Tese - Modeling Od Road Vehicle Lateral Dynamics

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Chapter 4 Two Degree-of-Freedom Vehicle Mode l

4.5.26 Frequency Response

It is also interesting to examine th e frequency response of th e vehicle. A driving

event where frequency response may be of particular interest is a slalom t e s t where th e

vehicle is driven t h r ough regularly spaced cones by means of a sinusoidal steering input.

The frequency of th e input required to negotiate th e slalom depends upon th e vehicle speed

and th e cone spacing. The performance of th e vehicle in th e slalom may be influenced by

th e magnitude of th e input frequency relative to th e natural frequency of th e vehicle.

Sinusoidal steering inputs may also be used in emergency maneuvers such as a double lane

change. Examining th e frequency response of th e vehicle may provide an indication of it s

performance in such a maneuver. Since it is generally desirable to minimize th e response of

a vehicle to disturbances such as side winds and road side s lope , frequency response

t echn iques can be used to examine th e response of th e vehicle to periodic disturbance

inputs .

Phase la gs in response to steering input require th e driver to adjust his input to

obtain th e desired r e sponse , making th e vehic le more difficult to drive. Smaller phase lags

t e n d to improve vehiclecontrollability.21

The frequency response of th e sample vehicle with

a forward velocity of 100 km/hr is examined using th e bode plotting capability of th e

MATLAB Controls Too lb ox . T h e g ain a nd phase responses of vehicle sideslip angle and

yaw velocity to steer ang le , aerodynamic side force, and road side slope are plotted in

F ig u re 4 .7 t h r o u g h Figure 4. 12. The MATLAB script DOF2LFreq.m, which is listed in

Appendix C.5, is used to facilitate plotting of th e frequency response. The script generates

g ain a nd phase versus input frequency fo r th e tw o degree-of-freedom model.

DOF2LFreq .m calls th e scripts DOF2Control.m, which sets program execution paramete r s ;

DOF2Param.m, wh ich s ets vehicle and input magnitude paramete r s ; and

58



Chapter 4 Two Deg r e e- o f-F re edom Veh i cl e Mode l

DOF2DependParam.m, which calculates vehicle parameters which depend o n o th er

parameters. These scripts are listed in Append ix C .l t h r o u g h Ap p en d ix C .3 .

The frequency response of a r oa d vehicle changes as forward velocity changes. Fo r

most of th e responses th e magnitude of th e gain changes while th e genera l shape of th e

curves remain approximately constant. There is little change in th e phase plot fo r most of

th e responses. The sideslip angle / steer angle response is th e only response which

experiences significant change in th e shape of th e g ain a nd phase plots as forward velocity

changes. The sideslip angle / steer angle frequency response is influenced strongly by th e

magnitude of th e forward speed relative to th e t a n g en t speed of th e vehicle. T his is a result

of th e sideslip angle / steer angle zero changing sign at th e t a ng en t speed. The sideslip angle

/ steer angle frequency response is plotted in Figure 4. 13 and Figure 4. 14 f o r f o rwa rd

speeds of 30 km/hr and 49.84 km/hr respectively.

A t 30 km/hr th e gain is flat up to approximately 1 Hz at 0.33 deg/deg and th e phase

goes from0

at 0. 1 Hz to at 10 0 Hz. The phase response is t yp ica l of a second order

system with a nega tive zero. A t 4 9.8 4 km/hr, th e t an g en t s p e ed , th e gain approaches zero

as th e frequency approaches zero as expected from th e definition of t a ng en t speed.

However, there is a significant peak in th e gain at approximately 2 Hz, which is the

undamped natural frequency at 49.84 km/hr. The phase goes from90

at 0. 1 H z to at

10 0 Hz, crossing zero at th e undamped natural frequency. The frequency response at 100

km/h r is shown in Figure 4.7. The phase goes from 180 at 0.1 Hz to at 100 Hz.

There is a180

phase lead at low frequency because above th e t a ng en t s peed a positive

steady-state steer angle produces a negative sideslip angle as shown in Section 4.5.18.

Also of interest is th e yaw velocity / road side slope phase response. The phase

goes from0

at 0.1 H z to at 100 Hz. The phase response of th is t r ans fe r function

differs from th e others due to th e lack of a zero.

59



Chapter 4 Tw o Degree-of-Freedom Vehicle Mode l

A t low frequencies th e gains fo r each input and output combination approach th e

values of th e steady-state step input response gains shown in Tabl e 4 .3 fo r a forward

velocity V = 100 km/hr.

60



Chapter 4 Two Degree-of-Freedom Veh ic le Mod e l

O )

o>CD

CO

O

D )CD2.

CDCOco

1 .6

1.2

0 .8

0 .4

0 .0

v i

T , -i^^^ t 1 i t- - _ _ _

r- -

n--i--i- T-i-i-i-r-

_ _ ~

t- - ~i~ "

r- \-

i ~i- i~ i-

0. 1

180

90

- 9 0

0 .1

1 10

Frequency (Hz)

1 10

100

------!---

+ __i_-|_

-+^^J

-- i * i t i t l l + l t i -- I l l

: ! :'

i' ' '

i : : : j'

IT

Frequency (Hz)Figure 4.7: Sideslip Angle / Steer Angle Frequency Response, V= 100 km/hr

100

3.0E-04

"& 2.0E-04CD

" I 1.0E-04CD

0 . 0E+00

0. 1

Frequency (Hz)10 100

CD

2_

CDCOCO

- 4 5

- 9 0

0 .1 10

Frequency (Hz)Figure 4.8: Sideslip Angle /Ae ro Side Force Frequency Response, V = 100 km/hr

100

61




CD

CD

2.

C

co

O

0 .1 1 10

Frequency (Hz)100

CD

CDCOCO

-4 5-

- 9 0

0. 1 1 10 100

Frequency (Hz)

F ig u re 4 .9 : Sideslip Angle / R o a d S id e S lo p e Frequency Response, V = 100 km/hr

CD

-52

co

CO

0 .20

0 .15

0 .10

0 .05

0 . 00

0 .1


CD

2,

CDCOCO

-4 5

- 9 0

__ _ _ __,_

_ _

T--,-

-, -r -i -i "I -i -^c- - -

r-

p r i -i i i-i-|------|---|---i-T-i-r-TT

0. 1 1 10

Frequency (Hz)

F igu re 4 .10 : Yaw Velocity / Steer Angle Frequency Response, V = 1 00 km /h r

100

62



Chapte r 4 Tw o Degree-of-Freedom Vehicle Mode l

8.0E-06

| 6.0E-06

2. 4.0E-06

O 2.0E-06

0 .0E+00

- -

---,--- ,- -

j-

i t irT^~

----n---i~~T-T-i-rTTr~___in i i t

0 .1


O )CD

2.

CDCOCO

1 80

135

9 0

45 \-

0

0 .1 1 10

Frequency (Hz)

F igu re 4 .11 : Yaw Velocity /Ae r o Side Force Frequency Response, V-100 km/hr

100

2 .5E-04

.g2 .0E -04

^ 1 . 5E -04CO

1 .0E -04 r

O 5.0E-05

0 . 0E+00

0 .1


CDTO

CDCO

CO

0

-4 5

-90

- 1 3 5

-180

0 .1


Figure 4 .12 : Yaw Velocity / R o a d Side Slope Frequency Response, V = 100 km/hr

63




0 .4^m^

O )CDT3 0 .3O )CD

2, 0 .2C

co

O 0. 1

0 .0

- - - - -

1- - -

r-

-\-

-\- \~

r t tt- - _ _ -i_

~ ^ ^ t~

~i-n-i-rTTr- - - - -

1- - -

r-

-i-

-i- i~

r t t

._____,___r __ ! i | -r ~r i - - - - - | - - - r

-

-|-

i- i ^W"

tt t~ ~ ~ -

-,---

r-

-i-

-j- |-

r t t

0. 1


CDTO

CDCOCO

-4 5

-9 0

0. 1

i t--i--|-|-t-]-it-----|----|--i ^c i-

r n -i -i -}

10

Frequency (Hz)

Figure 4 .13 : Sideslip Angle / S t e e r Angle Frequency Response, V = 30 km/hr

100

0 .3,, .

O)

CD

0 .2CDD-H ^

c

"co 0 .1

CD

2,

CDCOCO

0 .0

------I---

+ __l__l_l_ + _l_l + _ _ _ ,_ _ _ -f̂ l̂ - - - 1 - + -I -I -I -I _ - _

- - t - - - - f __ l __ l _H

0. 1

90

45

0

-4 5

- 9 0

0. 1

1 10

Frequency (Hz)

10

100

Frequency (Hz)Figure 4.14: Sideslip Angle / Steer Angle Frequency Response, V = 49.84 km/hr

100

64



Chap te r 4 Tw o Degree -o f -Freedom Vehicle Mode l

4.5.27 Simulation

Another t o o l t ha t is very useful in analyzing vehicle lateral dynamics is numerical

simulation. Simulation is done by integrating th e differential equations of motion wi th

respect to t ime and can be used to predict th e response of th e vehicle to arbitrary control and

disturbance inputs. Non-linearities are generally much easier to handle with numerical

simulation t han with th e analytical t e c h n i q u e s used up to t h i s point in th is chapter.

To maintain consistency with th e non-linear model simulation which fol lows in

Section 4.6, th e lateral velocity v is used as a state variable fo r th e linear model simulation

instead of th e sideslip angle (3. Since th e equations of motion were originally derived in

t e rm s of yaw velocity and lateral velocity an d t h e n simplified to be in t e rm s of yaw velocity

and sideslip ang le , th e model has been returned to its or ig ina l , more gen e r a l , form.

The equations o f motion in their gene ra l , non-linear form are given by Eq . ( 4.9 ).

W ith th e smal l s teer angle assumption used fo r th e linear model th e equations become

Fy f+Fy r+Fy a+Fyg=m(v + u r)

aFyf~bFyr

~ (c ~a^Fya

= IJ

Expressions fo r t he tir e slip ang le s , t ire lateral forces, and gravitational side force

are derived i n S e cti on 4.5.3 and Sec ti on 4 . 5. 4 using th e small angle assumption and are

repeated here fo r convenience.

v + a r

a f= 8

u, (4.19)

v - b r

ar=

u

Fyf=Cfaf

Fy=mgQ (4.23)

65



Chapter 4 Two Degree -o f -Freedom Vehic le Mode l

Outputs from th e simulation are t ime histories of lateral ve loc i t y, yaw v e l o c i t y,

vehicle sideslip ang le , t ire slip ang le s , and lateral acceleration. Inputs can be a step s teer,

ramp step s teer, ramp square s teer, sine s teer, step aerodynamic side force, or step road

side slope.

The simulation is imp lemen te d i n th e MATLAB script DOF2LSim.m, which is

l is ted i n Appendix C.6. Integration of th e differential equations of motion is done using th e

built-in MATLAB function o d e 2 3 , which uses second and th i rd order Runge -Ku t t a

f o r m u l a s . 3 4

The function ode23 returns th e state variables v and r over th e t im e interval

specified fo r th e simulation.

A t each t ime step th e ode23 funct ion calls th e function DOF2LDE .m which

calculates th e state derivatives v and r based upon th e instantaneous values of th e state

variables v and r. First, th e instantaneous steer angle is calculated by th e function

SteerAngle.m, which is l is ted i n Appendix C.4, based upon th e current time, th e t y p e of

input selected, th e magnitude of th e input, and th e values of th e input duration parameters.

Any arbitrary steer input, including steer inputs measured experimentally during vehicle

testing, could easily b e im p lem en te d in th is f unc ti on . Use of measured steer i npu t da ta

facilitates experimental validation of th e model.

After th e steer angle is ca l cu l a t ed , th e t ire slip angles a re calculated from th e current

values of th e state variables v and r which are passed as parameters into DOF2LDE.m. The

t ire lateral forces are t h e n calculated from th e slip angles using Eq. (4.21). Finally, th e state

derivatives are calculated as

Fyf+Fyr+Fya+Fygv= -

u r

m

. aFyf -bFyr- (c -a )Fya(4-88>

r=

L

66




Eq. (4.88) are obtained by solving Eq. (4.87) fo r v and r. A listing of DOF2LDE .m is

provided in Appendix C.7.

To illustrate th e effect of forward velocity on r e s p o n s e , simulations are performed

at lo w speed (3 0 km/hr), at th e ta n g en t speed (49.84 km/hr), at normal highway speed (100

km/hr), and at high speed (150 km/h r) . The simulations are run until steady-state is

reached. Initial conditions fo r th e simulations are zero. Lateral v e l o c i t y, yaw ve loc i t y,

sideslip ang le , front t ire slip ang le , rear t ire slip ang le , and lateral acceleration are plotted

fo r each input studied.

The inputs used fo r th e simulation are a1

step s teer, a1

ram p s te p s teer wi th a

ramp t ime of 0.2 sec , a1

ramp square steer with a r amp t ime of 0 .2 sec and a dwel l t ime

of 1.0 sec , a1

s in e s te er with a period of 1 sec , a 10000 N step aerodynamic side force,

and a1

step road side slope. The ramp step s teer, ramp square s teer, and sine steer inputs

are shown in Figure 4.15. The steer input is a posi tive steer ang l e , indicating a r ight rum.

Ramp Step S t e e r Input Ramp Square Stee r Input

CD

;o

<

CD

CDI

CO

1.0

0 .5

0 .0

1

Time (s)

Sine Stee r Input

O )CD

2,<D

D )c

<

CD

CD

1.0

0 .0

OT - 1 - 0

0

CD

S 1 .0

< 0.5

CD

CD

55 0 .0 E l1 2

Time (s)

1 2 3Time (s)

Figure 4.15: Simulat ion Steer Angle Inputs

67



Chapter 4 Two Degree-of-Freedom Vehi cl e Mod e l

The a e r o d y n am i c side force input is applied in th e positive y - d i r e c t i o n , and th e road side

slope is positive, which means t h a t th e road s lopes down to th e right.

Simulation results fo r th e step steer input are plotted in Figure 4. 16 t h r ough

Figure 4.21. The response t ime s increase with forward speed. The steady-state lateral

velocity and sideslip angle are positive below th e t a n g en t s p e ed , zero at t he t an g e nt s p e ed ,

and negative above th e t a ng en t speed. This agrees with th e definition of th e t a ng en t speed

presented in Section 4.5.18. Above th e t a ng en t speed th e la teral veloci ty and sideslip angle

also initially begin to increase from zero becoming positive and t hen decrease to negative

values. This is a result of th e system zero being positive when th e forward speed is greater

t han th e t a ngen t speed. As explained in Section 4.5.25 a sys tem wi th a positive zero is a

nonminimum-phase system and typically exhibits th e ty pe of step response shown here,

initially in th e opposite direction to th e steady-state value. The front tire slip angles show

response similar to th e sideslip ang le , but with initial values of du e to th e1

step steer.

The l at er al acceleration has a non-zero initial value due to th e rate of change of lateral

velocity when th e step steer occurs. The steady-state values of yaw ve loc i ty, sideslip ang le ,

front t ire slip ang le , rear t ire slip ang le , and lateral acceleration agree with th e steady-state

response gains presented in Table 4.3. In add i t i on , th e steady-state sideslip angles and yaw

velocities agree wi th th e values approached at lo w frequency in th e frequency response

plots of Figure 4.7, Figure 4.10, Figure 4.13, and Figure 4.14.

Ramp step steer simulation results are plotted in Figure 4.22 t h r ough Figure 4.27.

The ramp step steer response is similar to th e step steer response and lags it slightly as

expected. The steady-state values are identical to th o se of th e step response. The lateral

veloci ty and sideslip angle still exhibit th e non-minimum phase system response above th e

t a ng en t s p e ed , bu t th e magnitude of th e initial response is less t han it is fo r th e step steer.

Unlike with th estep

input, th e front t ireslip

angle and lateral acceleration are

initiallyzero

68




fo r th e ramp step input. A t 3 0 km /h r and 49.84 km/hr t he re are peaks in th e front t ire slip

angle response at 0 .2 sec which is when th e ramping of th e steer inpu t is completed.

The responses to th e ramp square steer input are plotted in F ig ure 4 .2 8 t h r o ugh

Figure 4.33. The ramp square steer response is ident ical to th e ra m p ste p response up until

th e t ime t h a t th e steer input is ramped b ac k d own to zero. A t th e lower speeds th e responses

reach steady-state before th e ramp down. However, at 150 km/hr th e ramp down occurs

before steady-state has been reached. As with th e ramp step response th e front t ire slip

angles experience overshoot at 30 km/hr and 49.84 km/hr as th e ramp up is completed.

The sine steer resu lt s a re shown in Figure 4.34 t h r o u g h Figure 4.39. The sine steer

input had a frequency of 1 Hz. From visual inspection of th e plots it is seen t ha t th e steady-

s ta te y aw velocity and sideslip angle g ai ns and phases agree wi th t h o s e obtained fo r 1 H z

from th e frequency response in Figure 4.7, Figure 4.10, Figure 4.13, and F ig u re 4 .1 4 .

The response amplitude increases with forward velocity in a ll cases except th e lateral

velocity and sideslip angle. As expected from th e definition of t a ng en t speed , at 49.84

km/h r th e lateral velocity and sideslip angle amplitudes are less t han t h o s e at 3 0 km /h r.

Results from th e aerodynamic side force step input simulation are provided in

Figure 4.40 t h r o u g h F ig ur e 4 .4 5. T h e magnitude of th e responses increases with forward

velocity. Since th e center of aerodynamic pressure is located behind th e neutral steer po in t ,

a positive aerodynamic side force produces a negative yaw velocity. The steady-state yaw

ve loc i t y, sideslip ang le , front t i re slip ang le , rear t ire slip ang le , and lateral acceleration

responses at 100 km/h r agree with th e steady-state gains in Table 4.3. In addi t ion, th e

steady-state yaw veloci ty and sideslip ang le a t 10 0 km/hr agree with th e frequency response

gains of Figure 4.8 and Figure 4. 1 1 as th e input frequency approaches zero. The sideslip

ang l e s , front t ire slip ang le s , rear t i re slip ang le s , and lateral acceleration curves have higher

slopes at lower speeds

indicatingthat th e response i s f as te r at lower speeds.

69



Chap te r 4 Tw o Degree-of-Freedom Vehicle Mode l

Road side slope step input results are plotted in Figure 4.46 through Figure 4.51.

The lateral veloci ty, yaw ve loc i t y, and lateral acceleration responses increase with fo rward

ve loc i ty, while th e sideslip ang l e , front t ire slip ang le , and rear t ire slip angle decrease.

Again, th e steady-state yaw v e l o c i t y, sideslip ang le , front t ire slip ang le , rear t ire slip ang le ,

and lateral acceleration at 100 km/h r obtained with th e simulation agree with th e steady-state

gains of Table 4.3, and th e steady-state yaw velocity and sideslip angle agree with

frequency response gains of F ig u re 4 .9 and Figure 4.12 as th e inpu t frequency approaches

zero.

The l in e ar tw o degree-of-freedom model is useful fo r characterizing and predicting

th e response of th e automobile to control and disturbance inputs. Although th is model

great ly simplifies th e vehicle s y s t em , much can be learned about road vehicle lateral

dynamics t h r o u g h its study. The effects of changing vehic le and t ire parameters on sys tem

response can quickly be studied. Power fu l linear analysis t e c hn i qu e s based on sys tem

t r a n s f e r functions can be readily applied to th e vehicle model to gain significant insight into

system behav io r. The results from th e linear model are generally valid fo r lateral

accelerations up to 0.35 g , which constitutes most of normal passenger car driving.

Beyond th is level a non-linear t ire model is required to accurately simulate t i re behavior at

high slip angles.

70



Chapte r 4 Tw o Degree -o f -Freedom Vehic le Mode l

0 .5

0 .0

_ - 0 . 5

I- 1 . 0

o

o

CD

co

CD

COJ

- 2 . 0

- 2 . 5

- 3 . 0

V = 30 km/hr

V = 49 .84 km/hr

V= 100 km/hr

V= 1 50 km/h r

0 .5 1 1 .5Time (s)

F igu r e 4.16: Linear Step Steer Latera l Velocity Response

0 .30

0 .25

0 .20

f0

8 -15

CD

>

o.io

0 .05

0 .00

'

> V = 1 50 km/h r

/ i/^ V = 1 00 km/h r

'/ ^ V = 49 .84 km/hr

i f s ^ ~

V = 30 km/hr

J0 .5 1 1. 5

Time (s)

Figure 4 .17 : Linea r Step Steer Yaw Velocity Response

71



Chapter 4 Two Degree-of-Freedom Vehic le Mod e l

0 .5

0 .0

- 0 . 5

C7> - 1 . 0

CD

T7

CD- 1 . 5

TOC

<Q . - 2 . 0

CO

CD

Tl

CO - 2 . 5

- 3 . 0-

- 3 . 5

- 4 . 0

, ,

V = 30 km/hr

V = 49 .84 km/hr

V = 100 km/h r

1

V = 150 km/hr

0 .5 1 .5Time (s)

F igu re 4.18: Linea r Step Steer Sideslip Ang le R e sp o n se

0 .0

- 0 . 5

- 1 . 0

^ - s .

O)CD

2. - 1 . S

CD

TO

<- 2 . 0

Q .

CO-?.5

a)

HH-*

r - 3 . 0

o

LL

- 3 . 5

-4.0-

- 4 . 5

V = 30 km/hr

V = 49 .84 km/hr

V= 1 00 km/h r

V= 15 0 km/hr

0 .5 1. 5Time (s)

Figure 4.19: Linea r Step Steer Fron t Tire Slip Angle Response

72



Chapter 4 Two Deg r e e- o f-F re edom Veh ic le Mode l

0 .0

- 0 . 5

- 1 . 0

^^

TO0)

2, -l.b

CD

TOC

<- 2 . 0

Q .

CO-?.S

CD

1-

m- 3 . 0

CD

DC

- 3 . 5

- 4 . 0

- 4 . 5

V = 30 km/hr

V = 49 .84 km/hr

V = 100 km/hr

V = 150 km/hr

0 .5 1 .5Time (s)

Figure 4.20: Linea r Step Steer Rea r Tire Slip Ang le Re spon s e

1 .4

1 .2

1 .0

c

o

ffl 0 .OJ

CD

OO< 0.

sCD

CO

-i 0 .4 -

0 .2

0 .0

V = 150 km/hr

V= 100 km/hr

V = 49.84km / h r " "

V = 30 km/hr1

0 .5 1 1 .5Time (s)

Figure 4.21: Linear Step Steer Lateral Acceleration Response

73




0 .5

0 .0

_ "-5

E,& - 1 . 0

o

o

CD

i " 1 - 5

CDI

coJ

- 2 . 0

- 2 . 5

- 3 . 0

V = 30 km/h r

\ ""V

V = 49 .84 km/hr

V= 100 km/hr

..

V = 150 km/h r

0.5 1 .5Time (s)

F igu r e 4.22: Linea r Ramp Step Steer Lateral Velocity Response

0 .30

0.25

0 . 2 0T3

8 0 .15

CD>

0 .10

0 .05

0 .00

,

s^ V= 150 km/hr

/fl / ^

V= 100 km/hr

Ifr

V = 49 .84 km/hr

/ !

V = 3 0 km /hr

0 .5 1.5Time (s)

Figure 4 .23 : L inea r Ramp Step Steer Yaw Velocity Response

74



Chapter 4 Two Degree-of -Freedom Vehi cl e Mod e l

TOCD

TO

<Q .

WCD

gCO

- 3 . 0-

0 .5 1 .5Time (s)

Figure 4.24: Linea r Ramp Step Steer Sideslip Angle Response

0 .0

- 0 . 5

- 1 . 0

^ - H ^

TOCD

2, - 1 . S

CD

TOC

<- 2 . 0

a.

CO-?.S

CDi

1-

C - 3 . 0

O

- 3 . 5

-4.0-

-4.5

V = 3 0 km /hr

V = 49 .84 km/hr

X^

^ ^ ^ ^

V= 1 00 km/h r

1 1

V = 150 km/hr'

0 .5 1 .5Time (s)

Figure 4 .2 5: L in ea r Ramp Step Steer Fron t Tire Slip Angle Response

75



Chapter 4 Two Deg re e -o f- F re ed om Veh ic le Mod e l

0 .0

- 0 . 5

- 1 . 0

^-^

TOa)2, -l.b

CD

TO

<- 2 . 0

a.

CO-2.5

a>

H

- 3 . 0

CD

oc

- 3 . 5

- 4 . 0

- 4 . 5

V = 30 km/h r

V = 49 .84 km/hr

V = 100 km/hr

.;> s

V= 150 km/hr

0 .5 1 .5Time (s)

Figure 4.26: Linea r Ramp Step Steer Rea r Tire Slip Angle Response

c

o

]3 0.

CD

OU

< 0.

B"5

1 .4

1 .2 -

1.0

8 -

0 .4

0 .2

0 .0

V= 150 km/hr

V= 100km/hr

V = 49 .84 km/hr"

V = 30 km/hr^^^

1 1 1

0 .5 1 1 .5Time (s)

Figure 4.27: Linear Ramp Step Steer Lateral Acceleration Response

76



Chapter 4 Tw o Degree-of-Freedom Vehicle Mode l

0 .5

0 .0

i- 1 . 0

u

o

CD

I" ' -5CD

COJ

- 2 . 0

- 2 . 5-

- 3 . 0

> V = 30 km/hr

"^V V = 49 .84 km/hr

\ ^Ss*^^ ] /

\ V= 100 km/hr

\

: \V = 150 km/hr

0 .5 1 .5Time (s)

2 .5

F igu r e 4.28: Linea r Ramp Square Steer Latera l Velocity Response

0 .30

0 .25

0 .20

~0 .15

o

o

2 0 .10

co>-

0 .05

0 .00

- 0 . 0 5

.

> ^V = 150 km/hr \

7 \ \l /? V = 100 km/hr A \

1 V\ / V = 49 .84 km/hr \ \W f

^*"^Nk \ \/]:*"'.1 >^ .>

0 .5 1. 5Time (s)

2 .5

Figure 4 .29 : L inea r Ramp Squ ar e S t ee r Yaw Velocity Response

77



Chapter 4 Two Degree -o f -Freedom Vehicle Mode l

0.5

0 .0

- 0 . 5

TOCD

- 1 . 0

CD

TOc

<

- 1 . 5

Q . - 2 . 0

CO

rn - 2 . 5

- 3 . 0

- 3 . 5-

- 4 . 0

^Z-~~^__'

V = 30 km/hr N^'

"^ l ' V = 49 .84 km/hr

W. !

\ ^ v ^V = 100 km/hr J

V

' \

\

\. V = 150 km/hr y

10 0 .5 1 1 .5 2 2.5

Time (s)

Figure 4.30: Linea r Ramp Squ ar e S te er Sideslip Ang le R e sp o ns e

0 .5

0 .0

- 0 . 5

- 1 . 0

"to - 1 - 5

<

- 2 . 0

TO

TO

<

9.

CO

S - 2 . 5

P - 3 . 0

- 3 . 5

-4.0

-4.5

c v oU Km/nrs

x^~~ r*"~ ' ^ '

\^ /V ; V = 49 .84 km/hr

\\ .

\ ^SV = 100 km/hr /

V

\

; \ V = 150 km/hr /

: \ , I

0 0 .5 1 1 .5 2 2 .5Time (s)

Figure 4.31: Linea r Ramp Square Steer Front Tire Slip Angle Response

78



Chapter 4 Two Degree -o f -Freedom Vehicle Mode l

0.5

0 .0

- 0 . 5

TO- 1 . 0

T3

TO - 1 . 5

C

<a. - 2 . 0

CO

2 - 2 . 5

Hi

CO- 3 . 0

rr

- 3 . 5

- 4 . 0

-4.5

Y s ^ : ^S

' /^

V V = 49 .84 km/hr j /

\ :^vV= 100 km/hr / \ /

\ v- ibUKm/nr ~/~

l .... i i

0 0 .5 1 1 .5 2 2 .5Time (s)

F ig ure 4 .3 2: L in ea r Ramp Square Steer Rea r Tire Slip Angle Response

1 .2

1.0

~ 0 .8

Co

j 0 .6

oo

< 0 .4

2

co

0 .2

0 .0

-0.2

js^~\.

/ V = 1 50 km/h r V

/ \ \

Sv = 100 km/hr \ i \

J / V = 49 .84 km / n r ; x

\^ ; X ^

i i 1

0 .5 1. 5Time (s)

2.5

Figure 4.33: Linear Ramp Squar e S t ee r Lateral Acceleration Response

79



Chapter 4 Tw o Degree-of-Freedom Vehic le Mod e l

1 .0

0 .5

~0 .0

o

>

2 - 0 . 5

CO

- 1 . 0

- 1 . 5

y\

V = 30 km/hnr~ \ \

>^-- j f - <5

\ \ ) V = 49 .84 km/hr V

\ ;V= 1 00 km/h r \

> ^V = 1 50 km /h r

0 0 .5 1 1 .5 2 2.5Time (s)

Figure 4.34: Linea r 1 Hz Sine Steer Latera l Velocity Response

0.25

0 .20

0 .15

W 0 .10

0 .05

o

o

0 .00

- 0 . 0 5

-0.10-

-0.15

-0.20

v ^ \

- 1/^sN'

//SV =

/ = 150 km/hr \I"

\

= 100 km/hr: /nA fss\\ n n49 .84 km/hr Is-^ \ \ T T^x \\w r\\\ r\\\

-30 km/hr -

^dr^^sAWu

i ^ r \ \ \ \

i i 1 i i

0 .5 1 .5Time (s)

2 .5

Figure 4 .35 : L inea r 1 Hz Sine Steer Yaw Velocity Response

80




1.5

1 .0

0 .5

TO

0 .0TOC

<a-

- 0 . 5CO

gCO

- 1 . 0

- 1 . 5

- 2 . 0

i f \. y \

1 \ ' 1

V = 30 km/hr flVV /

\\ : V = 49.84 km/hr \ |

W/ : W\ V= 100 km/hr \ /

V= 150 km/hri i i i

0 0 .5 1 1 .5 2 2 .5Time (s)

F igu re 4.36: Linear I Hz Sine Steer Sideslip Angle Response

1 .0

0 .5

0 .0

TO

TOC

<

- 0 . 5

CO

p

< -1.0

-

LL

- 1 . 5

- 2 . 0

V = 49 . 8 4 km/h r^y / \ V !sT* l

v = 30 km/h |>7 /v \ \ \ yy / h \

-Vs'/ /

"

~^\\"

'/ /

Yv = 100 km/h r :

V = 150 km/hr

j 1 1 i

0 0 .5 1 1. 5 2 2 .5Time (s)

Figure 4.37: L inea r 1 Hz Sine Steer F ron t Tire Slip Angle Response

81



Chapter 4 Tw o Deg r e e - o f - F r e e dom Vehic le Mode l

1. 5

1.0

0 .5 -

TO

0 .0TOC

<

- 0 . 5

CO

2F - 1 . 0

CO

DC- 1 . 5

- 2 . 0

- 2 . 5

1

__

V = 30 km/hr^^-f \ \ \

\ V = = 4 9 . 8 4 km/nr \\

// ! \

\ V = 100 km/hr \

\ M

V= 1 50 km /h r

0 .5 1 .5Time (s)

2 .5

Figure 4 .38 : Linea r 1 Hz Sine Steer Rea r Tire Slip Ang le R e s po n se

0 .5

0 .4

0 .3

3 0 .2Z

o

1 o .i

o

3 o .o

I " 0 . 1 hco

-0.2

- 0 . 3

- 0 . 4

f\

J \v = 150 km/hr

\- \

// \V = 100 km/hr L'

f ^ A^ ^ \ v = 49 ,84 km/hr"

'/its.' ~/Tr<~

V = 30 km/h r^^VVy; / /N\l^-

\ 1 / /

VI y

.1 1 1

0 .5 1. 5Time (s)

2 .5

Figure 4.39: Linea r 1 Hz Sine Steer Latera l Acceleration Response

82



Chapter 4 Two Deg r e e - o f - F r e edom Vehic le Mod e l

3 .0

2 .5

-52 2 .0E

o

-2 1 5CD

" J

>

2

* 1 .0

0 .5

0 .0

V = 150 km/h r

,

',

V = 100 km/hr"

i

i

i

V=

49 .84 km/h r

:

*

1 1 1

V = 30 km/hr

0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1.4 1 .6Time (s)

Figure 4.40: Linea r Step Aero Side Force Lateral Velocity Response

0 .00

-0.02

JO

T3co

-0.04

'oo

-0.06

>

> -0.08

-0.10

-0.12

0 0 .2 0 .4 0 .6_

0 .8 1 1.2 1 .4 1 .6

Figure 4.41: Linea r Step Aero Side Force Yaw Velocity Response

83



Chapter 4 Two Deg r e e - o f - F r e edom Vehicle Mode l

TO

TOC

<

Jo

CO

4 .0

3 .5

3 .0

2.5

2 .0

1 .5 -

1.0

0 .5

0 .0

V = 15 0 km/h rr

i

V = 100 km/hr

V = 49.84, km/h rJr

V = 30 km/h r

i

i

i

t i 1 i

i

i

0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1 .4Time (s)

Figure 4 .42 : Linear Step Aero Side Force Sideslip Ang le R e sp o ns e

3 .5

3 .0

CD<-D

1? 2 .0<

f1.5

p 1.0 -

0 .5

0 .0

1 .6

V= 1 50 km /h r

V = 100 km/hr

V = 49.84 km/hr|

V = 30 km/hr

V i i 1 1 J I i

0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1 .4 1.6Time (s)

Figure 4.43: Linear Step Aero Side Force F r on t Tire Slip Angle Response

84



Chapter 4 Two Deg r e e - o f - F r e e dom Vehic le Mod e l

4 .0

3 .5

^ 3 .0TO

2 .5TOC

<

J2 - 2 .0co

2F 1.5

CO

DC1.0 -

0 .5

0 .0

i

,V= 150 km/h r

i i

!V= 100 km/h r

! V = 49 . 8 4 km/h r

---

-/-/-i V = 30 km/h r

i i i \ 1

0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1.4 1 .6Time (s)

Figure 4 .44 : Linea r Step Aero Side Force Rea r Tire Slip Angle Response

0 .6

0 .4

0 .2

0 .0

o

\

oo

<

2

3 - o - 2

CO

-0.4

-0.6

V = 30 krr l/hr

V=

49 .84 km/hr

V= 100 km/hr

1 1 1 1

V= 150 km/hr

0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1.4 1. 6Time (s)

Figure 4.45: L inea r Step Aero Side Force La t er a l Acceleration Response

85



Chapter 4 Two Deg r e e - o f - F r e edom Vehic le Mode l

0 .05

0 .04 -

jo

~0 .03

+-*

'ao

>

2 0 .02

0.01

0 .00

;

V = 150 km/h r

i

. /j

V= 100 km/h r

i

V = 49 . 8 4 km/hr

l~r

i i i 1

V = 30 km/h r

0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1.4Time (s)

Figure 4 .46 : L inea r Step Road S id e S lo p e La t e r a l Velocity Response

1 .6

3 .5E-04

3 .0E -04

2 .5E-04 -

T3

2~2 .0E-04

+-*

"oo

S1 . 5E -04 -

co>-

1 . 0 E - 0 4

5.0E-05

0 . 0E+00

1

V= 1 50 km /h r

V= 1 00 km /h r

V = 49 .84 \ km/hr

V = 3 0 km /hr1

i i_. i i

0 0 .2 0 .4 0 .6 0 .8 1 1. 2 1 .4Time (s)

Figure 4 .47 : Linea r Step Road Side Slope Yaw Velocity Response

1 .6

86




0 .07

0 .06

0 .05 -

TO

0 .04 -

TOc

<

0 .03CO

T3

CO

0 .02

0.01

0 .00

V = 30 km/hr ; ; ;

j / 0 ^ t . ii . i I

/ Ai=

49.84k n r v h r ^ - '

V = 100k m / h r -

V = 150 km/h r

L i i i i

0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1 .4Time (s)

F igu re 4 .48 : Linear Step Road Side Slope Sideslip Angle Response

0 . 0 7

0 .00

V= 150 km/hr

1. 6

0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1 .4 1 .6Time (s)

Figure 4.49: L inea r Step Road Side Slope F ron t Tire Slip Angle Response

87



Chapte r 4 Two Degree-of-Freedom Vehic le Mode l

0 .07

0 .06 -

V = 1 00 km/h r

TO 0 .05

TOC 0 .04<

Q .

(f )

(1> 0 .03

H\

CCCD 0 .02

0.01

0 .00

V = 150 km/hr

0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1.4 1 .6Time (s)

Figure 4.50: L inea r Step Road S id e S lo pe Re a r Tire Slip Ang l e Response

0 . 018

0 . 016

0 . 014

S 0 . 012co

2 0 . 010

|0 . 008

2% 0 . 006CO

0 . 004

0 . 0 02 h

0 . 000

-" W = 30

\ \v =

km/hr" ~ "

\ v :

= 49 .84 km/hr

vVcV = 1 0 0 l cm/hr !

V = 1 50 km /h rV ^W '"^*H^^

V^ ' ^^^. -i

'

0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1 .4 1 .6Time (s)

Figure 4.51: Linea r Step Road Side Slope Latera l Acceleration Response

88



Chapte r 4 Two Degree -o f -Freedom Vehic le Mode l

4 .6 No n - L i n e a r Mo d e l

A s previously noted th e linear vehicle model is valid fo r la teral accelerations up to

approximately 0.35 g. This is primarily a resul t of t ire l at eral fo rce being l inear with respect

to slip a ng le a t sm all slip ang le s , and hence small lateral accelerations. Beyond 0.35 g when

higher slip angles are being attained a non-linear t i re model is usual ly necessary to

accurately predict t i re lateral forces.

As d i scu s sed in Chapte r 3 many models of t ire behavior exist. The t ire model

chosen fo r t h i s work is called t ire data nondimensionalization and was originated by Hugo

Radt . This tire model is d i scu ss ed i n detai l in Section 3.4.

In th is section th e equations describing th e non-linear tw o degree-of-freedom

vehic le model are presented. Simulat ion of th e model is performed fo r selected steering

inputs and the results are compared with th e simulation of th e linear model.

4.6.1 Mode l Equat ions

The equations of motion fo r th e non-linear tw o degree-of-freedom vehic le a re

derived in Sec ti on 4 .3 and are repeated here .

F^ cos 8 + F + F + F =m(v + u r)

(4.9)aF^ cos 5 - bFyr

- (c -

a)Fya = I^r

Express ions fo r th e t ire slip angles are derived in Section 4.4.

t f = ^ f ) - i,

- atan | - 8

,v -M

(4'14>

a, = a t a n

u j

The t i re lateral f or ce is given by th e following expressions as described in

Sec t ion 3 .4 and repeated here fo r convenience. From t h e s e equations th e tire lateral force F

can be calculated based upon t he tir e vertical load

Fzand t he t ir e

slipangle a.

89



Chapte r 4 Two Degree -o f -Freedom Vehic le Mode l

CC=B3 + C3FZ (3.3)

Hy=B5 + C5Fz (3.5)

_=Qtan ( a ) (3 6)

_ / x_ FatanfB.a) ,^x

y/ = ( l-E1)a+- ^-^- (3.10)i

0 =C, atan(f l^) (3.9)

Fy=DlSin(tJ) (3.8)

Fy=FuFz (3.11)

4.6 .2 Simulat ion

Simulat ion of th e non-linear tw o degree-of-freedom vehicle model is implemented

in th e MATLAB script DOF2NLS im .m . This script is listed in Appendix C.8 and is very

similar to DOF2LS im .m which performs simulation of th e linear model. As with th e linear

s imu l a t i o n , th e scripts DOF2Control.m, DOF2Param.m, and DOF2DependParam.m are

c al le d a t th e beginning of DOF2NLS im .m to set s imu l a t i o n , veh ic l e , and t i re parameters.

The built-in MATLAB function ode23 is used aga in to integrate th e differential equations of

motion which are contained in th e funct ion DOF2NLDE .m . This function calculates th e

state derivatives v and r based upon th e instantaneous values of th e state variables v and r

and th e current steer angle. DOFTNLDE .m is l is ted i n Appendix C.9. The state derivatives

are calculated as

m

laF^ cos(5)-

2bFyr- (c -

a)Fya(4'89)

r ==

90




These expressions are obtained by solving Eq. (4.9) fo r v and r .

The most significant di ffe rence be tween th e non-linear and l inear model simulations

is in th e calculation of t ire lateral forces. The t ire lateral forces are calculated by th e

MATLAB function NLTire.m which is lis te d in Appendix A.3 . This script t a k e s t he tir e

vertical load and slip angle as arguments and returns t h e t ir e lateral force. Note t h a t with the

non-linear t ire model th e lateral forces F^ and F,r are fo r only one tire, while fo r

simplification in th e linear model they are fo r tw o tires. Thus here they are multiplied by th e

factor of tw o in Eq. (4.89) to get th e l a te ra l fo rces fo r tw o t i res . NLTire .m is called at each

t im e step by DOF2NLDE.m, which also calls th e funct ion SteerAngle .m to calculate th e

instantaneous steer angle.

For comparison with th e linear mod e l , simulation of th e non-linear model is

performed fo r th e step steer input and th e r amp square steer input. As with th e linear

mo d e l , simulations are performed fo r forward velocities of 30 km/hr, 48 .94 km/hr, 10 0

km/hr, an d 15 0 km /h r. T h e step steer and ramp square steer inputs are identical to t h o s e

used fo r th e linear mod e l , having a magnitude of 1. Tire parameters fo r th e non-linear t ire

model a re g iv en in Table 3.1. These parameters are a result of th e curve fitting of empirical

tire data done in Sect ion 3.4. The values of th e linear t i re cornering stiffnesses used in

throughout Sec ti on 4 .5 are derived from t hese p a r am e t e r s , so th e linear t i re model and non

linear t ire model a gr ee a t small slip angles. Vehic le parameters are identical to th os e used in

the l inear simulation. Results from th e simulations are provided in F igu re 4 . 52 t h r ough

Figure 4.63. Included on t h e s e plots as dashed lines are th e linear simulation results fo r

comparison.

Lateral ve loc i t y, yaw ve loc i t y, sideslip ang le , front t ire slip ang le , rear t ire slip

angle, and lateral acceleration results fo r both th e non-linear and linear simulations are

presented in F ig u re 4 .5 2 through Figure 4.57 fo r th estep

steer input. The li ne a r and non-

91



Chapte r 4 Two Deg r e e - o f - F r e edom Vehic le Mode l

linear lateral acceleration results agree within 1% over th e complete durat ion of th e

simulation fo r forward velocities of 30 km/hr and 49 .84 km/hr. These speeds correspond

to steady-state lateral accelerations of 0.05 g and 0.14 g respectively. A t 100 km/hr, which

produces a 0.55 g steady-state lateral accelera t ion, th e linear model lateral acceleration

resu lt s exceed those of th e non-linear by 4.3% during th e transient and 1.0% once steady-

state is reached. A t th is speed th e tire slip angles reach slightly more than 2. A t t h e s e slip

angles th e t ire lateral force versus slip angle curve is still very nearly a straight line. Thu s

fo r th is tire and vehicle th e linear t ire model is reasonably accurate and useful fo r lateral

accelerations in excess of 0.5 g. However, at 15 0 km/hr th e l inear model lateral

accelerations exceed t h o s e of th e non-linear model by over 27%. A t th is speed th e no n

linear model predicts a steady-state lateral acceleration of 0.95 g while th e linear model

predicts 1.20 g. The t ire slip angles have exceeded6

where th e l a te ral fo rce versus slip

angle curve is approaching it s peak. The linear t ire approximation is not sufficiently

accurate at slip angles of th is magnitude.

A t high speeds th e linear model predicts th a t th e magnitudes of lateral ve loc i t i e s ,

sideslip ang l e s , and t i re slip angles are below t h o s e t h a t th e non-linear model predict s and

t ha t t he yaw vel oc it ie s and lateral accelerations are above t h o s e of th e non-linear model. The

linear model also predicts faster response t h a n th e non-linear model. At 15 0 km/hr th e non

linear model predicts overshoot in all of th e quantities ex am in ed , while th e linear model

predicts no overshoot.

Non- l inea r and linear simulation results fo r th e ramp square steer inpu t are plotted

in F ig u re 4.58 through Figure 4 .6 3. T h e differences between th e non-linear and linear

models fo r th is input are similar to t h o s e of th e step steer input. T he tw o models agree very

well fo r forward velocities of 30 km/hr and 4 9 .8 4 km / hr. As with th e step steer input, at

th e higher speeds th e linear model predicts peak lateral veloci ty, sideslip ang le , and t ireslip

angle magnitudes below t h o s e of th e non-linear model and predicts peak yaw velocities and

92




lateral accelerations above those of th e non-linear model. Differences in peak la teral

acceleration reach 25%. Again, the linear model predicts faster response than th e non-linear

model. In pa r t i c u l a r , at 150 km/hr th e response of th e non-linear model lags th e linear

model by approximately 0.5 seconds after th e steer input is ramped back down to zero.

Here differences between th e linear and non-linear lateral accelerations reach nearly 100%.

Compar i son of th e linear and non-linear model simulations shows that at low slip

angles and lateral accelerations th e linear vehicle and t i re models can produce results

comparable to th e non-linear model. E ve n fo r th e 10 0 km/hr case where slip angles exceed

2

and th e lateral acceleration reaches 0.55 g th e linear model produces results t h a t are

acceptable fo r most engineering purposes. When t ire slip angles and lateral accelerations

become h ig h it is necessary to have a non-linear tire model to obtain accurate results.

However, s in ce most driving is done at lo w slip angles and lateral acce l e r a t i ons , th e linear

mod el a nd th e l inear analysis t e c hn i qu e s presented in Section 4.5.6 t h r o u g h Sec ti on 4 . 5. 26

can be used both to study vehicle behavior and to design vehicles to have desirable

performance characterist ics over a wide varie ty of operating conditions.

93




CO

E

o

o

>

2

To

0 .5

0 .0

- 0 . 5

-1.0

- 1 . 5

- 2 . 0

- 2 . 5

- 3 . 0

- 3 . 5

- 4 . 0

- 4 . 5

V = 30 km/h r

V = 49 . 8 4 km/hr

= 100 km/h r'

[ V

Linear

No r l-Linear

\ s\ *s

Linear

V= 150 km/hr

.

Non-Linear

>

1 2 3 4Time (s)

Figure 4 .52 : Non -L inea r Step Steer La t e r a l Velocity Response

0 .30

0 .25

;

0 .20

co

o 0 .15o

>

>-1 0 h

0 .05

0 . 00

/.

\ Linear

\/ 1 <^ n km/hr

Ifij

^ ^ ^ ! Non-Linear

;V= 1 00 km/h r

V = 49 .84 km/hr

i i i

Time (s)

Figure 4 .53 : Non-L inea r Step Steer Yaw Velocity Response

94



Chapte r 4 Two Degree-of-Freedom Vehic le Mod e l

1.0

0 .0

- 1 . 0-

- 2 . 0-

TO

TOC

<

- 3 . 0

CO

CO- 4 . 0

- 5 . 0

- 6 . 0

V = 30 km/h r1 *

V = 49 . 8 4 km/hr

\ ^^^ ^ _' V = 100 km/h r

, ,

V.j

'

;

i Linear

V= 150 km/hrt i

^-''

; Non-Linear

012

3 4Time (s)

F igu re 4 .54 : Non-Linear Step Steer Sideslip Angle Response

1.0

0 .0 -

- 1 . 0

TO

o

-2.0

TO

C

<Q . -3.0

CO

r- - 4 . 0

4_d

c

o

u- - 5 . 0

- 6 . 0

- 7 . 0

V = 30 km/hr1 '

, p

V = 49 . 8 4 km/hr1 '

V = 1 00 km/h r

\ *" **- _ Linear

-,- -

V = 150 km/hr---

^ , Non-Linear

1 1 1

012

3 4Time (s)

Figure 4.55: Non-Linear Step Steer Fron t Tire Slip Angle Response

95



Chapter 4 Two Degree-of-Freedom Vehic le Model

1.0

0 .0

-.- 1 - 0

TO

CD - 2 . 0

TO

C

<j?- - 3 . 0

co

2p - 4 . 0

CO

c- 5 . 0

- 6 . 0

-7 . 0

V = 30 km/hri

v"̂! 1

'. V = 49.84 km/h ri i

\ '

\**"' "~" " ~~ r

V = 100 km/hr

V^* - ^ . Linear

x.1 v. , -v = i50km/nr

-

,i

_^ -

"~

Non-Linear

012

3 4Time (s)

Figure 4.56: Non-Linear Step Steer Rea r Tire Slip Angle Response

1 .4

1.2

1 .0 -

Co

nS 0 .8

o

< 0 .6

2

3 0 .4

0 .2

0 .0

/

/ ,

Linear

V= 150 km/hr

/ ^

'

. . . J - j / - L

; Non-Linear

//

-I f.

'

__, --i-'

S V= 1 00 km/h r

i^

V = 49.84 km/hr" '

V = 30 km/hr

i123 4

Time (s)

Figure 4 .5 7: N o n- Lin ea r Step Steer Lateral Acceleration Response

96




0 .5

0 .0

- 0 . 5

" c o "

- 1 . 0

b'*~^r

-1.5

oo

> - 2 . 0

m

co - 2 . 5

- 3 . 0-

- 3 . 5

- 4 . 0

;V = 30 km/h r

" ^ vV = 49 .84 km/hr

y.

^^ . 4

V = 1 00 km/h r

/ /

f : / :

/ /

--\n ; Linear ,

\_V,H /

7 / ;

; \^ / V = 150 km/h r

Non-Lineari i i. 1

0 0 .5 1 1 .5 2 2 .5 3 3.5Time (s)

Figure 4.58: Non -L inea r Ramp Squa re S te e r La t e r a l Velocity Response

co

oo

0 .30

0 .25

0 .20

0 .15

g 0 .10

co

0 .05

0 .00

-0.05

0 0 .5 1 1 .5 2 2.5 3 3 .5Time (s)

Figure 4.59: Non-Linear Ramp Square Steer Yaw Velocity Response

97



Chapte r 4 Two Degree-of-Freedom Vehicle Mod e l

1.0

0 .0

- 1 . 0

- 2 . 0

TO

TO

<

-3.0

CO

gCO

- 4 . 0

- 5 . 0

-6 . 0

V = 30 km/h r

< S ^^VvV = 49 .84 km/hr

'

^'"*^'- """ r

^

L -

'- -

'

/

V /

\ \ Linear / /

Nv /v= 150 km/hr

^__^ j

'1

1 Non-Linear

0 0 .5 1 1 .5 2 2 .5 3 3.5Time (s)

Figure 4 .60 : Non-Linear Ramp Square Steer Sideslip Ang le R e sp o ns e

1.0

0 .0

to -i.o

c -2.0

<

Q .

CO- 3 . 0

2 - 4 . 0

-5.0

- 6 . 0

\/ = 3Dj h < V- . :_^_

y ' / ^ r ^ ~ \ ^V V = 49 .84 km/hr

. ^

> "

X

\X v = 100 km/hr /^ / /

:\ j / / :

: \ / /Y _ Linear 1 ; / ;

V / /\ s^ / : / : ;

Nw r ^ \ j = 15 0 km/hr

Non-Linear

i i i i i 1l

0 0 .5 1 1 .5 2 2 .5 3 3 .5 4Time (s)

Figure 4 .61 : Non-Linear Ramp Square Steer Fron t Tire Slip Angle Response

98



Chapter 4 Two Degree-of-Freedom Vehic le Mode l

1 .0

0 .0

1 .0 -

TO -

? - 2 . 0

<Q .

CO- 3 . 0

CO

- 4 - 0

- 5 . 0

- 6 . 0

^-_- V = 30 km/hr -

; ;

^^sj_' "^ y

ffr /--

/ /

\ V = 49 .84 km/hr

\\v = 1 00 km /h r /

\ ^ < ^ \ */X ^ : / : 7\ / : /

., \_, / ,/

\\ Linear /\N '

/

1 f

/ V = 1 50 km/h r

; Non-Lin ear1 1

3co

S 0 .6

oo

<

2

15

0 0 .5 1 1 .5 2 2 .5 3 3.5 4Time (s)

Figure 4 .62 : Non-Linear Ramp Square Steer Rea r Tire Slip Ang le R e sp o ns e

1 .2

1 .0 h

0 .8

0 .4 -

0 .2

0 .0

-0.2

0 0 .5 1 1 .5 2 2 .5 3 3 .5 4Time (s)

F igu r e 4.63: Non -L inea r Ramp Square Steer Lateral Accelerat ion Response

99



5 Conclus ion

In the ear ly part of t h i s century as th e top speeds of automobiles increased vehicle

dynamics became an important consideration fo r engineers. Manufac tu re r s had to meet

higher and higher standards of per fo rmance , particularly in th e areas of safety and comfort.

Mathematical modeling of vehicle dynamics has become an excellent w ay fo r engineers to

study vehicle behavior and to reduce th e tim e and cost to develop vehicles which meet

performance goals. There is a g re at deal of literature on th e to p ic of vehicle dynamics .

Lateral vehicle dyn am i cs in particular h as b ee n a t op i c of great interest due to it s

relationship with safety. Two areas of f oc us in th e literature concerning th e modeling of

la teral dynam ic s h av e b e en th e tw o degree-of-freedom vehicle model and models of t ire

behavior. Since t i res play an extremely impor tan t role in th e l at era l dynamics of road

veh ic l e s , sufficiently accurate representation of t ire mechanics is essential fo r vehicle

models.

In Chapte r 3 an overview of t i re l a te r al f o rc e mechanics was given. Tw o

representations of t i re lateral forces were used. In th e linear t i re model th e lateral force was

considered to be a linear function of th e t ire slip angle. The non-linear t ire model utilized a

method called t ire data nondimensionalization to predict lateral force. In th is method

exper imenta l ly measured t ire lateral force versus slip angle curves fo r several vertical loads

are normalized and curve fit. Tire lateral force can t h e n be predicted as a non-linear function

of both vertical load and slip angle.

In Chapte r 4 th e equations of motion fo r a tw o degree-of-freedom vehicle model

were derived from basic principles of Newton ian mechanics. The model was t hen

deve loped in two forms, linear and non-linear. The l inear vehicle model utilized th e linear

tire model. Transfe r funct ions were written relating both yaw velocity and sideslip angle to

the inputs of s t ee r ing , aerodynamic side force, and road side slope angle. Expressions fo r

10 0



Chapter 5 Conc lus ion

steady-state step inpu t response gains were derived from t he t ra n s fe r funct ions. Severa l

other measures of steady-state stability were derived including th e understeer gradient and

t a ngen t speed. Express ions fo r t r a n s i e n t response characteristics such as natural frequency,

damping r a t io , and pol es and zeros were developed . Num er ica l simulation of th e response

of th e model to step s teer, ramp-step s teer, ramp-square s teer, sine s teer, step aerodynamic

side force, and step road side slope inputs was performed. It was seen that th e steady-state

and t r a n s i e n t response characteristics of th e vehicle were very dependen t upon its forward

speed. In par t i cu la r, when th e forward speed was above t he t an g e nt speed of th e veh ic l e ,

th e zero assoc ia ted wi th sideslip angle response to steer input became positive. The effect

of t h i s on th e vehicle w as see n clearly in th e frequency response and in the simulation. For

some combinations of speed and input magnitude th e linear model predicted lateral

accelerations higher t h a n were actually possible due to th e assumption of linear tire

behavior. In all cases tested th e steady-state response ga ins , frequency r e s pon s e , and

s imula tion resul ts were in agreement.

The non-l inear veh ic le model used a th e non-linear tire model fo r predict ing t ire

lateral forces during simulation. This model was seen to predict reasonable responses a t

high slip ang le s and lateral accelerations. Compar i son with th e linear model showed t ha t fo r

th e vehicle studied th e linear mode l was reasonably accurate fo r most engineering purposes

up to slip angles of2

and lateral accelerations of 0.5g. It was seen t h a t fo r accurate

modeling of vehicle response a t high slip angles and lateral accelerations a non-linear

representation of th e t i r e s was necessary.

101



References

1. Gillespie, Thomas D. Fundamenta l s o f Vehicle Dynamics. Warrendale, PA : SAE,

1992 .

2 . Lanchester, F. Will iam. "Some Ref lec tions Pecu lia r to th e Design of anA u t o m o b i l e . "

Proceedings o f th e Institution o f Mechanica l Engineers, Vol. 2, 1908, p. 187-257 .

3 . Olley, Maur ic e . "Su spen si on andHandling."

Detroit, MI: Chevrolet EngineeringCenter, 1937 .

4 . Olley, Maurice. "Notes onSuspensions."

Detroit, M I: Chevrolet Engineering Center,1961 .

5 . Olley, Maur ic e . "Su spen si on s Notesn."

Detroit, MI: Chevrolet Engineering Center,1966 .

6 . Segal, Leonard . "Theoret ical Predict ion and Experimental Substant ia t ion of th e

Response of th e Automobi le to SteeringControl."

Proceedings o f the Automobile

Division o f the Institution o f Mechanica l Engineers, 1956-1957 .

7 . Whitcomb, David W . and W illiam F. Milliken. "Des ign Implications of a Genera l

Theory of Automobi le Stability andControl."

Proceed ings o f th e Automobile Divis ion

o f the Inst i tut ion o f Mechanica l Engineers, 1956-1957 .

8 .

Bastow,D . and G Howard . C ar Suspension an d Handling. Warrendale, PA: SAE,

1993 .

9 . Cole, D.E . Elementary Vehicle Dynamics. A nn Arbor, MI: University of Michigan,1972 .

10 . Dixon, John C. Tyres, Suspension an d Handling, Cambridge, Eng land : Cambr idge

University Press, 1991 .

11 . Ellis, John R. Vehicle Dynami c s . London : Bu s in e ss Books, 1969.

12 .Ellis,

John R. Road VehicleDynamics, Akron,

OH : J.R.Ellis,

1989.

13 . Milliken, W ill iam F. and Doug L. Mill iken. Race C ar Vehicle Dynamics. Warrendale,PA : SAE, 1995.

14 . Mola, Simone . Fundamen t a l s o f Vehicle Dynamics, Detroit, MI: General Motors

Institute, 1969 .

15 . Reimpell, Jornsen and He lmu t Stall. The Automotive Chassis: Engineering Principles.

Warrendale, PA : SAE, 1996.

16 . Taborek, Jaroslav J. Mechan i c s o f Vehicles. Cleveland, OH: Penton, 1957.

102



Refe rences

17 . Wong, Jo Yung . Theory o f Ground Vehicles . New York : John Wiley & Sons, Inc.,1993.

18. Bundorf, R.T. and R.L. Leffert. ' T h e Cornering Compliance Concep t fo r Descr ip t ion

of Vehicle Direc t iona l ControlProperties."

SAE Paper No. 760713, Oct. 1976.

19 . Allen, R . Wade, Theodore J. Rosenthal, and Henry T. Szostak. "Steady State and

Trans ien t Analysis of Ground VehicleHandling."

SAE Paper No. 870495, 1987.

20 . Heydinger, Gary J. " Improved Simulation and Validat ion of Road Veh ic le HandlingDynamics."

Ph.D. Dissertation, Ohio State University, Columbus, Ohio, 1990 .

2 1. Xia, Xunmao . "A Nonlinear Analysis of Closed Loop Driver/Vehicle Per fo rmance

with F o u r Wh e el SteeringControl."

Ph.D. Dissertation, Department of Mechan ica l

Engineering, Clemson University, Clemson, SC, Dec . 1990.

22 . Trom, J.D., J .L. Lopex, and M.J . Vanderploeg. "Modeling of a Mid-Size PassengerCar Using a Multibody Dynamics

Program."

Transact ions o f the ASME, J o u r n a l o fMechanisms, Transmissions, an d Automation in Design, Vol. 109, Dec . 1987.

23 . Kortum, W . and W. S ch ie hle n. "Genera l Purpose Veh ic le S y stem Dyn am i cs Sof tware

Based on MultibodyFormalisms."

Vehicle System Dynamics, No. 14, 1985, p. 229-

263 .

24 . Clarke, S .K . (E d.) . Mechan i c s o f Pneumat ic Tires, DOT HS-805952, US

Gove rnmen t Printing Office, Washington, DC, 1981.

25 . Gim, Gwanghun and Parviz E. Nikravesh. "An Ana ly ti ca l Mode l of Pneumatic Tyresfo r Vehicle Dynamic Simula tions. Par t 1: Pure

Slips."

Internat ional Jou rna l o f Vehicle

Design, Vol . 11, No. 6, 1990 .

26. Bakker, Egbert, Lars Nyborg, and Hans B. Pacejka. "Tyre Modelling fo r U se in

Veh ic le Dyn am ic sStudies."

SAE P ap er No . 870421, 1987.

27 . Radt, Hugo S. and D .A . G lemm in g. "Normalizat ion of Tire Force and Momen tData."

Tire Science an d Technology, Vol . 21, No. 2, Apr.-June 1993.

28 . Allen, R. Wade, Raymond E. Magdaleno, Theodore J. Rosenthal, David H . Klyde,and Jeffrey R. Hogue . 'Ti re Modeling Requirements fo r Vehicle DynamicsSimulation."

SAE Pape r No. 950312, Feb . 1995.

29 . Society of Automo t i v e Engineers . "Vehicle DynamicsTerminology."

SAE J670e,1976 .

30 . Radt, Hugo S. "A n Efficient Method fo r Treating Race Tire Force-MomentData."

SAE Pape r No. 942536, Dec . 1994.

3 1. Meriam, James L. and L . Glenn Kra ige . Engineering Mechanics : Dynamics. N ew

York : John

Wiley& Sons, 1992.

10 3



Refe rences

32. Katz, Joseph. Race C ar Aerodynamics : Designing fo r Speed. Cambridge,Massachuset ts : Rober t Bentley, Inc., 1995.

33. Franklin, Gene F., J. David Powell, and Abbas Emami-Nae in i. Feedback Cont ro l o fDynamic Systems. N ew York: Addison-

Wesley Publishing Company, Inc., 1994 .

34. MATLAB Reference Guide. The MathWorks, Inc., 1994.

104



Append i x A Tire Mode l MATLAB P rog r ams

A.l M ag i cF i t.m

%Mag i c F i t Curve Fitting o f Tire Data to Magic Formula

%

% Finds p a r ame t e r s for Mag i c Formula cu rve f it o f t i re lateral force o r

% aligning moment v s . slip a n g l e data r e a d f r om file TireSlip.dat

%

% Created 4/21/96

% J. Kiefer

% Ini t ia l izat ion

c l e a r all

ele;

% Load Data from File

load TireSlip.dat

t = TireSlip(:,l) ;

y = TireSlip(:,2) ;

% Find Curve F it Parameters

xO = [ . 7 4 0 7 1.35 1.00 - 0 . 5 ] ;

x = l e a s t sq (' M a g i c E r r o r '

, xO , [] , [ ] , t, y)

% Construct F it Function

t l = l i n s p a c e ( 0 /max ( t ) ,10) ;

p s i=

(l-x(4))*tl + x ( 4 ) / x ( l ) * a t a n ( x ( l ) * t l ) ;

theta = x ( 2 ) * a t a n ( x ( l ) * p s i ) ;

F = x ( 3 ) * s i n ( t h e t a ) ;

% Plot Data an d F it Function

p lo t ( t l , F , t, y, 'o')

t i t l e([' Tire Data Magic Formula Fi t

(B='

num2str(x(l) ) ',C='

num2str(x(2) ) . .

',D=-

num2str(x(3)) ',E= '

num2s t r (x(4) ) ') '])

x l a b e l ('t

'

)

y l abeK ' y ' )

g r i d

105



Appendix A Tire Model MATLAB Programs

A . 2 M ag i cE r r o r.m

f un c t i on e= M ag i cE r ro r ( x , t, y)

%MagicError Error in M a g ic Formula Cu r v e Fi t%

%e =MagicEr ro r (x , t, y)

%

%

Calculates v e c t o r o f e r r o r s o f Mag i c F o rmu l a cu rve fi t g i v e n pa rame te r s

x an d data (t, y)

Inputs:

% x(l) B

%x(2) c

%x(3) D

% x(4) E

% t

% y% Outputs:

% e

%

% Created 4/21/96

% J. Kiefer

Vector of curve f it parameters

Vector o f i ndependen t data

Vector o f d e p e n d e n t data

Vector o f e r r o r s b e tw e e n data an d f it function

p s i = (l-x(4))*t + x(4)/x(l)*atan(x(l)*t);theta = x(2)*atan(x(l)*psi);F =

x(3)*sin(theta);

e=

y- F ;

10 6



Appendix A Tire Mod el MATLAB Programs

A . 3 NLTi r e .m

function Fy= NLT i r e ( F z , alpha)

%NLTire N b n Linear Tire Model Lateral Force

%

%Fy = NLTi r e (Fz , alpha)%

% Calculates t i re lateral force f r om inputs o f t i r e v e r t i c a l load an d slip

% ang le . Based on Radt's t i re data n o n d im e n s i o n a l i z a t i o n model an d th e

% Magic Formula mode l . Force is for on e t i r e . Called by th e function

% D0F2NLDE .m .

%

% Inputs:

% a lpha Tire slip a n g l e ( rad)% F z Tire v e r t i c a l load (N)% Outputs:

% Fy Tire lateral force (N)

%

% Created 2/18/96

% J. Kiefer

g l o b a l Bl C I Dl El B3 C3 B5 C5;

% Normalization Parameters

C c = B3 + C3*Fz ; % N/deg/N Corne r i ng c o e f f i c i e n t

mu= B5 + C5*Fz ; % N /N Friction c o e f f i c i e n t

% Normalized Sl ip Angle

alphaN = Cc.*tan (a lpha) . / m u * 1 8 0 / p i ;

% Normalized Lateral Force

ps iFN= (l-El)*alphaN + E l /B l * a t a n (B l * a l p h aN ) ;

thetaFN = C l * a t a n (B l * p s i FN ) ;

Fy N = Dl*sin( t h e t a FN ) ;

% Lateral Force

Fy = - F yN . *m u . * F z ;

10 7



Append i x B Tw o DO F Mode l Mathema t i c a Sess ion

Stability Derivat ives

SDRules = {Y b -> Cf + Cr, Yr -> (a Cf - b Cr)/V#

Yd -> - C f , N b -> a Cf - b Cr, Nr -> (aA2 Cf +

bA 2 Cr)/V, Nd -> -a C f}

a Cf - b Cr

{Y b -> Cf + Cr, Yr ->, Y d ->

-Cf, N b -> a Cf - b Cr,

V

2 2

a Cf + b Cr

Nr ->, Nd -> -(a Cf)}

V

Transformed Equations of Mot ion

A = {{s-Yb/(m V), l -Yr/(m V)},

{-Nb/Izz, s - N r / I z z } } ;

Matr ixForm [A ]

Y b Yr

s- 1

m V m V

N b Nr-(-) -( ) + s

Iz z Iz z

Bl = {Yd/(m V), Nd/Izz};

M a t r ix F o rm [Bl]

Y d

m V

N d

Iz z

B2 = {l/(m V), (a -c) / Izz};

M a t r ix F o rm [B2]

1

m V

a-

c

Iz z

108



Append ix B Tw o D OF Model Mathematica Session

B 3 = { g /V, 0};

MatrixForm[B3]

g

v

B 4 = {l/(m V), (a-d)/Izz>;Matr ixForm [ B 4 ]

m V

a - d

Iz z

Transfer Function Denomina to r

Ds = Co l l e c t [Det [A ] ,s ]

N b 2 Nr Y b Nr Y b N b Yr

+ s + + s (-( ) )Iz z Iz z m V Iz z m V Iz z m V

Transfer Function Numerators

Nbd =

Co l l e c t [Det [Transpose [ R e p l a c e P a r t [Transpose [A],Bl,l]]],s]

N d Nr Yd s Y d N d Yr

Iz z Iz z m V m V Iz z m V

Nba = Co l l e c t [Det [Transpose [ReplacePart [

Tr a n s p o s e [A],B2,l]]] ,s]

a c Nr s a Yr c Yr

Izz Izz Iz z m V m V Iz z m V Iz z m V

Nb t = Co l l e c t [Det [Transpose [Rep lacePar t [

Tr a n s p o s e [A] ,B3 , l ] ] ] , s ]

g Nr g s

-( ) +

Iz z V V

10 9



Appendix B Two DOF Model Mathematica Session

Nrd = Co l l e c t [Det [Transpose [ReplacePart [

Tr an spo s e [A ] , Bl, 2 ] ] ] , s]

N d s N d Y b N b Y d

+

Iz z Iz z m V Iz z m V

Nr a = Co l l e c t [Det [Transpose [ReplacePart [Transpose [A],B2,2]]],s]

a c N b a Y b c Y b

Iz z Iz z Iz z m V Iz z m V Izz m V

N rt = Co l l e c t [Det [Transpose [ReplacePart [Tr a n s p o s e [A ] , B3 , 2 ] ] ] , s]

g N b

Iz z V

Nr n = Co l l e c t [Det [Transpose [ReplacePart [Transpose [A ] ,B4 , 2 ] ] ] , s]

ad N b a Y b d Y b

Iz z Iz z Iz z m V Iz z m V Izz m V

Transfer Functions

Sideslip Angle

Gbd = Nbd/Ds

N d Nr Y d s Y d N d Yr

Iz z Iz z m V m V Iz z m V


+ s + + s (-( ) )Iz z Iz z m V Iz z m V Izz m V

Gba = Nba/Ds

a c Nr s a Yr c Yr

Iz z Izz Izz m V m V Izz m V Iz z m V


+ s + + s (-( ) )

Iz z Iz z m V Iz z m V Iz z m V

110



Ap p en d ix B Two D OF M odel Mathemat ica Session

Gbt = Nbt/Ds

g Nr g s

( ) +

Iz z V V



Yaw Velocity

Grd = Nrd/Ds

N d s N d Y b N b Y d

+




Gra = Nra/Ds

a c N b a Y b c Y b

Iz z Iz z Iz z m V Iz z m V Iz z m V

Nb 2 Nr Y b Nr Y b N b Yr+ s + + s (-( ) )

I z z Iz z m V Iz z m V Iz z m V

Grt = Nr t /Ds

g N b


Iz z V ( + s + + s (-( ) ) )Iz z Iz z m V Iz z m V Iz z m V

Grn = Nrn/Ds

ad N b a Y b d Y b

Iz z Iz z Iz z m V Iz z m V Iz z m V


+ s + + s (-( ) )


111



Ap p en d ix B Two DOF Mode l Mathemat ica Session

Steady State Step-Input Respon s e Gains

Sideslip Ang l e

Sbd = Simplify [Limit [ G b d , s->0] ]

-(m Nd V)- Nr Y d + N d Yr

m N b V + Nr Y b - N b Yr

Sba = Simplify [Limit [ G b a , s ->0 ] ]

-Nr - a m V + c m V + a Y r - cY r


Sb t = Simplify [Limit [Gbt, s ->0] ]

g m Nr

_ ( )

m Nb V + Nr Y b - N b Yr

Y aw Velocity

Srd = Simplify [Limit [Grd, s - > 0 ] ]

-(Nd Yb) + N b Y d


S ra = Simplify [Limit [Gra, s->0] ]

N b -a Y b + c Y b

m N b V + Nr Yb- N b Yr

S r t = Simplify [Limit [Grt, s - > 0 ] ]

g m N b


S rn = Simplify [Limit [Grn, s ->0 ] ]

N b - a Y b + d Y b


F ron t Tire Slip Ang l e

Safd = Simplify [Sbd + a /V Sr d - 1 ]

2 2

(m N b V + m N d V + a N d Y b + Nr V Y b -a N b Y d + Nr V Y d

N b V Yr - Nd V Y r) / (V (-(m N b V)- Nr Y b + N b Y r) )

112



Ap pe nd ix B Two DOF Mode l Mathemat ica Session

Safa = Simplify [Sba + a /V Sra]

2 2 2

(- (a N b) +NrV+amV - c m V +a Yb-acYb-aVYr+

c VY r)

/ (V (-(m N bV)

- Nr Y b + N bY r) )

S a f t = Simplify [Sbt + a /V Srt]

g m (a N b - Nr V)

V (m N b V + Nr Y b - N b Y r)

Re ar Tir e Slip Ang l e

Sard = Simplify [Sbd - b/V Srd]

2

m N d V - b N d Y b + b N b Yd + Nr V Yd - N d V Yr

V (-(m N b V)- Nr Y b + N b Y r )

Sa ra = Simplify [Sba - b/V Sra]

2 2

(bNb + N rV + amV - c m V - a b Y b + b c Y b - a V Y r +

c V Y r) / (V (-(m N b V)- Nr Y b + N b Y r) )

S a r t = Simplify [Sbt - b/V Srt]

g m (b N b + Nr V)

V (-(m N b V)- Nr Y b + N b Y r)

Path Curva tu re

Sc d = S im p l i f y [ l/ V Srd]

- (N d Yb) + N b Y d


Sea = Simplify [1/V Sra]

N b -a Y b + c Y b


Se t = S im p l i f y [ l/ V Srt]

g m Nb

V (m N b V + Nr Yb- N b Y r )

113



Append ix B Two D OF Mode l Mathema t i c a Session

Lateral Accelerat ion

SAd = Simplify [V / g Srd]

V (-(Nd Yb) + N b Yd)

g (m Nb V + Nr Yb - Nb Y r)

SAa = Simplify [V / g Sra]

V (N b -

a Y b + c Yb)

g (m N b V + Nr Y b - N b Y r)

SAt = Simplify [V /g Srt]

m N b V


Steer Ang l e Re spon s e to P a th R a d iu s

deltaR = d e l t a / . So lve [Scd == 1/R / delta, delta] [ [1, 1 ] ]

2

- (m N b V ) - Nr V Y b + N b V Yr

_ ( )

-(Nd R Yb) + N b R Y d

Terml = C o e f f i c i e n t [Expand [deltaR] ,V, 2 ] VA 2

2

m Nb V

-(Nd R Y b) + N b R Yd

TermlS = Simplify [Terml / . SDRules]

2

(a Cf - b Cr) m V

a Cf Cr R+

b CfCr R

TermlSa = Nume r a t o r [TermlS] / Simplify [Denominator [

TermlS] / . a ->L-b ]

2

(a Cf - b C r) m V

Cf Cr L R

11 4



Append i x B Two DOF Model Mathema t i c a Session

Terms 2 3 = E x p a n d N um e r a t o r [Simplify [Coeff ic ien t [Expand [

de l t aR ] , V ] V]]

- (Nr V Yb) + Nb V Yr

Nd R Yb- Nb R Y d

Terms23S = Simplify [Terms23 / . SDRules / . a->L-b]

L

R

de l taRl = Terml + Terms23S

2

L m N b V

- +

R - (N d R Yb ) + N b R Yd

deltaR2 = TermlSa + Terms23S

2

L (a Cf - b C r) m V

-+

R Cf Cr L R

Understeer Gradient

Kus = Co e f f i c i e n t [Simplify [ d e l t aR l R g] ,VA2]

g m N b

-(Nd Y b) + N b Y d

Kus l = Simplify [Kus / . SDRules]

(a Cf - b Cr) g m

(a + b ) Cf Cr

StabilityFac tor

Kl = Simplify [K / . So lve [Srd == V/(L (1+K VA2)), K][[l]]]

2

m N b V + L N d Y b + Nr V Y b - L N b Y d - N b V Yr

2

L V (-(Nd Yb) + N b Yd)

115



Append ix B Two D OF Mode l Mathematica Session

K 2 =Simplify [Numerator [Kl] - Coe f f i c i e n t [Numerator [Kl] ,

VA2 ] VA 2 / . SDRules /. a->L-b] + Coef f ic ient [

Numera to r [Kl] , VA2 ] VA 2 / Denominator [K l ]

m N b

L (-(Nd Yb) + N b Yd)

K 3 = Simplify [K 2 / . SDRules]

(a Cf - b C r) m

a Cf Cr L + b Cf Cr L

K 4 = Numerator [K3 ] /Simplify[ (Denominator [K3] /,

a -> L - b)]

(a Cf-

b C r) m

2

Cf Cr L

Neutral Steer Point

d l = Simplify [d / .

N b

a ---

Yb

d2 = Simplify [d l /,

(a + b ) Cr

Solve [Numerator [Srn] = = 0 , d ] [ [1 ] ] ]

SDRules]

Cf + Cr

d3 = Simplify [d2 / . a

Cr L

> L b ]

a ) / L

Cf + Cr

S t at ic Ma rg in

SM = (d l

N b

-( )

L Y b

SMI = Simplify [S M /.

-( a C f) + b Cr

Cf L + Cr L

SDRules]

116



Ap p en d ix B Tw o D OF Model Mathemat ica Session

Tangent Speed

Vtan = v / . So lve [Sbd deltaR == 0,V][[2,1]]

Nr Y d - Nd Yr

-( )

m N d

Vtan l = S q r t [Simplify [ E x p a n d [ (V / . Solve [V == Simplify [Vtan / . SDRules] ,V] [ [2 ,1 ] ]) A2] ] ]

b (a + b) Cr

Sqrt[-( )]a m

Vtan2 = Sq r t [Simplify [Numerator [ V t a n l A 2 ] / . a -> L-b] /Denominator [Vtan l A 2 ] ]

b Cr L

Sqrt[-( ) ]a m

Critical Speed

V c r i t = v / . Solve [Denominator [Srd] == 0,V][[1]]

Nr Y b - N b Yr

-( )

m N b

V c r i t l = S q r t[Simplify [ (V / . Solve [V

==

Vc r i t/

SDRules,V] [[2,1]])A2]]

2

(a + b ) Cf Cr

Sqr t [ ]- (a Cf m ) + b Cr m

Vc r i t 2 = Sqr t [Simplify [Numerator [VcritlA2] / . a -> L-b]/Denom in a t o r [V c r i t l A 2] ]

2

Cf Cr L

Sqrt [ ]- (a Cf m ) + b Cr m

Characteristic Speed

Vch a r = v / . So l v e [ d e l t aR == 2 L/R, V] [ [2, 1 ] ]

(-(Nr Y b) + N b Yr + Sqr t [-4 LmNb (2 N d Y b - 2 N b Yd) +

2

(Nr Y b - N b Y r) ]) / (2 m Nb)

117



App en d ix B Two DOF Model Mathematica Session

Vcharl = Sq r t [Simplify [ (V / . Solve [V == Vchar /-

SDRules, V ] [[2,1]])A2]]

(a + b ) Cf Cr (a + b - 2 L)

Sqrt[ ]-( a Cf m ) + b Cr m

Vchar2 = S q r t [Simplify [Numerator [Vcharl A2] / . a -> L-b] /

Denominator [Vcharl A 2 ] ]

2

Cf Cr L

Sqrt[-( ) ]-( a Cf m ) + b Cr m

Yaw Radius of Gyration

kz = Sqrt[Izz/m]

Iz z

Sqrt[ ]m

Geometry to Inertia Ratio

GIR = LA2/kzA2

2

L m

Iz z

Total Cornering Fac tor

T C F = Cf Cr/mA2

Cf Cr

2

m

Characteristic Equation

Ds == 0


+ s + + s (-( ) ) == 0


a2 = Coef f i c i en t [ D s , sA2]

1

118




al = Coef f i c i e n t [Ds, s]

Nr Yb

-( )Iz z m V

aO = Ds - a2 sA 2 - al s

N b Nr Y b N b Yr

+ _


Undamped Natural Frequency

wn = Simplify [ Sq r t [a O ] ]

in N b V + Nr Y b - N b Yr

Sqrt[ ]Iz z m V

w n l =Simplify [w n / . SDRules]

2 2 2 2

a Cf Cr + 2 a b Cf Cr + b Cf Cr + a Cf m V - b Cr m V

Sqrt[ ]2

Iz z m V

Damping Ratio

zeta = Simplify [a l Iz z m V/(2 Sqrt[wnA2 (Izz m V)A2])]

- (m Nr V + Iz z Yb)

2 Sqrt [Izz m V (h i N b V + Nr Y b - N b Y r) ]

zetal =Simplify [zeta /. SDRules]

2 2

- (Cf Iz z + Cr Iz z + a Cf m + b Cr m ) /

2 22

(2 Sqr t [Izz m (a Cf Cr + 2 a b Cf Cr + b Cf Cr + a Cf m V -

2

b Cr m V ) ] )

119




Poles

p o l e s = So lve [Ds==0, s] ;

si = s /- p o l e s [[1,1]]

2

(m Nr V + Iz z Y b -Sqrt[(-(m Nr V) - Iz z Yb)

-

4 Iz z m V (m N b V + Nr Y b - N b Y r) ] ) / (2 Iz z m V)

sla = Simplify [s i / . SDRu l e s ]

2 2

( (Cf + C r) Iz z + (a Cf + b C r) m-

2 2 2

Sqr t [ ( - (Cf Izz)-

Cr Izz - a Cf m - b Cr m ) -

2 2 24 Iz z m (a Cf Cr + 2 a b Cf Cr + b Cf Cr + a Cf m V -

2

b Cr m V ) ] ) / (2 Iz z m V)

s2 = s / . p o l e s [[2,1]]

2

(m Nr V + Iz z Y b + Sqrt[(-(m Nr V)- Iz z Yb)

4 Iz z m V (m N b V + Nr Y b - N b Y r) ] ) / (2 Iz z m V)

s 2 a = Simplify [s2 / . SDRules]

2 2

( (Cf + C r) Iz z + (a Cf + b C r) m +

2 2 2

Sq r t [ ( - (C f Izz)- Cr Iz z -

a Cf m- b Cr m )

2 2 24 Iz z m (a Cf Cr + 2 a b Cf Cr + b Cf Cr + a Cf m V -

2

b Cr m V ) ] ) / (2 Iz z m V)

Zeros

Zbd = s / . Solve [Nbd == 0 , s][[l,l]]

- (m N d V)-

Nr Y d + N d Yr

_ ( }Iz z Y d

120



Append ix B Two DOF Model Mathemat ica Session

Zbdl = Simplify [Zbd / . SDRules]

2 2

abCr+b C r + a m V

Iz z V

Zba = s / . So lve [Nba == 0 , s][[l,l]]

-Nr - a m V + c m V + a Y r - cY r

_ ( )

Iz z

Zba l = Simplify [Zba / . SDRules]

2 2 2

acCf+abCr+b Cr-bcCr+amV - c m V

Iz z V

Zb t = s / . Solve [Nbt = = 0, s][[l,l]]

Nr

Iz z

Z b t l = Simplify [Zbt / . SDRules]

2 2

a Cf + b Cr

Iz z V

Zrd = s / . Solve [Nrd == 0, s][[l,l]]

- (N d Yb) + N b Y d

_ ( )

m N d V

Z r d l = Simplify [Zrd /- SDRules]

(a + b ) Cr

a m V

Zra = s / . Solve [Nra == 0 , s][[l,l]]

N b -

a Y b + c Y b

-( )

(a -

c) m V

Z r a l = Simplify [Zra /. SDRules]

-( c Cf) + a Cr + b Cr -

c Cr

a m V -

c m V

12 1



Ap p en d ix B Two DOF Model Mathemat ica Session

Z rt = Solve [Nrt == 0 , s]

{{}}

12 2



A p pen dix C Tw o D O F Mod e l MATLAB P rog r ams

C.l DOF2Control.m

%DOF2Con t ro l 2 DOF Mo d e l Execution Control

%

Controls e x e c u t i o n o f 2 DOF mod e l . Sets c o n t r o l input type ( s t e p , step

ramp, r am p step / r am p down, o r s ine steer) . Sets s imu l a t i o n p a r a m e t e r s .

% Created 1/11/96

% J. Kiefer

% Control Input Ty p e

step= 1;

r am p = 2 ;r ampsqua r e

=3 ;

s i n e= 4 ;

% Step s t e e r

% Ramp step s t e e r

%

Ramps q u a r e s t e e r

% Sine s t e e r

input = 1; % Select wh i c h c o n t r o l input to us e

% Simulation Parameters

tO = 0 . 0 ; % s

t r = 0 . 2 ; % s

td = 1.0; % s

ts = 1 . 0 ; % s

tf = 4 . 0 ; % s

to l = le-5; %

I n i t i a l time for s t e e r input

Ramp time

EWe l l time

Period for s i n e s t e e r

Final time for s imu l a t i o n

Simulation accuracy (default = l e -3 )

123



Append ix C Two DOF Mo del MATLAB Programs

C .2 D O F 2P aram .m

%DOF2Param 2 DOF Model Independent Parameters an d Simulation Control

%

Sets independendent vehicle, tire, control, an d disturbance pa rame te r s

fo r 2 DOF mode l .

% Created 1/7/96

% J. Kiefer

% Initialization

c l e a r a l l ;

clc;

g l o b a l m Izz L a b c u Cf Cr dO Fzf Fzr Fyg Fya tO tr td ts tf i n pu t ;g l o b a l Bl CI Dl El B3 C3 B5 C5;

% Constants

g = 9.81; % m/s~2

% Vehicle Independent Parameters

m= 1775; % kg

Izz = 1960; % kg -m^2

f = 0 .52 ; %

L = 2 .372 ; % m

u = 100; % km / h r

% Control an d Disturbance Inputs

dO = = 1 ; % degtheta = 0; % de gFya = 0; % N

c= 1 .25 ; % m

%

% Linear Tire Model Parameters

Cf = =- 1 2 3 0 . 5 ; % N/deg

Cr = =- 1 1 5 5 . 5 ;

% N/deg

Acceleration d u e to gravity

Gross v e h i c l e mass

Y aw inertia

Fraction o f we i g h t on front a x l e

W he e l b a se

Vehicle forward speed

Steer input magn i t ude

Side s l o p e

Aerodynamic s i d e force

Distance f rom front a x l e to

a e r o d y n am i c s i d e force

Front cornering s t i f f n e s s (one tire)

Rear cornering s t i f f n e s s ( o n e tire)

% N o n Linear Tire Mo d e l Parameters

% Normalized Lateral Force Magic Formula Parameters

Bl = 0 .5835 ;CI = 1 . 7 1 6 6

Dl = 1 . 0 0 0 5

El = 0 . 2 5 1 7

% Co rn e r i n g Coefficient Parameters

B3 = 0 .333 ;

C3 =- 1 . 3 5 2 e - 5 ;

% Friction Coefficient Parameters

B5 = 1 .173 ;

C 5 = - 3 . 6 9 6 e - 5 ;

% Unit Conversions

u = u*1000/3600; % m/s Vehicle forward speed

124



Append ix C Tw o DOF Mo del MATLAB P ro gram s

dO = d0*pi /180; % ra d Step s t e e r input

Cf =Cf*180 /p i*2 ; % N/rad Front t i r e cornering s t i f f n e s s ( two tires)

Cr = Cr*180 / p i * 2 ; % N/rad Rear t i r e cornering s t i f f n e s s ( two tires)

12 5



Append i x C Two DOF Mo de l MATLAB Programs

C .3 DOF2DependParam.m

%DOF2DependParam 2 DOF Mo d e l Dependent Parameter Calculation

%

% Calculates v a l u e s o f d e p e n d e n t pa rame te r s for 2 DO F model .

%

% Created 1/7/96

% J. Kiefer

% Dependent Parameters

a = ( l - f ) *L ;b = f*L;V = u;

Fy g =

m*g*sin(theta*pi/180) ;

Fzf = m*g*f/2*cos(theta*pi/180);

Fzr = m*g*(l-f)/2*cos(theta*pi/180);

% Stability DerivativesYb = Cf + C r;Yr = (a*Cf-b*Cr) /V ;Y d = -Cf ;

Nb =

a*Cf-b*Cr;

Nr = (a~2*Cf+b~2*Cr) /V ;N d =

-a*Cf ;

% m Distance f rom front t i r e to C.G.

% m Distance f r om r e a r t i r e to C . G .

% m/ s Vehicle speed

% N Side s l o p e lateral force

% N Front t i r e n o rm a l load ( o n e tire)% N Rear t i r e n o rm a l load (one tire)

% N/rad D a m p i n g - in -s ides l ip% N-s/rad Lateral force / yaw coupling

% N/rad Control force

% N-m/rad Directional stabi l i ty

% N-m-s/rad Y aw damping% N-m/rad Control moment

12 6



Append ix C Two DOF Model MATLAB Programs

delta = 0;if t < tO + 2*tr + td

delta = d0*(t0+td+2*tr-t)/tr;

en d

if t < tO + tr + td

delta = dO ;

en d

if t < tO + t r

delta = d0*(t-t0)/tr;en d

if t < tO

delta = 0;en d

en d

% Sine Steer

if input == 4

delta = d0*s i n (2*p i* ( t - t 0 ) / t s ) ;

if t < tO

delta = 0;en d

en d

12 8



Append i x C Two DOF Mode l MATLAB Programs

C .5 DOF 2 L F r e q . m

%DOF2LFreq F r e q u e n c y Response o f Linear 2 DO F M o d e l

% For a Single Set o f Parameters

%

% Generates bode p l o t data for linear 2 DO F model for o utp uts o f

% sideslip a n g l e an d yaw speed, an d for inputs o f s t e e r a n g l e cont ro l ,

% ae rodynamic s i d e force d i s t u r b anc e , and r o a d s i d e s l o p e disturbance.

%

% Created 2/4/96

% J. Kiefer

D0F2Param; % Set i n d e p e n d e n t p a r ame t e r s

D0F2Control ; % Set e x e c u t i o n c o n t r o l p a r ame t e r s

D0F2DependParam; % Calculate d e p e n d e n t pa rame te r s

% Transfer Function Denominator

D = [1 -Nr/Izz-Yb/ (m*V) Nb/Izz+(Nr*Yb-Nb*Yr) / (Izz*m*V) ] ;

% Transfer Function Numerators

N bd = [Yd/(m*V) (Nd*Yr-Nr*Yd-Nd*m*V) / (Izz*m*V) ] ;

N ba = [1 / (m*V) (c-a) /Izz+ ( ( a - c ) *Yr-Nr ) / (Izz*m*V) ] ;

N b t = [g/V - g *N r / ( I z z *V ) ] ;

Nrd = [Nd/Izz (Nb*Yd-Nd*Yb) / (Izz*m*V) ] ;

Nra = [ ( a - c ) /Izz ( (c-a)*Yb+Nb) / (Izz*m*V) ] ;

Nrt = [g*Nb/ (Izz*V) ] ;

% Bo d e Plot Data

w = lcgspace(-l,2) *2*p i ;

[Mbd,Pbd,w] = b o d e (Nbd, D,w) ;

[Mba,Pba,w] = b o d e (Nba, D,w) ;

[Mbt,Pbt,w] = b o d e (Nbt, D,w) ;

[Mrd,Prd,w] = b o d e (Nrd, D,w) ;

[Mra ,Pra ,w] = b o d e (Nra, D,w) ;

[Mr t ,P r t ,w] = b o d e (Nr t , D,w) ;

129



Append i x C Tw o D OF Model MATLAB Programs

C .6 DOF2LS im .m

%D0F2LSim Simulation of Linear 2 DOF Mo d e l Response to Control an d Disturbance

% Inputs

%

% Performs s i m u l a t i o n of linear 2 DO F model r e sponse to c o n t r o l an d

% disturbance inputs. Determines yaw speed, lateral speed, sideslip angle,

% front an d r e a r t i r e slip angles, front an d r e a r t i r e lateral f o r c e s , an d

% lateral a c c e l e r a t i o n . Plots these r es po n se s v e r su s time. Reads data f rom

% D0F2Param, D0F2Dep e n dP a r am .

%

% Created 1/7/96

% J. Kiefer

D0F2Param;

D0F2Con t r o l ;

D0F2DependParam;

% Set i n d e p e n d e n t p a r ame t e r s

% Set e x e c u t i o n c o n t r o l p a r ame t e r s

% Calculate d e p e n d e n t p a r ame t e r s

% Perform s i m u l a t i o n

[t,x] = ode23(,DOF2LDE',0,tf, [0 0]',tol);v = x ( : , l ) ;

r = x ( : , 2 ) ;

% Steer Angle

delta = z e r o s ( length ( t) , 1) ;

for i = l:length(t)

delta(i) = SteerAngle (t ( i) , i npu t , t0, tr, td, ts , tf , dO) ;

en d

% ra d Steer ang le

% Vehicle an d Tire Slip Angles

beta = v/u;

a lphaF = (v+a*r ) /u-delta;

a lphaR = (v-b*r)/u;

% ra d

% ra d

% ra d

Vehicle sideslip ang le

Front t i res slip a n g l e

Rear t i res slip ang le

% External Forces an d Moments

Fyf = C f * a l p h aF ;

Fy r = C r * a l p h aR ;

% N

% N

Front t i res lateral force

Rear t i res lateral force

% State Derivatives

v d o t = ( Fy f + Fy r + Fya + Fyg) /m -

u*r;

r d o t = ( a * F y f - b*Fyr - ( c - a ) *Fya) /Izz;

% Lateral Acceleration

ay = v d o t + u*r; % m / s ^ Lateral a c c e l e r a t i o n

% D o Plots

s u b p l o t (2 , 2 ,1 )

p l o t ( t , v )

g r i d

t i t l e (' Lateral Speed

'

)xlabeK'Time (s ) ')ylabel

(' Speed (m/s) ')

subplot (2 , 2 , 2)

p l o t ( t , r * 1 8 0 / p i )

130



Append ix C Two DO F M odel MATLAB Programs

gr id

t i t l e ( 'Yaw Speed')

xlabelCTime (s ) ')

ylabel( 'Speed ( d eg / s )'

)

subp lo t(2 , 2 , 3)

plot(t,beta*180/pi,t,alphaF*180/pi, ,t,alphaR*180/pi, '- .'

, t ,del ta*180/pi ,

'

:'

)g r i d

t i t l e ( 'Vehicle S ide s l i p Ang l e , Tire Slip Ang l e s , Steer Ang l e ' )xlabeK 'Time (s )

'

)y l a b e l (

'

Slip Angle (deg)'

)

s u b p l o t (2 , 2 , 4)

plot ( t ,ay/g)

gr id

t i t l e ('

Lateral Acceleration'

)

xlabel( 'Time (s ) ')

y l a b e l ( 'Acceleration (g )'

)

13 1




C .7 DO F 2 L D E . m

function x d o t= DOF2NLDE ( t , x )

%DOF2NLDE N on Linear Differential Equations for 2 DOF Mo d e l

%

%xdo t = D0F2NLDE( t , x )

%

% Determines derivatives o f lateral speed an d yaw speed g i v e n time an d

% s t a t e v e c t o r. N on linear t i re an d no n linear slip ang les . Used w i t h ode23

% for s imula t ion .

% Inputs:

% t

% x(l)

% x(2)

% Outputs:

% xdot ( l )

% x d o t (2)%

% Created 2/18/96

% J. Kiefer

T im e (s )Lateral speed (m/s)Yaw speed ( r ad / s )

Derivative o f lateral s p e e d (m / s ^2 )

Derivative o f yaw speed ( rad/s ' '2 )

g l o b a l m Izz L a b c u dO Fzf Fzr Fyg Fya tO t r td ts tf i n pu t ;

delta = SteerAngle(t, i n pu t , tO , tr,td,ts,tf,dO);

a lphaF = atan( ( x ( l )+a*x (2 ) ) /u ) -delta;

a lphaR = atan( (x ( l ) -b*x (2 ) ) /u ) ;

[Fyf , Mzf] = NLTire(Fzf, alphaF);

[Fyr, Mzr]= NLTi r e ( F z r , alphaR);

x d o t = [ - u*x (2 ) + (2*Fyf*cos ( d e l t a ) +2*Fyr+Fya+Fyg)/m

(2*a*Fyf * c o s (de l t a ) - 2 * b * F y r + (a -c) *Fya) /Izz] ;

13 2



Append ix C Two DO F M odel MATLAB P ro gr am s

C .8 DOF 2N L S i m . m

%DOF2NLSim Simulation o f Non-Linear 2 DO F M o d e l Response to Control an d Disturbance

% Inputs

%

% Performs s im u la tio n o f no n linear 2 DOF model r e sponse to c o n t r o l an d

% disturbance inputs. Determines yaw speed, lateral speed, sideslip angle,

% front an d r e a r t i r e slip angles, front an d r e a r t i r e lateral f o r c e s , an d

% lateral a c c e l e r a t i o n . Plots these r es po n se s v e rs u s time. Reads data f r om

% D0F2Param, D0F2Dep endP a r am .

%

% Created 2/18/96

% J. Kiefer

D0F2Param; % Set i n d e p e n d e n t p a r ame t e r s

D0F2Con t r o l ; % Set e x e c u t i o n c o n t r o l p a r ame t e r s

D0F2DependParam; % Calculate dependent p a r ame t e r s

% Perform s imula t ion

[ t ,x ] = ode23(,DOF2NLDE',0,tf, [0 0]',tol);

v = x ( : , l ) ;

r = x(:,2) ;

% Steer Angle

delta = z e r o s (length(t) , 1) ;

for i = l:length(t)

delta(i) = SteerAngle (t(i) , i npu t , tO , t r , t d , t s , t f ,d0) ; % ra d Steer ang le

en d

% Vehicle an d Tire Slip Angles

beta = atan(v/u) ; % ra d Vehicle sideslip a n g l e

a lphaF = atan( (v+a*r ) /u ) -delta; % ra d Front t i r e s slip ang le

a lphaR = atan( ( v -b* r ) /u ) ; % ra d Rear t i res slip a n g l e

% Ex te rn a l For ce s an d Mom e n t s

Fyf = NLTire(Fzf, alphaF); % N Front t i r e la teral force ( o n e tire)

Fyr = NLTire(Fzr, alphaR) ; % N Rear t i re lateral force (one tire)

% State Derivatives

v d o t = (2*Fyf.*cos (delta) + 2*Fyr + Fya + Fyg) /m -u*r;

r d o t = (2*a*Fyf.*cos (delta)- 2 * b * F y r - (c-a)*Fya)/Izz;

% Lateral Acceleration

ay = v d o t + u*r;% m/s^2 Lateral a c c e l e r a t i o n

% D o Plots

s ubp l o t ( 2 , 2 , l )

p l o t ( t , v )

g r i d

t i t l e('

Latera l Speed ' )x l a b e K 'T i m e (s ) ')

ylabel ('Speed (m/s) ')

s u b p l o t (2 , 2 , 2)

p l o t ( t , r * 1 8 0 / p i )

133



Append i x C Tw o DOF Mode l MATLAB Programs

g r i d

t i t l e ( 'Yaw Sp e e d ' )

xlabelCTime (s ) ')

y l a b e l ('Speed ( deg / s )

'

)

subp lo t ( 2, 2 , 3)

plot(t,beta*180/pi,t,alphaF*180/pi, ''

, t , a l p h aR*180 / p i , '- .'

, t ,del ta*180/pi ,

': ')

g r i d

t i t l e ( 'Vehicle Sideslip Ang l e , Tire S l i p Ang l e s , Steer Ang l e ' )

xlabel( 'Time (s ) ')

y l a b e l ('

Sl ip Angle (deg)'

)

s u b p l o t (2 , 2, 4)p l o t (t, ay /g )g r i d

t i t l e (' Lateral Acceleration '

)

xlabe l ( 'Time (s )'

)y l a b e l

('Acceleration

(g )

'

)

13 4




C .9 DOF 2NLDE . m

f un c t i on x d o t =D0F2NLDE( t , x )

%D0F2NLDE N on Linear Differential Equations for 2 DO F Mo d e l%

%xdot=

D0F2NLDE(t ,x)%

% Determines derivatives of lateral speed an d yaw speed g iven time an d

% s t a t e v e c t o r. N on linear t i r e an d no n linear slip a n g l e s . U s e d w i t h ode23% for simulation.

%

% Inputs:

% t T im e (s )% x(l) Lateral speed (m/s)% x(2) Yaw speed ( rad/s)% Outputs:

% xdot(l) Derivative o f lateral speed (m/s^2)% x d o t (2) Derivative o f yaw speed (rad/sA2)%

% Created 2 / 1 8 / 9 6% J. Kiefer

g l o b a l m Izz L a b c u dO Fzf Fzr Fyg Fya tO tr td ts t f i n pu t ;

delta = S t e e rAn g l e ( t , inpu t , tO , tr,td,ts,tf,dO);a lphaF

=atan((x(l)+a*x(2))/u)-delta;

alphaR=

atan( (x ( l ) -b*x(2 ) ) /u ) ;

Fyf =NLTi r e (Fz f , alphaF);

Fyr =NLTi r e (Fz r, alphaR);

x d o t = [ -u*x(2) + ( 2*Fy f*co s (de l t a ) +2*Fyr+Fya+Fyg)/m

( 2 * a *F y f * c o s (de l t a ) -2*b*Fyr+ ( a - c ) *Fya) / I zz ] ;

135



App e n d i x D Relevan t Li te ra tu re

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13 8




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13 9




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