duplo a completo

7/27/2019 Duplo a Completo

1/216

FACULTYOFMECHANICAL,MATERIALSANDAUTOMOTIVEENGINEERING

92420CapstoneII

2007BajaProject Suspension

Submittedby:

Student EmailAddress(NonUwindsor)

WilliamBombardier ([email protected])

AhmadFadel ([email protected])

XiangdongDing ([email protected])

IanFunkenhauser ([email protected])

BrianZuccato ([email protected])

MikeBowie

([email protected])

YeTao ([email protected])

BoHuang ([email protected])

August3rd2007

Submitted To: Dr. Bruce Minaker


2/216

2007 Baja ProjectSuspension

i

1) Table of Contents

1) Table of Contents............................................................................................................. i

2) List of Tables ................................................................................................................. iv

3) List of Figures................................................................................................................. v

4) List of Equations............................................................................................................ ix

5) Nomenclature.................................................................................................................. x

6) Introduction to Suspension Kinematics and Kinetics..................................................... 1

6.1) Suspension Kinetics................................................................................................. 1

6.1.1) Vehicle ride modeling (vertical dynamics) ...................................................... 2

6.1.2) Vehicle handling............................................................................................. 11

6.2) Suspension Kinematics.......................................................................................... 18

6.2.1) Track width and tire scrub.............................................................................. 186.2.2) Instant center and roll center position............................................................. 19

6.2.3) Camber angle.................................................................................................. 21

6.2.4) Caster angle and caster trail............................................................................ 24

6.2.5) Kingpin angle and scrub radius ...................................................................... 25

6.2.6) Toe angle, roll steer and bump steer............................................................... 27

6.2.7) Aligning torque or self centering moment...................................................... 30

6.2.8) Anti-dive/anti-squat........................................................................................ 30

6.2.9) Motion ratio and wheel rate............................................................................ 33

6.2.10) Roll stiffness................................................................................................. 34

6.2.11) Vehicle ride height........................................................................................ 35

6.2.12) Understeering/Oversteering characteristics of vehicle ................................. 35

6.3) Spring rate determination ...................................................................................... 37

7) 2007 Suspension Kinematics........................................................................................ 38

7.1) Choosing the dimensions of the vehicle ................................................................ 38

7.2) Choosing the suspension points............................................................................. 40

7.3) Choosing the suspension geometry angles ............................................................ 41

7.4) Choosing the inner suspension points ................................................................... 427.5) Choosing the steering tie rods lengths................................................................... 47

7.6) Choosing the strut mounting points....................................................................... 48

7.7) Design front and rear suspension to be consistent................................................. 49

8) 2007 Suspension kinetics.............................................................................................. 52

8.1) Handling analysis on 2006 vehicle........................................................................ 52


3/216


ii

8.2) Approach to designing 2007 suspension kinetics.................................................. 53

8.3) 2007 front and rear suspension shocks .................................................................. 54

8.4) The required spring rates based on the Olley criteria............................................ 54

8.5) CarSim model for 2007 vehicle............................................................................. 56

8.6) Necessary combination of Elka Suspensions springs............................................ 64

8.7) Evaluation of spring rate in CarSim ...................................................................... 65

8.8) Ride, bounce, pitch and wheel hop frequencies .................................................... 67

8.9) Prediction of vehicle performance in regards to the dynamic events.................... 72

9) Suspension Component Design.................................................................................... 74

9.1) Choice of Materials........................................................................................... 74

9.2) Front Suspension System.................................................................................. 75

9.2.1) Control Arms ............................................................................................ 75

9.2.2) Finite Element Analysis............................................................................ 769.2.3) Joints ......................................................................................................... 78

9.2.4) Steering tie rod and bump stop ................................................................. 80

9.3) Rear Suspension System........................................................................................ 82

9.3.1) Control Arms .................................................................................................. 82

9.3.2) Finite Element Analysis.................................................................................. 84

9.3.3) Joints............................................................................................................... 84

9.4) Installation ............................................................................................................. 85

10) Shocks (Dampers & Springs) ..................................................................................... 88

10.1) Chosen shocks ..................................................................................................... 88

10.2) Adjustable Damping............................................................................................ 89

10.3) Progressive spring rates ....................................................................................... 90

11) Hubs & Uprights......................................................................................................... 93

11.1) Background & Research...................................................................................... 93

11.2) Concepts & Brainstorming.................................................................................. 93

11.3) CATIA Modeling ................................................................................................ 94

11.4) FEA...................................................................................................................... 95

11.5) Materials & Manufacturing Procedure Used....................................................... 9511.6) Finished Product .................................................................................................. 95

11.6.1) Testing .......................................................................................................... 95

11.7) Recommendations for Improvements.................................................................. 96

12) Tires and Rims............................................................................................................ 97

12.1) Background and Research .................................................................................. 97


4/216


iii

12.2) Concepts and Brainstorming .............................................................................. 97

12.3) CATIA Modeling ............................................................................................. 105

12.4) Additional Analysis .......................................................................................... 105

12.5) Materials and Manufacturing Procedures Used................................................ 106

12.6) Finished Product ............................................................................................... 106

12.6.1) Product Assembly and Maintenance ......................................................... 106

12.6.2) Testing ........................................................................................................ 106

12.7) Recommendations for Improvement ................................................................ 107

13) Suspension tuning and testing .................................................................................. 108

13.1) Suspension kinematics adjustment and measurement ....................................... 108

13.2) Dynamic tuning of the suspension..................................................................... 111

13.3) Problems during testing..................................................................................... 112

14) Strain gage testing .................................................................................................... 11614.1) Background & Research.................................................................................... 116

14.2) Concepts & Brainstorming................................................................................ 116

14.3) CATIA Modeling .............................................................................................. 118

14.4) FEA.................................................................................................................... 119

14.5) Additional Analysis ........................................................................................... 120

14.6) Materials & Manufacturing Procedure Used..................................................... 121

14.7) Recommendations for Improvements................................................................ 121

15) Suspension Prototype ............................................................................................... 122

15.1) Background & Research.................................................................................... 122

15.2) Concepts & Brainstorming................................................................................ 122

15.3) ADAMS Modeling ............................................................................................ 123

15.4) Additional Analysis ........................................................................................... 127

15.5) CATIA & FEA .................................................................................................. 127

15.6) Materials & Manufacturing Procedure Used..................................................... 129

15.7) Finished Product ................................................................................................ 130

15.7.1) Product Assembly & Maintenance ............................................................. 130

15.7.2) Testing ........................................................................................................ 13215.8) Recommendations for Improvements................................................................ 133

16) References and contacts............................................................................................ 139

16.1) Contacts ............................................................................................................. 139

16.2) Websites............................................................................................................. 140

16.3) Books and professional papers .......................................................................... 141


5/216


iv

17) Appendixes ............................................................................................................... 143

17.1) Appendix A........................................................................................................ 143

The derivation of the half car model ....................................................................... 143

17.2) Appendix B........................................................................................................ 145

The derivation of the bicycle model ........................................................................ 145

17.3) Appendix C........................................................................................................ 148

17.4) Appendix D........................................................................................................ 154

Critical speed calculations of 2006 vehicle ............................................................ 154

17.5) Appendix E ........................................................................................................ 156

17.6) Appendix F ........................................................................................................ 157

Predicted spring rates............................................................................................. 157

17.7) Appendix G........................................................................................................ 158

Acceleration Plots ................................................................................................... 158Acceleration and Cornering ................................................................................... 160

Braking.................................................................................................................... 161

Braking and Cornering ........................................................................................... 162

S Shaped Plots......................................................................................................... 163

2007 Jump Performance ......................................................................................... 165

Cornering ................................................................................................................ 166

17.8) Appendix H........................................................................................................ 167

17.9) Appendix I ......................................................................................................... 180

17.10) Appendix J....................................................................................................... 193

Spreadsheets to record the data during testing........................................................ 193

17.11) Appendix K...................................................................................................... 197

17.11.1) Rear suspension assembly Bill of Material: ................................................. 197

17.11.2) Front suspension assembly Bill of Material ................................................. 201

17.12) Appendix L...................................................................................................... 205

2) List of Tables

Table 1: Summary of vehicle dimensions......................................................................... 40

Table 2: Static Suspension Angles.................................................................................... 42

Table 3: Estimated cornering stiffness of the 2006 tires................................................... 52

Table 4: Critical speed of 2006 vehicle ............................................................................ 52

Table 5: Weight of the vehicle and weight distribution.................................................... 54


6/216


v

Table 6: Ride frequencies of the 2005 vehicle.................................................................. 55

Table 7: Required spring rates for 2007 vehicle based on Olley criteria.......................... 55

Table 8: The main and auxiliary springs required to obtained the appropriate ride

frequencies ................................................................................................................ 64

Table 9: Spring rate evaluation results.............................................................................. 66

Table 10: The frequencies of the vehicle.......................................................................... 67

Table 11: Summary of material properties ....................................................................... 74

Table 12: 2003 Testing Data............................................................................................. 97

Table 13: Tire Pressure ................................................................................................... 107

3) List of Figures

Figure 1: Vehicle axis system............................................................................................. 2Figure 2: The quarter car model.......................................................................................... 3

Figure 3: Bounce/pitch model............................................................................................. 5

Figure 4: The half car model............................................................................................... 8

Figure 5: The front and the rear suspension amplitudes as a function of time ................... 9

Figure 6: Eigenvalues verses vehicle speed for an understeering vehicle........................ 15

Figure 7: Oversteering and Understeering Vehicle .......................................................... 16

Figure 8: The lateral force verses the slip angle ............................................................... 17

Figure 9: Vehicle track width ........................................................................................... 19

Figure 10: The roll axis of the vehicle .............................................................................. 19

Figure 11: The effect of the jacking forces....................................................................... 20

Figure 12: Roll center position of a double A-arm type of suspension ............................ 21

Figure 13: Definition of camber angle (note in the figure one is looking at the vehicle

from the front)........................................................................................................... 22

Figure 14: The effect camber has on the tire contact patch .............................................. 22

Figure 15: The effect of the camber angle on the cornering curve................................... 23

Figure 16: Caster angle and caster trail............................................................................. 24

Figure 17: Kingpin angle (steering inclination angle) and scrub radius........................... 26Figure 18: Toe angle (note the view in the figure is the top view)................................... 27

Figure 19: The necessary steps to locate the tie rod position to have no toe angle change

with suspension travel............................................................................................... 29

Figure 20: The pitch center ............................................................................................... 31

Figure 21: Anti-dive suspension geometry....................................................................... 32


7/216


vi

Figure 22: Anti-squat suspension geometry ..................................................................... 32

Figure 23: Motion ratio..................................................................................................... 33

Figure 24: The lateral force verses the vertical force for a given slip angle..................... 36

Figure 25: Vehicle Dimensions ........................................................................................ 38

Figure 26: Rear end of vehicle.......................................................................................... 39

Figure 27: Front and Rear Uprights .................................................................................. 40

Figure 28: Wheel hub........................................................................................................ 41

Figure 29: ADAMS/Car suspension modeling ................................................................. 43

Figure 30: Anti Squat Angle ............................................................................................. 44

Figure 31: Anti Squat Reaction......................................................................................... 44

Figure 32: Longitudinal wheel travel................................................................................ 45

Figure 33: Roll Center Height and Swing Arm Length.................................................... 46

Figure 34: Camber Gain.................................................................................................... 46Figure 35: Steering tie rod length ..................................................................................... 47

Figure 36: Tie rod clearance with control arm.................................................................. 48

Figure 37: Motion Ratio.................................................................................................... 48

Figure 38: Roll Center Lateral Position............................................................................ 50

Figure 39: Roll Center Vertical Position .......................................................................... 50

Figure 40: Roll Stiffness ................................................................................................... 51

Figure 41: Track Width Change ....................................................................................... 51

Figure 42: The three interfaces in CarSim........................................................................ 57

Figure 43: Vehicle model in CarSim ................................................................................ 57

Figure 44: The mass, Inertia and vehicle dimensions screen in CarSim .......................... 58

Figure 45: The powertrain model in CarSim .................................................................... 59

Figure 46: The brake model in CarSim............................................................................. 60

Figure 47: The steering model in CarSim......................................................................... 61

Figure 48: The front suspension kinematics model in CarSim......................................... 62

Figure 49: The front suspension compliance model in CarSim........................................ 63

Figure 50: Motion amplitude ratio for front excitation..................................................... 68

Figure 51: Pitch/Excitation amplitude ratio for front excitation....................................... 68Figure 52: Motion of the rear unsprung mass/excitation amplitude for front excitation.. 69

Figure 53: Motion of the front unsprung mass/excitation amplitude for front excitation 69

Figure 54: Motion amplitude ratio for rear excitation ...................................................... 70

Figure 55: Pitch/Excitation amplitude ratio for rear excitation ........................................ 70

Figure 56: Motion of the front unsprung mass/excitation amplitude for rear excitation.. 71


8/216


vii

Figure 57: Motion of the rear unsprung mass/excitation amplitude for rear excitation ... 71

Figure 58: The hill created to simulate the hill climb....................................................... 72

Figure 59: Suspension and traction course ....................................................................... 73

Figure 60: Front lower control arm................................................................................... 75

Figure 61: Front upper control arm................................................................................... 76

Figure 62: Front lower control arm FEA .......................................................................... 76

Figure 63: Front upright FEA ........................................................................................... 77

Figure 64: Front suspension assembly FEA ..................................................................... 78

Figure 65: Laser cut tabs................................................................................................... 78

Figure 66: Pivot joint construction ................................................................................... 79

Figure 67: Caster adjustment mechanism......................................................................... 80

Figure 68: Camber adjustment mechanism....................................................................... 80

Figure 69: Steering tie rod ................................................................................................ 81Figure 70: Steering stop.................................................................................................... 81

Figure 71: Schematic of rear lower control arm............................................................... 82

Figure 72: Rear control arms ............................................................................................ 83

Figure 73: Aluminum rear upper control arm................................................................... 83

Figure 74: Rear suspension assmebly FEA ...................................................................... 84

Figure 75: Hiem joint........................................................................................................ 85

Figure 76: Upright to control arm pivot............................................................................ 85

Figure 77: Front control assembly .................................................................................... 86

Figure 78: Rear control assembly ..................................................................................... 87

Figure 79: Elka Suspensions coil over shock ................................................................... 88

Figure 80: Rebound and compression damping adjustment ............................................. 89

Figure 81: Suspension springs with the crossovers .......................................................... 91

Figure 82: Load versus displacement of Elka Suspension with longer sides of collars

facing up.................................................................................................................... 92

Figure 83: Load versus displacement of Elka Suspension with shorter sides of collars

facing up.................................................................................................................... 92

Figure 84: Final Catia model ............................................................................................ 94Figure 85: Rear Assembly FEA........................................................................................ 95

Figure 86: Proposed test setup .......................................................................................... 96

Figure 87: Tire internal cord scenarios ............................................................................ 98

Figure 88 Tire contact patch reactions.............................................................................. 98

Figure 89 Contact patch aligning moment........................................................................ 99


9/216


viii

Figure 90 Internal Pressure Model.................................................................................... 99

Figure 91 Lateral force, traction force affect on slip %.................................................... 99

Figure 92 Aligning moment for vertical loads and slip angles....................................... 100

Figure 93 Lateral forces for brake forces at different slip angles ................................... 101

Figure 94 Lateral forces and aligning moments for different traction forces ................. 101

Figure 95 Vertical and longitudinal reactions for tire roll over a bump ......................... 103

Figure 96 Tire natural frequency vibration modes ......................................................... 103

Figure 97 Rolling loss factor graph ................................................................................ 104

Figure 98 CATIA Model of Rim .................................................................................... 105

Figure 99 CATIA Model of Rear Suspension Assembly ............................................... 106

Figure 100: Caster angle measurement........................................................................... 109

Figure 101: Toe angle measurement............................................................................... 110

Figure 102: Camber angle measurement ........................................................................ 111Figure 103: Track with measurement ............................................................................. 112

Figure 104: The protection layer on the control arms..................................................... 113

Figure 105: The bend in the control arm ........................................................................ 113

Figure 106: Angle iron to reinforce the rear control arms.............................................. 114

Figure 107: The wear in the bushings............................................................................. 114

Figure 108: Timken tapered needle roller bearings ........................................................ 115

Figure 109: Strain gauge testing specimen ..................................................................... 117

Figure 110: Bending of test specimen ............................................................................ 117

Figure 111: Axial test on specimen ............................................................................... 118

Figure 112: Specimen modeled in Catia......................................................................... 118

Figure 113: 2004 lower control arm model .................................................................... 118

Figure 114: 2004 lower control arm FEA for 500 lb loading......................................... 119

Figure 115: Cantilever FEA simulation.......................................................................... 119

Figure 116: Axial FEA simulation.................................................................................. 120

Figure 117: 2007 control arm gauging locations ............................................................ 120

Figure 118: Tailing arm and Semi trailing arm .............................................................. 122

Figure 119: Semi trailing arm......................................................................................... 123Figure 120: Tailing arm .................................................................................................. 123

Figure 121: New semi trailing arm................................................................................. 124

Figure 122: Camber angle comparison........................................................................... 124

Figure 123: Roll centre comparison................................................................................ 125

Figure 124: Toe angle comparison ................................................................................. 125


10/216


ix

Figure 125: Anti Squat comparison ................................................................................ 126

Figure 126: Wheel travel track comparison.................................................................... 126

Figure 127: Prototype Suspension Assembly 1 .............................................................. 128

Figure 128: Prototype Suspension Assembly 2 .............................................................. 128

Figure 129: Rear Lower Control Arm FEA.................................................................... 129

Figure 130: Rear Upper Control Arm FEA .................................................................... 129

Figure 131: Prototype front view.................................................................................... 130

Figure 132: Prototype back view .................................................................................... 130

Figure 133: Prototype top view....................................................................................... 131

Figure 134: prototype side view ..................................................................................... 131

Figure 135: Joint and axis control................................................................................... 132

Figure 136: Camber checking 1...................................................................................... 132

Figure 137: Camber checking 2...................................................................................... 133Figure 138: Semi trailing arm 1 ...................................................................................... 134

Figure 139: Semi trailing arm 2 ...................................................................................... 134

Figure 140: Tailing arm 1 ............................................................................................... 135

Figure 141: Tailing arm 2 ............................................................................................... 135

Figure 142: New Semi trailing arm 1 ............................................................................. 136

Figure 143: New Semi trailing arm 2 ............................................................................. 136

Figure 144: Other suspension 1 ...................................................................................... 137



4) List of Equations

Equation 1: The equations of the quarter car model ........................................................... 2

Equation 2: The natural frequencies of the unsprung and sprung mass ............................. 3

Equation 3: The natural frequency of the both the unsprung and sprung mass in hertz..... 4

Equation 4: The amplitudes of displacements of both masses (unsprung and sprung) ...... 4Equation 5: Bounce and pitch equations of motion (neglecting damping)......................... 5

Equation 6: Motion ratios at each of the natural frequency................................................ 5

Equation 7: Natural frequencies in bounce and in pitch ..................................................... 6

Equation 8: Equations of motion in bounce and pitch........................................................ 6

Equation 9: Bounce and pitch damped natural frequency .................................................. 7


11/216


x

Equation 10: The half car model equations ........................................................................ 8

Equation 11: Sprung and unsprung mass.......................................................................... 10

Equation 12: The equations used in the bicycle model..................................................... 11

Equation 13: Equations of motion in steady state cornering ............................................ 12

Equation 14: Vehicle yaw rate as a function of the steering angle................................... 12

Equation 15: The cornering radius as a function of the kinematic cornering radius ........ 12

Equation 16: The kinematic turning radius....................................................................... 12

Equation 17: Body slip angle............................................................................................ 13

Equation 18: The body slip angle as a function of the steering angle .............................. 13

Equation 19: The limit of the /ratio for an understeering vehicle................................ 13

Equation 20: Critical speed of an oversteering vehicle .................................................... 14

Equation 21: Characteristic speed of an understeering vehicle ........................................ 14

Equation 22: Solution to the transients associated with the bicycle model ...................... 15Equation 23: Magic tire Formula...................................................................................... 17

Equation 24: Tire cornering stiffness................................................................................ 18

Equation 25: Condition for proper Ackermann steering................................................... 29

Equation 26: Aligning moment......................................................................................... 30

Equation 27 : Wheel rate................................................................................................... 34

Equation 28: Roll stiffness as a function of ride rate........................................................ 34

Equation 29: 3 cases to determine whether the vehicle will oversteer or understeer based

on the bicycle model ................................................................................................. 35

Equation 30: Ride frequency ............................................................................................ 37

Equation 31: The spring rate of 4 springs in series........................................................... 64

Equation 32: Caster angle from measurements .............................................................. 108

Equation 33: Toe angle measurement............................................................................. 110

5) NomenclatureR Actual cornering radius

0R Low speed cornering radius (kinematic), obtained when cornering without lateral

slip.

m Mass of the vehicler Yaw rater& Rate of change of vehicle yaw rate

u Vehicles forward velocity

a Distance between the center of mass and the front axle

b Distance between the center of mass and the rear axle

fC The cornering stiffness of both of the front tires

rC The cornering stiffness of both of the rear tires


12/216


xi

f Front tire slip angle

r Rear tire slip angle

zF Normal load at the tire

yF Lateral Force at the tire

yz Lateral force coefficient

Camber angle

C Cornering stiffness

E Tire belt compression modulus

tb Tire belt thickness

w Tire belt width

tr Rim radius

s Sidewall vertical deflection when loaded (unitized percent)

ta Tire aspect ratio (height/width)

lata Lateral acceleration

Steering angle

* Limit steering angle (based on a lateral acceleration of 0.5gs)v Lateral velocity

v& Rate of change of the lateral velocity

I Yaw inertia

Body slip angle

fF Lateral force on both of the front tires

rF Lateral force on both of the rear tires

x The coordinate direction from the center of gravity to the front of the car. Also

this coordinate rotates with the vehicle, rotating frame of reference.

y The coordinate direction from the center of gravity to the side of the vehicle (the

lateral direction). Also this coordinate rotates with the vehicle, rotating frame of

reference.tu Forward velocity of the tire

tv Lateral velocity of the tire

f Front tire slip angle

r Rear tire slip angle

fvt Lateral velocity of the front tire

trv Lateral velocity of the rear tire

frut Forward velocity of the front right tire

flut Forward velocity of the front left tire

wt Vehicle width

t Time

sm Meters per second

2s

m Meters per second squared

g Gravity Constant (9.81m/s^2)

deg Degree

srad Radians/second


13/216


xii

ssec, Second

m Meter

X Displacement of the vehicle in the x direction (forward)Y Displacement of the vehicle in the y direction (lateral)

m Mass of the vehicle

a Distance between the center of mass and the front axleb Distance between the center of mass and the rear axle

I Pitch inertia

t Time

fk Front suspension spring constant (for both of the front suspensions)

rk Rear suspension spring constant (for both of the rear suspensions)

fC Front damping coefficient (for both of the front suspensions)

rC Rear damping coefficient (for both of the rear suspensions)

tfk Front tire spring constant (for both of the front tires)

trk Rear tire spring constant (for both of the rear tires)

sm Sprung massum Unsprung mass

ufm Portion of the unsprung mass associated with the front of the vehicle

urm Portion of the unsprung mass associated with the rear of the vehicle

yr Radius of gyration in pitch

sZ Vertical motion of the vehicle body

Vehicle pitch motion

fZ Vertical motion associated with the unsprung mass at the front of the vehicle

rZ Vertical motion associated with the unsprung mass at the rear of the vehicle

fh Disturbance (excitation) motion at the front of the vehicle

rh Disturbance (excitation) motion at the rear of the vehicle

n Natural frequencyFrequency of excitation

Damping ratio

1f Approximate body motion frequency

2f Approximate wheel hop frequency

2s

m Meters per second squared

deg Degree

srad Radians/second

ssec, Second

m Meter

zH Hertz

lb Pound

in Inches

mN Newtons per meter


14/216


xiii

sm

N Newtons per meter per second


15/216


1

6) Introduction to Suspension Kinematics and Kinetics

Vehicle dynamics is the study of all forms of transportation (trains, airplanes,

boats, and automobiles). However vehicle dynamics as we know it is the study of the

performance of the automobile in all of its motions (ride, acceleration, cornering, and

baking). The vehicles suspension plays a key roll in each of these motions. The study ofa vehicles suspension can be broken into two major categories: suspension kinetics and

suspension kinematics. Suspension kinetics is a dynamic and a vibration analysis on the

vehicle and suspension systems. Suspension kinematics involves analyzing the motion ofthe tires as the suspension compresses and extends. Each of these two divisions will be

analyzed in depth in the following sections.

6.1) Suspension Kinetics

Suspension kinetics is an analysis that is important to the overall performance of

the vehicle because it is what determines if the vehicle is capable of absorbing ground

loads; it is what judges the comfort of the driver, it is what determines if the vehicle willroll or not; and it is what determines the resonant frequency of the chassis, the shock and

the tire; it is what determines the handling performance of the vehicle. The vehicle will

see a wide range of vibrations because of the speeds it travels and the boundaries it

travels on, thus it is important to analyze the resonant frequency of the suspensioncomponents and the chassis. The ride quality (or vertical dynamics) of a vehicle can be

analyzed using the half car model. The handling performance of the vehicle can be

analyzed using the bicycle model. However before each of these models are considered

it is important to define the vehicle axis and the appropriate rotations about each of theaxis.

The conventional axis system is placed at the center of mass of the vehicle withthe x axis pointing towards the front of the vehicle, the y axis pointing towards the right

side of the vehicle, and the z axis pointing towards the bottom of the vehicle. The x axis

is known as the longitudinal axis, the y axis is known as the lateral axis, and the z axis isknown as the vertical axis. The rotation about the x axis is know as roll, the rotation

about the y axis is known as pitch and the rotation about the z axis is known as yaw

(Figure 1: Vehicle axis system).

Vehicle ride modeling is the study of the motions transmitted to the vehicle

chassis, and thus the motions felt by the passengers in the vehicle. The motionstransmitted to the vehicle chassis come from the vibration of the suspension as it absorbsthe motion coming from the disturbance at the ground. It is these vibrations that cause

the passengers to feel uncomfortable when they are riding in a vehicle. Therefore,

vehicle ride problems arise from the vibrations of the vehicle body (chassis). One of themain objectives of the suspension system is to control the vibrations of the vehicle body

in order to provide a comfortable ride for the driver.


16/216


2

Figure 1: Vehicle axis system

6.1.1) Vehicle ride modeling (vertical dynamics)

Mechanical vibrations in a vehicle represent a very complex field, and usually

require multiple degrees of freedom to accurately predict the vertical performance of the

vehicle. However, there exist two simplified models which when combined give anaccurate approximation as to the ride quality of the vehicle. These include the quarter car

model (corner model) (used to predict the motion of a single suspension unit) and the

bounce/pitch model (used to predict the motions of the sprung mass of the vehicle).

These models combined produce the half car model (four degrees of freedom model).The vertical performance of the vehicle is directly linked to the sprung mass, the

unsprung mass, the pitch inertia, the suspension stiffness, the tire stiffness, the damping

in the tires, the damping in the suspension units, and the excitation frequency. Before the

half car model is introduced, the quarter car model and the bounce/pitch models will beintroduced.

The quarter car model is a model that models the motion of a single suspension

system (it models one corner of the car) (Figure 2: The quarter car model). The sprung

mass in this model represents some portion of the total sprung mass of the system. Thetire is excited because of the shape of the path it is following (the shape is not flat,

especially for an off road track). Applying Newtons 2nd

law of motion the equations of

motion that govern the quarter car model are as follows (Equation 1: The equations of the

quarter car model).

( ) ( )( ) ( ) 00221212221211 0

zkzCzkzCzzkzzCzmzzkzzCzm

ttttssu

sss

+=++++=++

&&&&&&

&&&&

Equation 1: The equations of the quarter car model


17/216


3

Figure 2: The quarter car model

This is a two degree of freedom system, thus there will be two natural frequencies (the

unsprung and sprung mass will each have a resonant frequency). The wheel hop

frequency is the frequency associated with the unsprung mass it is usually around 10Hz.The body motion frequency is the frequency associated with the sprung mass and it is

usually around 1 to 1.25 Hz. Note, the damping ratios in most suspension systems is

relatively low, therefore the majority of the time the undamped natural frequency will bereally close to the damped natural frequency thus the damped natural frequency is usually

calculated by neglecting any damping in the system. The following equation can be usedto calculate the natural frequencies of the system. Note the natural frequencies are

calculated by neglecting damping in the system and neglecting any excitations (Equation2: The natural frequencies of the unsprung and sprung mass).

[ ] 0det0

0

0

0

2

2

1

2

1

=

=

+

+

Mk

z

z

kkk

kk

z

z

m

m

tss

ss

u

s

&&

&&

Equation 2: The natural frequencies of the unsprung and sprung mass

Note the above equation in matrix form leads to an eigenvalue problem by assuming thedisplacement of each mass to be harmonic (z = Zcos(t)). By solving the determinant

will lead to the natural frequencies of both masses in the system. The frequencies can beapproximated by the following equations (Equation 3: The natural frequency of the both

the unsprung and sprung mass in hertz).


18/216


4

hop)(wheel2

1

motion)(body2

1

2

1

u

ts

s

ts

ts

m

kk

f

m

kk

kk

f

+

=

+=

Equation 3: The natural frequency of the both the unsprung and sprung mass in hertz

Some important observations can be made by solving the above equations. The first

observation is that the sprung mass is well isolated at high frequency, however it will be

poorly isolated at low frequencies, and in some cases, at low frequencies the amplitude of

the sprung mass can be amplified in such a way that it is greater than that of theexcitation amplitude.

Damping will have an effect on the amplitudes of motion even though it does not

have a significant effect on the natural frequencies. The easiest way to solve for theamplitudes is to use a complex number approach (assume z = Ze

it). If this is taken into

consideration the equations of motion will be as follows (Equation 4: The amplitudes ofdisplacements of both masses (unsprung and sprung)).

[ ]

[ ] 012

2

1

0

2

12

0

0

Ztik

kCiMz

z

eZtik

ez

zkCiM

t

ti

t

ti

+++=

+=

++

Equation 4: The amplitudes of displacements of both masses (unsprung and sprung)

Note, the result will be a complex number because of the phase lag between the motion

and the disturbance (this is because of the damping in the system, note the i term next to

the C in the equation of motion above). The amplitude is simply the sum of the squaresfo the real and imaginary parts of the answer obtain from the above equation

( 22 imaginaryrealZ += ). The usual way to solve the equations to obtain the

amplitudes is to assume the excitation is one, and calculate the amplitudes of theunsprung and sprung mass with respect to this input over a wide range of frequencies.

This will allow the amplitude ratios to be obtained over a wide range of frequencies. The

difference between the motion of the sprung and unsprung mass represents the

suspension shock travel, and the distance between the travel of the unsprung mass and theexcitation is the tire deflection. Tire deflection is a measure of handling because it is the

normal force that generates the necessary friction to propel the vehicle forward (ie if the

normal force is fluctuating up and down the tire is being prevented from griping theroad). Therefore, it can be seen that a stiffer suspension will hurt the tires capability from

gripping the road. The unsprung mass has almost no effect at low frequencies, but at

higher frequencies a lower unsprung mass will lead to lower tire deflections and thusbetter handling performance of the vehicle. At mid range frequencies, a lower spring rate


19/216


5

leads to a reduction in tire deflections, and thus improves tire grip. However, a lower

spring rate allows for increased body motions which are detrimental to vehicle handling.

The bounce/pitch suspension model models the vehicle motions separately from

the wheel motions (Figure 3: Bounce/pitch model).

Figure 3: Bounce/pitch model

The equations of motion that govern this system can be obtained by applying Newtonssecond law of motion in both pitch and bounce to the system (note damping will be first

neglected so that the natural frequency can be obtained) (Equation 5: Bounce and pitch

equations of motion (neglecting damping)).

( ) ( )

( ) ( )2

20

0

ys

rfys

rfs

rmI

bzbkazakrm

bzkazkzm

=

=++

=+++

&&

&&

Equation 5: Bounce and pitch equations of motion (neglecting damping)

The equations of motion are coupled as can be seen above. If it is assumed that thedisplacements are harmonic then the natural frequencies can be obtained (z = Zcos(t)

and =cos(t)). The following is the equation that would be obtained from assuming

the motions are harmonic for the natural frequencies (Equation 7: Natural frequencies inbounce and in pitch). The motion ratios can be obtained at each of these frequencies by

substituting each of the results back into the equation of motion (Equation 6).

1

2

2

2

1

2

1

2

D

DZ

D

DZ

=

=

Equation 6: Motion ratios at each of the natural frequency


20/216


6

( ) ( )

( )

( )

( )2223

2

1

2

2

22

31212,1

1

1

1

4

1

2

1

bkakrm

D

akbkm

D

kk

m

D

r

DDDDD

rf

ys

fr

s

rf

s

y

+=

=

+=

++=

Equation 7: Natural frequencies in bounce and in pitch

The bounce and pitch natural frequencies are usually very close to one another. They are

usually between 1 to 1.5 Hz. The bounce and pitch equations of motion can be re writtento include damping in the equation of motion (damping is important when it is desired to

obtained the amplitudes of motion) (Equation 8: Equations of motion in bounce andpitch).

+

+=

+

+=

=

=

=

+

+

frfr

frrf

frfr

frrf

ys

s

CaCbaCbC

aCbCCCC

kakbakbk

akbkkkK

rm

mM

zKM

zCM

z

zK

zC

zM

22

22

2

11

0

0

0

&

&

&&

&&

&

&

&&

&&

Equation 8: Equations of motion in bounce and pitch

The above equations of motion can be solved to obtain the natural frequencies and

amplitude ratios, as well as the amplitudes for a given frequency. This can be done by

reducing the equations from second order to first order (Equation 9: Bounce and pitch

damped natural frequency).


21/216


7

[ ]

[ ]

[ ]

[ ]Aeigenvalues

s

sz

z

AIs

e

s

sz

z

Ase

s

sz

z

eZez

Assume

z

z

Az

z

CMKM

I

z

z

stst

stst

xx

=

=

=

==

=

=

0

&

011

2222

&

&

&

&

&&

&&

&

&

Equation 9: Bounce and pitch damped natural frequency

It is important to note that the eigenvalues will be complex numbers because of the phase

change; however the natural frequency is just the sum of squares of the real andimaginary values. The damping ratio is the negative of the real part divided by the

natural frequency (=-a/n). The amplitudes at all frequencies can be solved by

assuming a value for either the pitch angle or the bounce and then solving the other value

over a wide range of frequencies.

The bounce/pitch model and the quarter car model are two of the most powerful

models to predict the vertical motion of the vehicle. These two models can be combined

to create the half car model. This model couples the motions of the front and rear

suspension through the motion of the sprung mass (both bounce and pitch). This modelallows the wheel hop frequencies to be obtained for both the front and rear suspensions at

the same time. As well as the pitch and body motion frequencies can be obtained. The

half car model predicts the motions of the both the front and both the rear suspensionunits at once. There are certain assumptions used in this model, and these include that the

tires on either side of the vehicle have the same effect on the dynamics, and the width of

the vehicle is assumed to be constant. Also, it is assumed that the springs are linear, andthat the damping can be modeled as viscous dampers. The model consists of four

coupled equations used to find the motions associated with the sprung mass and both the

unsprung masses (Equation 10: The half car model equations) (Figure 4: The half car

model) (Appendix A).


22/216


8

( ) ( )( ) ( )

( ) ( )( ) ( )

( )( )

( )( )( )( )

=

+

+

+

+

+

+

+

+

trr

tff

r

f

s

trrrr

tffff

rfrfrf

rfrfrf

r

f

s

rrr

fff

rfrfrf

rfrfrf

r

f

s

ur

uf

s

kh

kh

Z

Z

Z

kkbkk

kkakk

bkakkbkabkak

kkbkakkk

Z

Z

Z

CbCC

CaCC

bCaCCbCabCaC

CCbCaCCC

Z

Z

Z

m

m

I

m

0

0

0

0

0

0

000

000

000

000

22

22

&

&

&

&

&&

&&

&&

&&

Equation 10: The half car model equations

Figure 4: The half car model

The following is a discussion on the important parameters that are applied in the half car

model.

The Suspension Stiffness and Damping

The suspension stiffness is one of the most important parameters when

considering the vertical performance of the vehicle. It is generally best to have a

moderate spring rates. This is because low spring rates reduce the tire deflection which

increases the tire grip, however it also allows for increased body motions (in roll and inpitch) which are harmful to the overall handling performance of the vehicle. The

opposite is true for high spring rates. Therefore, there should be a compromise between

implementing high and low suspension stiffness. Also, according to Maurrie Olley thefollowing set of rules should be followed when designing a suspension system for the

comfort of the passenger, and they are:

1. Front suspension should have a 30% lower ride rate than rear suspension


23/216


9

2. Pitch and bounce frequencies should be close together, bounce frequency shouldbe 1.2 times the pitch frequency

3. Neither the bounce nor the roll frequency should be greater than 1.3Hz.

The reason for this is that the front of the vehicle will ride over the bump (or

disturbance) first creating an excitation in the front suspension, and then seconds later therear suspension will ride over the bump creating an excitation in the rear suspension. Ifthe two suspension rates are identical the phase lag between the front and the rear

suspensions will create an undesirable motion in pitch. There have been studies that have

shown that the driver/passenger is/are very uncomfortable in pitch motion, it tends to

cause neck muscle strains. Therefore, by increasing the suspension rate in the rearsuspension allows for the rear of the vehicle to catch up to the front of the vehicle

(Figure 5: The front and the rear suspension amplitudes as a function of time).

Figure 5: The front and the rear suspension amplitudes as a function of time

It can be seen from the figure above that there exists a phase lag between the front and

the rear excitations, and that by having a rear suspension rate higher than the frontsuspension rate allows for the rear excitation to catch up to the front excitation.

The Tire Stiffness and Damping

The tires stiffness and the tires viscous damping coefficient are important to theride quality of the vehicle, but more importantly to the handling performance of the

vehicle. In typical passenger car vehicles the stiffness of the tires is of an order ofmagnitude greater than the suspension stiffness. It is typically the tire deflection that is

important for the handling performance of the vehicle, because the tire deflection is one

of the parameters in which decides the tires grip capabilities. As the deflection of the tireincreases, the grip capabilities of the tire will decrease. It is very important to not allow

the tire to lose contact with the ground, because if it does the car will not be controllable


24/216


10

in handling. Typically, the damping coefficient of the tires is neglected because it is

generally very low compared to the other parameters in the system, and neglecting itresults in a small error in the analysis

The Sprung and Unsprung Mass

The mass of the vehicle is an important parameter in the analysis of the verticaldynamics of the vehicle. The mass of the vehicle is one of the main parameters in which

will decide the deflections of both the front and the rear tires, and the suspension units

when they are excited. The mass of the vehicle is divided into two parts the sprung mass

and the unsprung mass. The sprung mass consists of everything the suspension unitshave to support, and these include the chasis, and the engine. The unsprung mass

consists of everything the tires have to support, and these include the front and rear axles.

Typically the sprung mass is of an order of magnitude greater than the unsprung mass.Therefore the following formula can be used to calculate the sprung mass and the

unsprung mass based on the mass of the vehicle (Equation 11: Sprung and unsprung

mass).

( )( ) kgkgms

kglbs

kg.lbs

mm

mmm

mmm

u

uu

us

65.135667.13510

67.13511

453592403290

11

10

==

=

==

+=

+=

Equation 11: Sprung and unsprung mass

When implementing the half car model the unsprung mass has to be further divided intothe unsprung mass supported by the front tires of the vehicle, and the unsprung mass

supported by the rear tires of the vehicle.

The Pitch Inertia

The pitch inertia is the inertia that arises in the rotation of the front and rear of the

vehicle with respect to the center of mass. The pitch inertia is usually calculated using

the radius of gyration. It is important in the study of the ride quality of the vehicle

because it is one of the significant parameters in which determine the amount ofdeflection a vehicle will have in pitch. Generally, in order to have good ride quality in

pitch the radius of gyration should be around 1.2m, and the ratio of the radius of gyrationsquared to the location of the front axle from the center of mass times the location of the

rear axles from the center of mass ( ( )( )bary

2

) should be between 0.8 and 1.2. Thesevalues provide a desirable ride in pitch because the center of oscillations in pitch and roll

will be close to the front and the rear axle, thus allowing the motion in pitch created atone axle to somewhat cancel out the motion in pitch created at the other axle, and


25/216


11

therefore minimizing the overall motion in pitch felt by the driver.

The half car model leads to a good prediction of the vertical performance of the

vehicle or the ride quality of the vehicle.

6.1.2) Vehicle handling

The Handling performance of an automobile is important to the all around

performance of the vehicle. The handling performance will determine how the car willexecute in turning corners; its lateral performance. There are many important parameters

that determine the lateral performance of a vehicle, these include but are not limited to

the location of the center of mass, tire cornering stiffness, the steering angle, the lateralvelocity, the forward vehicle velocity, the lateral acceleration, the rotational speed (yaw

rate), the body slip angle, and the tire slip angle. The model usually used to predict the

lateral performance of the vehicle is the linear bicycle model.

There are certain assumptions used in this model, and these include that the tires

on either side of the vehicle have the same effect on the dynamics, and the width of the

vehicle is assumed to be constant. The model consists of two coupled equations used tofind the lateral acceleration and the rate of change of the vehicles yaw rate while

assuming the forward vehicle speed is held constant (its in the control of the driver)

(Equation 12: The equations used in the bicycle model) (Note, for a clarification of themodel see the derivation in Appendix B).

( ) ( )

( ) ( )

=

+

++

+

f

f

rfrf

rfrf

aCC

rv

u

CbCa

u

bCaC

mu

u

bCaC

u

CC

rv

Im 220

0&

&

Equation 12: The equations used in the bicycle model

Once the above equations of motion are solved for the yaw rate, lateral velocity, lateraldisplacement and the vehicle yaw several other parameters can be solved for, and certain

characteristics of the vehicle can be determined. Also certain cases can be analyzed in

detail, and one such case is steady state cornering (lateral acceleration and rate of changeof the yaw rate are equal to zero) (Equation 13: Equations of motion in steady state

cornering). Solving the equations of motion in steady state leads to the following

important equations (Equation 14: Vehicle yaw rate as a function of the steering angle)(Equation 15: The cornering radius as a function of the kinematic cornering radius).


26/216


12

( ) ( )

( ) ( )

=

+

++

f

f

rfrf

rfrf

aC

C

r

v

u

CbCa

u

bCaC

muu

bCaC

u

CC

22

Equation 13: Equations of motion in steady state cornering

( )( ) rf

rf

CCba

bCaCmuba

ur

+

+

=2

Equation 14: Vehicle yaw rate as a function of the steering angle

( )( )

rf

rf

CCba

bCaCmu

R

R2

2

0

1+

=

Equation 15: The cornering radius as a function of the kinematic cornering radius

The second equation (equation 15) is important because it describes the path the

understeer/oversteer characteristics of the vehicle. If the vehicle was cornering with nolateral slipping than the vehicle would corner about a perfect circular path with a radius

of R0 (R0 is known as the kinematic turning radius) (Equation 16: The kinematic turning

radius).

Equation 16: The kinematic turning radius

The kinematic turning radius is the radius in which the driver is aiming for the vehicle to

follow. Examination of equation 15 reveals that if aCf < bCr than the vehicle will

understeer. If the vehicle understeers, the radius of the path will increase with vehicle


27/216


13

speed. In order to maintain the desired path of the vehicle the driver will have to increase

the steering angle with vehicle speed. If aCf > bCr the vehicle oversteers, and thecornering radius will decrease with vehicle speed. The driver will have to decrease the

steering angle as the speed of the vehicle increases in order to maintain the desired path

of the vehicle. If aCf = bCr the vehicle neutral steers and will turn on the kinematic

turning radius. The radius of curvature will be independent of vehicle speed. It is alsoimportant to note that understeer/oversteer characteristics is also affecting by theinclination of the roll axis and the front and rear suspension roll stiffness as will be seen

in the suspension kinematics section. The cornering stiffness of the driving wheels will

change as the traction (driving) force increases (as the traction force increases the lateral

force will decrease (friction circle)). For a front wheel drive vehicle this effect is to forcethe vehicle to understeer, and for a rear wheel drive to force the vehicle to oversteer.

When the vehicle is cornering it does not point in the direction it is traveling in,this is known as body slip. The vehicle will experience a body slip angle (Equation 17:

Body slip angle).

u

v=tan

Equation 17: Body slip angle

Using the steady state bicycle model the body slip angle can be solved for as a function

of the steering angle (Equation 18: The body slip angle as a function of the steeringangle).

( )( )( ) rf

rf

r

CCba

bCaCmuba

Cba

amub

+

+

+

= 2

2

Equation 18: The body slip angle as a function of the steering angle

At low speeds, the /ratio will be positive which indicates that the rear wheels will track

inside the front wheels. However, at high speeds the opposite will be true; the rearwheels will track outside the front wheels. For an understeering vehicle the /ratio will

tend to a limit; at high speeds it will be a constant (Equation 19: The limit of the /ratiofor an understeering vehicle).

rf

f

it bCaC

aC

=

lim

Equation 19: The limit of the /ratio for an understeering vehicle

An oversteering vehicle will have larger slip angles than an understeering vehicle, and the/ratio will tend to infinity at a critical vehicle speed. The vehicle will become unstable


28/216


14

at the critical vehicle speed (Equation 20: Critical speed of an oversteering vehicle).

( )

( )rfrf

criticalbCaCm

baCCu

+=

2

Equation 20: Critical speed of an oversteering vehicle

If the critical speed is reached the driver is capable of stabilizing the vehicle with steeringinputs. The r/ ratio will also go to infinity at the critical speed for an oversteering

vehicle; however for an understeering vehicle the r/ratio will reach a maximum at the

characteristic speed (the highest amount of yaw rate for a given steering angle will occurat this speed) (Equation 21: Characteristic speed of an understeering vehicle).

( )

( )frrf

sticcharacteriaCbCm

baCCu

+=

2

Equation 21: Characteristic speed of an understeering vehicle

The transient effects of vehicle cornering can be considered by solving the bicycle

model with zero steering angle; the model is solved assuming that the driver is not going

to react (the steering angle is zero). It is important that the transient effects die out overtime; that is the amplitude of vehicle oscillations tends to zero over time. If it does not go

to zero, then the vehicle will be unstable. It is best to use an eigenvalue approach when

solving the bicycle model to analyze the transient effects of the vehicle (Equation 22:Solution to the transients associated with the bicycle model). If s is smaller than zero the

vehicle will be stable. Analyzing the equation that determines the value of s will indicate

that if C is greater than zero than the vehicle will always be stable. This occurs for an

understeering vehicle. An understeering vehicle will always be stable. However, for anoversteering vehicle the value of C will become negative at the critical speed. This is

implying that an oversteering vehicle will be stable up until the critical speed, but once

the critical speed is reached the vehicle will become unstable. It is also important to notethat the solution can take on real and complex solutions. We are generally looking for

our vehicle to have a stable response (want s to be negative or a complex number with a

as being negative) indicating that the yaw rate and lateral velocity will decayexponentially to zero. If we have an unstable response the yaw rate and the lateral

velocity will increase when excited causing the vehicle to loose control. It is generally

better to design the vehicle so that it is an overall understeering vehicle because it is

guaranteed to be stable. Negative eigenvalues are basically indicating that the system iscapable of correcting itself (allow for the yaw rate and the lateral velocity to decay back

to zero) if excited without any input from the driver. The only difference between the

real and the imaginary parts is that in the imaginary part of the eigenvectors will fluctuateas they decay to zero, a frequency will exist (Figure 6: Eigenvalues verses vehicle speed

for an understeering vehicle).


29/216


15

( )( )

( )

( ) ( )

( ) ( )

( ) ( )

A

ACBBs

bCaCmuCCbaC

CCIuCbCamuBmIuA

u

CbCaIs

u

bCaC

muu

bCaC

u

CCms

CMs

CMs

XCMs

CXeMsXe

Xexassume

r

vx

CxxM

rfrf

rfrf

rfrf

rfrf

stst

st

2

4

followingthetoleadstdeterminantheSolving

0det

0

0

0

2

22

22

2

22

=

+=

+++==

++

++

+=+

=+

=+

=+

=

=

=+&

Equation 22: Solution to the transients associated with the bicycle model

Figure 6: Eigenvalues verses vehicle speed for an understeering vehicle

The eigenvectors are the associated response of the vehicle when it is operating at that


30/216


16

particular eigenvalue. As previously mentioned, these eigenvectors allow for the vehicle

to be a stable vehicle. Also as the vehicles speed is increased the s value decreases,indicating that the yaw rate and lateral velocity will approach zero at a slower rate. That

is as the vehicles speed increases itll take a longer time for the yaw rate and the lateral

velocity to approach zero.

The tire cornering st iffness

The tire cornering stiffness is an important parameter in determining the handling

performance of the vehicle. It is to some extent arbitrary; each tire has its own stiffness,and the tires on a vehicle can be changed. Therefore the cornering stiffness can be

chosen by the user to precisely predict turning (cornering) characteristics of the vehicle.

It is this parameter that will determine whether the car is an understeering (the actualcornering radius increases with vehicle speed) or an oversteering (the actual cornering

radius decreases with vehicle speed) automobile because the center of mass of the vehicle

is a fixed parameter (Figure 7: Oversteering and Understeering Vehicle). It is generally

better to have an understeering vehicle, because the vehicle is normally more stable. Inan oversteering case, the vehicle oversteers the turn, and the driver will be forced to

decrease the steering angle as he/she turns in order to stay on the desired path (the path

the vehicle takes when there is no lateral slipping).

Figure 7: Oversteering and Understeering Vehicle

There are also more chances that the vehicle spins on the spot (about its own z-axis). In

an understeering case, the car understeers and the driver is forced to increase the steeringangle in order to stay on the desired path. There are several ways to determine the tires

cornering stiffness. Two of these ways are by using the magic tire model and second by

using an estimation given the tires dimensions.

Magic tire model

The stiffness can be estimated as the slope of the linear range on the lateral forceverses slip angle diagram, which can be obtained from the magic tire model (Figure 8:

The lateral force verses the slip angle, on the following page).


31/216


17

Figure 8: The lateral force verses the slip angle

However, tests will have to be done on the tire in order to determine the necessarycoefficients to apply the magic tire formula (Equation 23: Magic tire Formula).

( )( ) ( )( )[ ]{ }

( )

11211111

131211

1098

5

4

3

76

21

0

1arctan2sin

arctan1arctansin

aFaa

aFaFaS

aFaaS

CDBCDB

a

a

FaBCD

aFaE

aFa

FD

aC

SSBESEBCDFy

z

zzv

zh

z

z

zyz

zyz

vhh

+=

++=

++=

=

=

+=

+=

=

=

++++=

Equation 23: Magic tire Formula

Tire Cornering Stiffness Obtained from the Tire Geometry

The tire cornering stiffness can also be obtained from the geometry of the tire byassuming that the tire is a cantilever beam. This cantilever beam is acted on by a self-

aligning moment and a shearing stress which act together to generate contact patch twist

during cornering. With some manipulation of the tire slip angle (deflection) obtainedfrom the cantilever beam an expression for the cornering stiffness can be obtained

(Equation 24: Tire cornering stiffness).


32/216


18

( )[ ]( ) ( )

+

++

=

tt

t

tt

ttt

t

war

swa

war

swawar

wEbC

1arccossin1arccossin

2

2

3

Equation 24: Tire cornering stiffness

6.2) Suspension Kinematics

Suspension kinematics is the study of the motions of the tire. It describes theorientation of the tire as a function of wheel travel and steering angle. The motions of the

tire are highly dependent on the type of suspension. In general there are two types of

suspension systems; solid axles and independent suspensions. A solid axle suspension isa suspension where the movement of one wheel is transmitted to the other wheel causing

them to move together. This type of suspension is essentially a dependent suspension,

the motion of the two wheels are correlated to one another. The biggest advantage of thistype is that the camber angle is not affected by vehicle body roll. The majordisadvantage of this type of suspension is the vibrations which are induced into the

system if the solid axle suspension also incorporates vehicle steering. Independent

suspension systems allow the left and right wheels to move independently; the movementof one wheel will have no effect on the other wheel. The advantages of independent type

of suspensions are: they provide better resistance to steering vibrations; they provide a

high suspension roll stiffness; steering geometry is easily controlled; suspensiongeometry is easily controlled; and they allow for higher wheel travel. The major

disadvantages are: the camber angle changes quite a bit over suspension travel; increased

unsprung mass; and the high cost of the system.

The study of suspension kinematics allows for several different suspension

parameters to be determined throughout suspension travel and steering angle. Some of

the most important parameters include: roll center position and instant center, camberangle, caster angle, toe angle, tire scrub, kingpin angle, scrub radius, caster trail, aligning

moment, vehicle ride height, track width, wheel rates, roll stiffness, roll axis,

understeer/oversteer characteristics, roll steer, bump steer, motion ratio, and anti-dive/anti-squat. The following will be a discussion of each of these parameters.

6.2.1) Track width and ti re scrub

The track width is a measure of the distance between the center of the tire contactpatches at the front and rear of the vehicle (Figure 9: Vehicle track width). The trackwidth will change as the wheels travel through the suspension travel, and this change is

known as tire scrub. The change in the track width is a measure of the location of the

instant center of motion of the suspension. As the track width is changing the tires areforced to push out or pull in at the ground, and thus the tires are forced to scrub against

the ground. Typically, if the suspension is in compression the tires will scrub out, and if

the suspension is in rebound the tires will scrub in. Tire scrub (track width change)


33/216


19

causes the rolling tires to slip and therefore generates lateral forces. Thus if one wheel

goes over a bump (causes the tire to scrub) there will be a disturbance in the lateraldirection; one side of the vehicle can start to see a larger lateral force than the other and

the vehicle may begin to yaw. Therefore it is important that the change in the track width

be kept to a minimum.

Figure 9: Vehicle track width

6.2.2) Instant center and rol l center position

The instant center is the point the wheel rotates about relative to the vehicle

chassis. It is a function of the geometry of the suspension system. The instant center is

important because it defines the position of the roll center. The roll center position is aposition where the lateral forces developed at the wheels are transmitted to the vehicle

sprung mass. This point will affect the behavior of both the sprung and unsprung mass

and thus effects the vehicles cornering characteristics. The roll center is defined as thepoint in the transverse vertical plane where the lateral forces may be applied to the sprung

mass without producing any suspension roll. The definition of roll center derives from

the fact that a vehicle will posses a roll axis (Figure 10: The roll axis of the vehicle).

Figure 10: The roll axis of the vehicle

The roll axis is the instantaneous axis where the unsprung mass will rotate relative to thesprung mass when a pure couple (moment) is applied to the unsprung mass. The roll

center is the intersection of the roll axis with the vertical plane at the front and rear of the


34/216


20

vehicle. Typically, the roll center position is located based on the suspension geometry

and then the roll axis is located by defining a line which connects the two roll centerstogether. The roll axis is also the instantaneous axis in which the whole vehicle rotates

with respect to the ground.

The amount of body roll depends on the height of the center of mass relative tothe roll center position. Therefore raising the roll center position closer to the center ofmass is equivalent to increasing the roll stiffness of the suspension. However, as the roll

center position is increased (roll center height measured from ground level is increased)

the amount of jacking forces will increase. The jacking forces are the forces that willtravel through the suspension components to the vehicle body; it is the force that is not

absorb by the suspension system. Thus as the amount of jacking forces increase, the

amount of forces absorbed by the shock will decrease. Forces generated at the tire have

two paths into the vehicle: a flexible path and a stiff path. The stiff path is through thesuspension components and the flexible path is through the suspension spring (Figure 11:

The effect of the jacking forces).

Figure 11: The effect of the jacking forces

Thus as the roll center is increased, the forces traveling thr

duplo a completo

Documents