slip angles under front and rear wheels

◼ Inserting the values of the side slip angles as a function of the velocity and the cornering stiffness of the wheels sets yields

◼ Or

◼ with

50

Understeer gradient

◼ The steering angle is expressed in terms of the centrifugal acceleration

◼ So

◼ With the understeer gradient K of the vehicle

51

Steering angle as a function of V

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

52

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

53

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

54

Lateral acceleration and yaw speed gains

◼ Lateral acceleration gain

◼ Yaw speed gain

55

Lateral acceleration gain

◼ Purpose of the steering system is to produce lateral acceleration

◼ For neutral steer, K=0 and the lateral acceleration gain is increasing constantly with the square of the speed : V²/L

◼ 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 perturbation produces an infinite lateral acceleration

56

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 and increases with the speed (slope 1/L)

◼ 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.

57

Yaw velocity gain

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

58

Sideslip angle at centre of mass

◼ Definition (reminder)

◼ Value

◼ Value as a function of the speed V

◼ Becomes zero for the speed

independent of R !

59

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

60

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

61

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 perturbation lateral forces F do not produce any steady-state yaw velocity

◼ That is:

62

Static margin

◼ It comes

◼ So the static margin writes

◼ A vehicle is

◼ Neutral steer if e = 0

◼ Under steer (K>0) if e<0 (behind the CG)

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

◼ Remember that

63

Static margin

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

64

Exercise

◼ Let a vehicle A with the following characteristics:

◼ Wheelbase 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 :

◼ Wheelbase 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

65

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

66

Exercise

◼ Compute:

◼ The 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 (%)

67

Exercise 1

◼ Data

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

2228,0=L

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=

68

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 = =69

Exercise 1

◼ Side slip angles under the front tires

²4992,91yf

c VF m N

L R= =

²/4893,4²

smR

Va

y== Nma

y1883,6424=

yfffFC =

3110 / degfC N =

radC

F

f

yf

f0281,06106,1 ===

70

Exercise 1

◼ Side slip angles under the rear tires

◼ Side slip angle at CG

²1431,3092yr

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 = − = − = −

71

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= =

72

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= = = =

−73

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

= = = =

74

Exercise 1

◼ Neutral maneuver point

◼ Static margin

75