Half-Car Model John O’Donnell, Barry O’Donnell, David Rogers

1

Contents I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

D e v e l o p m e n t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

A s s u m p t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

D e v e l o p m e n t P r o c e s s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

F i n a l M o d e l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

V e r i f i c a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1

S i m u l a t i o n a n d T e s t i n g o f M o d e l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2

E f f e c t o f N a t u r a l F r e q u e n c y o n P e r f o r m a n c e / C o m f o r t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2

T e s t i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3

1 . S p e e d B u m p : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3

2 . P o t h o l e : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4

3 . R e p e a t i n g S p e e d B u m p s : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4

R e s u l t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5

1 . S p e e d B u m p : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5

2 . P o t h o l e : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5

3 . R e p e a t i n g S p e e d B u m p s : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5

P r o b l e m s w i t h C r e a t i n g S p e e d B u m p s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5

V e r i f i c a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6

C o n c l u s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 7

B i b l i o g r a p h y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 7

2

A p p e n d i x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 8

[ 1 ] D e r i v a t i o n s : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 8

[ 2 ] C a l c u l a t i o n s o f s t i f f n e s s a n d d a m p i n g c o e f f i c i e n t s f o r s u s p e n s i o n : . . . . . . . 2 0

[ 4 ] I n p u t s u s e d t o c r e a t e s p e e d b u m p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1

[ 5 ] R e s u l t i n g s p e e d b u m p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1

[ 6 ] V a l u e s f o r t h e t w o i n p u t s t e p s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1

[ 7 ] D i s p l a c e m e n t a t 2 . 5 k m / h r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2

[ 8 ] A n g l e a t 2 . 5 k m / h r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2

[ 9 ] D i s p l a c e m e n t a t f r o n t a n d r e a r a t 2 . 5 k m / h r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2

[ 1 0 ] D i s p l a c e m e n t a t 5 k m / h r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3

[ 1 1 ] A n g l e a t 5 k m / h r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3

[ 1 2 ] D i s p l a c e m e n t a t 1 0 k m / h r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4

[ 1 3 ] A n g l e a t 1 0 k m / h r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4

[ 1 4 ] D i s p l a c e m e n t a t 2 0 k m / h r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5

[ 1 5 ] A n g l e a t 2 0 k m / h r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5

[ 1 6 ] D i s p l a c e m e n t a t 3 0 k m / h r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6

[ 1 7 ] A n g l e a t 3 0 k m / h r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6

[ 1 8 ] D i s p l a c e m e n t a t 4 0 k m / h r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 7

[ 1 9 ] A n g l e a t 4 0 k m / h r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 7

[ 2 0 ] D i s p l a c e m e n t f o r a p o t h o l e a t 5 k m / h r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 8

[ 2 1 ] A n g l e f o r a p o t h o l e a t 5 k m / h r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 8

[ 2 2 ] C a l c u l a t e d v a l u e s t o b e i n p u t t e d i n t o t h e p u l s e g e n e r a t o r s . . . . . . . . . . . . . . . . 2 9

[ 2 3 ] I n p u t s u s e d t o c r e a t e s e r i e s o f s p e e d b u m p s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 9

[ 2 4 ] R e s u l t i n g s p e e d b u m p s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 9

[ 2 5 ] D i s p l a c e m e n t f o r r e p e a t i n g s p e e d b u m p s a t 2 . 5 k m / h r . . . . . . . . . . . . . . . . . . . . . . . 3 0

[ 2 6 ] A n g l e f o r r e p e a t i n g s p e e d b u m p s a t 2 . 5 k m / h r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0

[ 2 7 ] I n p u t s i g n a l s f o r a s a m p l e t i m e o f 1 0 s e c o n d s a t 1 0 k m / h r . . . . . . . . . . . . . . . . . 3 1

[ 2 8 ] R e s u l t i n g s p e e d b u m p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1

[ 2 9 ] D i s p l a c e m e n t w i t h i n s u f f i c i e n t s a m p l i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2

[ 3 0 ] A n g l e w i t h i n s u f f i c i e n t s a m p l i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2

[ 3 1 ] S i g n a l B u i l d e r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3

[ 3 2 ] I n p u t s i g n a l s a n d r e s u l t i n g b u m p f o r t h e t a b l e t o p b u m p c r e a t i o n . . . . . . . 3 3

[ 3 3 ] R e s p o n s e s f o u n d b y A . K a d e r e t a l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4

3

Introduction

T h e a im o f t h i s p ro j ec t w as to m od e l a „ h a l f c a r ‟ s ys t em . T h i s s ys t em,

w h ich i s a l s o kn ow n a s a b i cyc l e m o d e l , c an b e m od e l l ed i n a n um ber

o f wa ys u s i n g v a r io us p r o gr am s o r m at h em at i ca l m et ho ds .

T h e tw o m ain m et ho ds th a t w er e m os t s u i t ed to t he p r o j ec t w e r e

S i mul in k an d S imM ech an i c s . Bo t h o f t he s e m et ho ds we r e av a i l ab l e o n

t h e M AT LA B p r o gr am . S im M ech an i cs i s a mo r e g r ap h i ca l ap p ro ach th a t

a l l o ws t h e p r o gr am to c r ea t e t h e re l ev an t f o rmu l ae , de r iv ed f rom th e

i np u t s s e l ec t ed b y t h e o p er a to r , w he r ea s S im ul i nk p ro v i d es a m o re

m at h em at i ca l app roach t han p r ac t i c a l .

T h e S i mul in k app l i c a t io n in t h e M AT LA B p r o gr am w as us ed f o r t h i s

p r o j ec t . T h i s ap pro ach w as d ec id ed u p on b ecaus e i t can b e ve r i f i ed

u s i n g m at h em at i c f o rm ul ae b a sed o n f i r s t p r in c i p l e s o f m e ch an i cs o f

m ach in e s , w h i ch i s u nav a i l ab l e f o r o th e r m eth o ds , l i k e t h e

S i mM ech an i cs m eth o d , f o r ex am pl e . P r ov id i n g th e s e f o rm ul ae w e r e

s ou nd and t ha t non e o f t h e m ajo r fo r ce s w e r e n egl ec t ed , t h e m od e l

p r od u ced w ou ld be accur a t e . T h e va l i d i t y o f t h e m od e l wo u l d be

d i f f i cu l t t o p ro v e fo r t he S i mM ech an ic s m et ho d .

Wi t h th e me th od se l ec t ed , t h e s ys t em w as f i r s t an a l ys ed to d e t e rm in e

t h e m ai n f o r ces a f f ec t i n g t h e r esp ons e o f t h e mo de l . E q ua t i ons w e r e

t h en d er iv ed to s im ul a t e t h es e fo r ces and a m a th em at i ca l m od e l w as

co ns t r u c t ed . T h i s m od e l w as t hen d ev e lo p ed fo r t h e S im ul i nk p r o gr am ,

w h e r e v a r i ou s in pu t s w e re app l i ed t o an a l ys e t h e r es po ns e o f t h e

s ys t em.

4

Development

Assumpt ions

T h e m od e l i s b a s ed o n t wo l i n ea r s ys t em s f o r f r on t and rea r su sp ens i on .

T h es e s ys t em s a r e w e l l e s t ab l i sh ed and r ead i l y p r o v en . Th e b od y o f t h e

ca r i s r ep r es en t ed b y a m as s co nn ec t ed to t h e f r on t and rea r su sp ens i on

s ys t ems . Th e d am pi n g co - e f f i c i en t and s p r in g s t i f fn e s s o f b o th th e t yr e

an d th e m ai n s uspen s i on e l em en t a r e t ak en in to accou n t i n t h e m od e l .

T h es e p a r am et e r s , t o ge t h er wi t h t h e mas s es o f t h e w h ee l s and t h e bo d y,

t h e i n e r t i a o f t h e m ass a r e i n c lu d ed i n t h e p r o j ec t a s t h e m aj o r f o rce s

i nv o l ved wi th t he m od e l .

S om e mi no r f o r ces h av e b een n eg l ec t ed f ro m th e mo d e l t o r ed u ce t h e

co mpl ex i t y o f t h e m at h em at i ca l mod e l . Th ese min o r f o r ces i n c l ud e

a s p ec t s l i k e f l ex in t h e bo d y, m o v em en t i n t h e b us h es o r b ack l a sh f ro m

gea r s ys t ems , e t c . T h es e f o r ce s a r e sm al l an d th e i r e f f ec t on t h e v eh i c l e

i s mi n i ma l an d h en ce th e y a r e l e f t ou t . T he mo un t i n g m et ho d fo r t h e

s us p en s io n i s a l so n eg l ec t ed w hi ch i s o n e o f t h e m or e im po r t an t mi no r

f o r ce e l em en t s .

T h e mo d e l p ro du ced f rom t h e p ro j ec t a s su mes th e s usp ens io n i s l i nea r

w h en i n a r e a l v eh i c l e t h e sus p en s i on w ou ld b e f ix ed t o a co mp on en t

co nn ec t i n g th e whee l t o t h e bo d y o f t h e v eh i c l e , a w i s hb on e i s an

ex ampl e o f t h i s com po n en t . T h i s w as n eg l ec t ed so t h e p a r am et e r s o f t h e

w i s hb on e co u l d b e n eg l ec t ed . T he m as s , d im en s i on s and in e r t i a o f t h e

w i s hb on e wo u l d n eed t o b e s ou gh t . Wh i l e t h e m ass an d d im en s i on s can

b e s i mpl y o b t a in ed , t he i ne r t i a wo u l d b e a v e r y co m p l ex m at h em at i ca l

o p e r a t i on . N eg l ec t in g t h i s a sp ec t r ed uced t h e com pl ex i t y o f t h e m od e l

w i t ho u t l o s i n g m u ch r ea l i sm . B y l e av in g t h i s e l em en t o u t o f t h e s ys t em

t h e v eh i c l e c an j us t b e a s s um ed t o h ave a s i mp le r s us p ens i on l ayo u t .

T h e b ump s to ps h av e a l so b een l e f t ou t o f t h e mo d e l b ecau s e i n c l ud i n g

t h em w ou ld h av e in v o lv ed a com pl ex p ro ce ss t h a t co u ld no t e as i l y b e

p r ov en . Wh en th e s us p en s io n t r av e l ru ns ou t an d th e b um p s t op o n the

s us p en s io n h i t s t he b um p s t op o f t he ca r b od y t h e su sp ens io n s ys t em

h as n o e f fec t w h a t s o ev e r on t h e r es po ns e o f t he v eh i c l e , e f f ec t i v e l y

m ak in g th e veh i c l e r i g id , ex cep t fo r t h e r e l a t i v e l y m in o r b ehav i ou r o f

t h e t yr e s . Th i s no n - l i n ear p a r am et e r w o u l d b e ve r y d i f f i cu l t t o q u an t i f y

an d p ro v e . T h e p ro b l em b ecom es m or e comp l ex w h en bo th s us p ens io n

e l em en t s h i t t h e bum p s to ps and t he no n - l i n ear p a r am et e r ch an ges f rom

t h e co nd i t i on w h er e j u s t o n e h i t s t h e b um p s to p . For t h e r e as on s

o u t l i n ed abo v e , t he mi no r f o r ce s and t h e ex cee d i n g l y co mpl ex f o r ces

h av e b een n eg l ec t ed . T h i s m ean s the mo d e l p ro du ced i s v a l i d wi th i n

ce r t a i n co nd i t i on s :

5

1 . S us pens io n do es n o t ru n ou t o f t r av e l

2 . S us pens io n con nec ted i n a l i n ea r m an ne r

3 . A l l b ea r i n gs an d f ix in gs a r e fu l l y r i g i d

4 . A l l b ea r i n gs an d f ix in gs a r e f r i c t i on l es s

5 . A l l m as s es h ave n o f l ex ( i . e . n o sp r ing s t i f fn es s )

6 . T h e p a r am et e r s d o n o t ch an ge du r in g o p e r a t i on o f t h e v eh i c l e

Development Process

F i g u r e 1 : H a l f - c a r m o d e l .

U si n g f i r s t p r i n c ip l es an d th e s ys t em s ho w n in F i gu r e 1 a bo v e 2

eq u a t i on s w e r e de r iv ed b y eq u a t i ng t h e u p w ar d f o r ce s wi th t h e

d o wn w ard fo r ces . E q ua t i on 1 re l a t es t h e m as s o f t h e bo d y o f t h e ca r

( m s ) an d t he up w ar d s acce l e r a t i on o f t he ca r a t t h e c en t re o f g r av i t y

( ) w i t h t he co e f f i c i en t s o f t h e s p r i n gs ( k s 1 ) , ( k s 2 ) and dam p er s ( c s 1 ) ,

( c s 2 ) i n t he s usp ens io n an d t he i r r e s pec t iv e d i sp l acem en t s and

v e lo c i t i e s . Eq u a t i on 2 r e l a t es t h e mom ent o f i n e r t i a o f t h e ca r ( I s ) an d

t h e an gu l a r a cce l e ra t i on ( ) w i t h t h e d i f f e ren ce i n t h e mom ent s a t t h e

f r o n t o f t h e ca r and th e r ea r o f t h e ca r . [1]

E q ua t i on 1 :

E q ua t i on 2 :

R ea r ran g in g th e abo v e equ a t io ns in o rd e r t o f i nd an d we ge t :

E q ua t i on 3 :

6

E q ua t i on 4 :

In t h e d ev e l op ment o f t h e mo de l , comm on b l o cks o f t e rms w e r e n o t i c ed

i n eq ua t i ons 3 and 4 . In o r d e r t o r ed uce wo r k lo ad an d m ai n t a in

s im pl i c i t y w i t h in th e m od e l , s eve r a l s ub s ys t ems w er e c r ea t ed wi t h in

t h e S im ul i nk p ackage . T he comm on b lo ck s and t h e i r co r r es po nd i n g

s ub s ys t e m a r e s how n b e l o w i n F i gu r e 2 .

F i g u r e 2 : S u b s y s t e m b l o c k s .

E q ua t i n g t h es e s ubs ys t ems as abo v e i t i s p oss ib l e t o r ed uce eq ua t i ons 3

an d 4 .

E q ua t i on 5 :

E q ua t i on 6 :

Wh en m od e l l i n g in S im ul i nk b egan we f i r s t a sk ed o u rs e l v es w h a t i s i t

w e a r e t r yi n g t o f i n d? I t w as d ec i ded t h a t t h e f i n a l go a l s w e r e t o b e th e

d i s p l acem en t a t t h e c en t r e o f g r av i t y o f t h e ca r an d th e an g l e o f t i l t o f

t h e ca r . T w o m a in ap p r o ach es w e r e t ak en wh en t r yi n g t o d ev e l op th e

m at h em at i ca l m od e l an d a r e d i s cus s ed b e l ow .

7

T h e f i r s t i nv o l v ed m o de l l i n g o u t equ a t io ns 5 and 6 d i r ec t l y h o w ev e r t h i s r e su l t ed in a h uge e r r o r d u e t o a

n um b er o f d i f f e r en t i a t i ons . T h e mo d e l w as a l s o i nc r ed ib l y d i f f i cu l t t o f o l lo w an d mo di f i c a t ion o f t h e s ys t em

t o r ed u ce th e e r ro r p r ov ed n ea r im po ss i b l e du e t o i t s comp l ex i t y. T h i s mo d e l i s s ho wn i n F i gur e 3 b e l ow .

F i g u r e 3 : F i r s t a t t e m p t a t h a l f - c a r m o d e l .

8

T h e s econ d ap pr o ach in vo l v ed t ak i n g t h e a l r ead y e s t ab l i sh ed q u a r t e r

c a r mo d e l , w h i ch w as de r iv ed in c l as s , and ex p and i n g i t i n o rd e r t o

f i nd t he d i s p l acemen t and an g l e o f t i l t a t t h e c en t r e o f g r av i t y o f t h e

ca r . T h e qu a r t e r c a r mo d e l i s sh o wn b e lo w in F i gu r e 4 an d i s d e r i v ed

f r om t h e sp r i n g -m as s -d am p e r eq u a t i on .

E q ua t i on 5 : w h e r e y r i s t h e d i s p l a c e m e n t o f t h e r o a d .

T h e f u l l qu a r t e r c a r m od e l co ns i s t s o f t w o o f t h e s e i n s e r i es w h e re t he

o u t pu t i s t h e d i sp l acem en t o f t h e ca r bo d y a t t h a t po i n t .

F i g u r e 4 : Q u a r t e r - c a r m o d e l .

U si n g eq u a t i on s 1 and 2 , t h e h a l f - ca r m od e l w as d ev e l op ed . A s

d i s cuss ed ea r l i e r , E q ua t i on 1 r e l a t e s t h e d i sp l acem en t o u t pu t s o f t he

f r o n t s ys t em an d r ea r s ys t em w i th th e o v e r a l l d i s p l acem en t o f t h e ca r a t

t h e c en t r e o f g r av i ty , w h i l e E qu a t io n 2 r e l a t e s t h e r es u l t i n g m om en t s a t

t h e f ro n t and r ea r o f t h e ca r i n o r d er t o es t ab l i sh t h e p i t ch .

Wh en fu l l y d ev e l op ed ( t h e f u l l h a l f - ca r m od e l i s sh ow n l a t e r i n F i gur e

6 ) t h e r e w as s t i l l a sm al l e r r o r wi t h i n th e s ys t em wh i ch cau s ed a

r amp in g e f f ec t i n t h e ou t pu t s . T he sou r ce o f t h i s e r ro r w as fo un d to b e

t h e d i f f e r en t i a t i o n i n t h e s us pens io n s ec t i on o f t h e qua r t e r c a r m od e l

w h ich caus ed a sm al l e r ro r wh i ch w as th en ad d ed by a n u m b er o f

i n t eg r a t io ns l a t e r on in t h e s ys t em . In o r d e r t o r e c t i f y t h i s , i n s t e ad o f

d i f f e r en t i a t i n g th e i n com in g d i sp l acem en t d u e t o t h e u ns p ru n g mass

( yu ) , t h e v e l o c i t y ( ) w as t ak en as a s eco nd ou t pu t f r om th e wh ee l and

t yr e s ys t em an d ins e r t ed i n t o t h e susp ens io n s ys t em as a s econ d in pu t .

T h i s i s sh o wn b e l ow in F i gu r e 5 .

9

F i g u r e 5 : R e m o v a l o f e r r o r .

Final Model

T h e f i n a l ha l f - ca r m od e l i s s ho wn b e lo w i n F i gu r e 6 . A s d i s cuss ed

ea r l i e r , i t con s i s t s o f t wo qu a r t e r - ca r m od e l s , t h e comm on su bs ys t em

b lo cks as sh o wn in F i gu r e 2 , and th e fo r ma t i on o f equ a t io ns 5 an d 6 .

T h e in pu t s ho w n b e lo w i s a s im p l e s t ep ho w ev e r mor e co mp l i ca t ed

i np u t s w e r e us ed to ob t a in th e re su l t s . T h e t r an sp or t de l a y i s u s ed to

v a r y t h e v e lo c i t y o f t h e ca r , w i t h t h e i np u t t o t h e r e a r o f t h e ca r b e i n g

d e l ayed b eh i nd th a t o f t h e f r on t o f t he ca r .

T h e t wo qu a r t e r - ca r s ys t em s a r e u sed t o ev a l u a t e t h e d i sp l acem en t s

( an d v e l o c i t i e s ) o f t h e s p r un g an d u ns p ru n g

m ass es a t t h e f r on t an d th e r ea r o f t he ca r . T h es e v a lu es a r e t hen f ed

i n t o th e s ub s ys t em b lo cks as d i s cus s ed ea r l i e r w h i ch r e su l t i n t h e

eq u a t i on s A , B , C an d D f r om ea r l i e r .

E q ua t i on 5 :

E q ua t i on 6 :

B y co m p ar in g eq ua t i on s 5 and 6 abov e to F i gu r e 6 b e lo w , i t i s c l e a r

w h a t i s h app en in g i n t h e g r een b l ock s to t h e r i gh t o f t h e d i ag r am.

E q ua t i on 5 s um s t he f ou r s ub s ys t em s A , B , C and D , mu l t ip l i e s t h em b y

- 1 and t h en d iv id es b y m s t o ge t t h e up w ard s acce l e r a t ion o f t h e ca r . B y

i n t eg r a t in g t h i s t wice w e can ge t t h e u p w ar ds d i sp l acemen t . S imi l a r l y,

E q ua t i on 6 sum s b lo cks B wi t h D an d A wi th C an d mu l t ip l i e s t h em b y

l 2 an d l 1 r e sp ec t iv e l y. T h e r es u l t i s d iv id ed b y t h e m om en t o f i ne r t i a I s

t o g iv e t h e an gu l a r a cce l e ra t io n a t t h e c en t r e o f g r av i t y. B y i n t eg r a t i n g

t h i s t wi ce th e p i t ch o f t h e ca r i s fo un d .

1 0

F i g u r e 6 : F i n a l h a l f - c a r m o d e l .

1 1

Veri f icat ion

A cas e s t ud y w as u s ed t o co n f i rm t h e equ a t io ns and m eth od o l o g y

n ecess a r y f o r t h e p r o j ec t . T he ca se s tu d y w as „ A h a l f c a r mo de l f o r

d yn am i c an a l ys i s o f v eh i c l es wi t h r and om pa r am e te r s ‟ by W . G oa e t a l .

[ re f e r en ce] T h e au t ho r s a l so us ed a h a l f c a r mo d e l b u t t h e d am pi n g

e f f ec t o f t h e t yr e s w e r e neg l ec t ed . T h i s fo r ce w as in c lu d ed in t h e

cu r r en t p r o j ec t .

T h e a i m o f t h e ab ov e s t ud y w as t o t ak e in to acco un t t he u n ce r t a i n t y o f

t h e pa r am e t e r s us ed i n t h e d yn a m ic an a l ys i s o f v eh ic l es . The

p a r am e te r s r e f e r r ed t o a re m as s , s p r in g s t i f fn e s s , d ampin g v a l ue s , e t c .

T h e i r co n cep t i s t h a t a l t ho u gh th e se v a lu es a r e a ccu r a t e an d r e l i ab l e

w h en th e veh i c l e i s f i r s t man uf ac t u r ed t h ey can b ecom e un r e l i ab l e a f t e r

a p e r i od o f t i m e o f o pe r a t io n o f t h e v eh i c l e . Fo r ex amp l e , t h e sp r i n g

s t i f fn es s and d amp in g co - e f f i c i e n t o f t h e t yr e a r e d ep end en t on t h e

p r e s s u r e i n s i d e th e t yr e w h ich ch an ges ov e r a r e l a t i v e l y s ho r t am ou n t

o f t i m e .

T h e d amp e r i n s i d e t h e m ain su sp ens i on s ys t em i s an ex ampl e o f t he

p a r am e te r ch an g i ng o v e r a l on ge r p e r i od o f t im e . T h e d amp e r us es an

o r i f i ce w i th a sma l l d i ame t e r t o r es t r i c t t h e f l u id f l o w con v er t i n g

k in e t i c en e r g y t o t he r ma l en e r g y. A s t h e f lu i d p as se s t h ro u gh th e

o r i f i ce co n t i nu ou s ly w h en t h e v eh i c l e m o v es i t g r ad u a l l y w ea r s d o wn

t h e o r i f i c e , ch ang i n g t h e d am pe r co - e f f i c i en t o f t h e sus p en s i on

g r ad u a l l y o v e r a l a r ge p e r i od o f t im e .

T h e p ap e r co nc lu des f r om t he r es u l t s t h a t t h e u n ce r t a i n t y o f t h es e

p a r am e te r s h as an e f f ec t o n th e r es pon s e o f t h e v eh i c l e , i . e . t h e n a t u r a l

f r eq uen cy o f t h e sys t em ch an ges w i th t h e chan g in g p a r am et e r s . Th i s

m eans t h a t a s t h e n a tu r a l f r equ en c y s h i f t s h i gh e r o r l o wer t h an t he

d e s i gn ed v a l u e th e v eh ic l e w i l l b ecom e u n co mf o r t ab l e f o r t h e

p a s s en ge rs .

1 2

Simulation and Testing of Model

Effect of Natural Frequency on Per formance/Comfort

T h e se t v a lu e f o r t h e n a t u r a l f r equ en cy f o r p a s s en ge r veh i c l es i s i n t h e

r an ge b e tw een 1 . 5 to 2 .1 Hz . 1 . 4Hz i s gen e r a l l y t h e n a t u r a l f r equ en cy o f

t h e hu m an b o d y a n d t he r e fo r e i s t he mo s t un co mf o r t ab l e v a l ue to

ch oo s e b ecau s e th e o s c i l l a t i o ns o f t h e v eh i c l e wi l l b e „ i n t un e ‟ w i th t h e

p ass en ge r , i . e . un de r r ep ea t ed os c i l l a t i o ns a t 1 . 4Hz t h e d i s p l acem en t

f e l t a t t h e p as s en ge r ‟ s s ea t wi l l be am pl i f i ed and g r ad ua l l y i n c r ea s e ,

t h ro wi n g th e p e r so n f r om th e s ea t . T h i s cou ld o n l y b e d es c r i b ed as

„ u n com f o r t ab l e ‟ f ro m t h e p as s en ge r ‟ s p o i n t o f v i ew . T he r e f o r e a v a l ue

h i gh er t h an t h i s i s n eed ed fo r t he mo de l . ( 1 . 5 -1 .6 Hz a r e v a lu es t yp i ca l

o f bu s es and o t h er l o w p e r fo rm an ce v eh i c l e s wh e r e p a s s en ge r com fo r t

i s a n eces s i t y) .

A v a l u e o f 2 . 2Hz i s com mo n am ong Fo r m ul a On e ca r s wh e r e t h e

p ass en ge r ‟ s com fo r t i s no t an i s s ue an d t h e p e r fo rm an ce o f t h e

s us p en s io n i s v i t a l ( i . e . max im um gr ip l ev e l s , e t c . ) , hence a v a lu e l e s s

t h an t h i s i s r equ i r ed . 1 .8 Hz w as cho sen t o p ro v i d e a r es po ns e t h a t i s a

co mp ro mis e b e t w een com fo r t and per f o rm ance . Va lu e s f o r t h e sp r in g

s t i f fn es s con s t an t an d d am pi n g coe f f i c i en t i n t h e su sp ens io n w er e

f o un d us in g t h e v a lu e o f 1 .8 Hz a s t h e n a tu r a l f r equ enc y. [ 3]

T hi s gav e th e v a l u es as s ho wn i n F i gure 7 b e l ow :

ku1 200,000 ks1 3240

ku2 200,000 ks2 3240

cu1 500 cs1 2520

cu2 500 cs2 2520

F i g u r e 4 : V a l u e s f o r s p r i n g s t i f f n e s s a n d d a m p i n g c o e f f i c i e n t s i n t h e t y r e a n d

s u s p e n s i o n .

1 3

Test ing

T h r ee d i f f e ren t t yp es o f ro ad d i sp l acem en t s w e re t e s t ed , a s peed b ump ,

a po th o l e and rep ea t in g sp eed b um ps .

1 . S p eed B u mp :

A s p eed b um p o f a w i d t h o f 1 . 5m was ch os en . T h i s w as f o un d f rom

r e s ea r ch and c l os e ly m a t ch es s peed bum ps l oca t ed in o u r a r ea .

Wi d th o f s p eed b um p = 1 .5 m

V ar io us co mp on en t s n eed ed to b e ca l cu l a t ed fo r e ach s p eed . As t h e

s p eed i n c r ea se s t h e t im e t ak en t o t r av e l o v e r t h e sp eed bu mp d ec r eas es

s o t h e t im e d e l ay b e tw een t h e f r on t o f t h e ca r and th e b ack o f t h e ca r

h a s t o b e ca l cu l a t ed . Al s o i n t e rms o f t he s i gn a l b e i n g ap p l i ed t o t h e

c i r cu i t , t h e l en g th o f t im e t h a t t h e s peed b um p a f f ec t s t h e ca r

d ec r eas e s , s o t h e p e r i od o f t he w ave d ec r ea s es and new f r eq u en c i es

n eed t o be ca l cu l a t ed th a t a re t o b e in p u t t ed i n to th e s ign a l gen e r a t o r

f o r each sp eed .

S p eeds o f 2 . 5k m pe r h o ur r an g in g u p t o 4 0km /h r w e r e ch os en b ecaus e

r ea l i s t i c a l l y c a r s w o u l d nev e r t r av e l ov e r a s peed bum p t r av e l l i n g o v e r

4 0 km/ hr f o r f e a r o f c au s in g d am age t o t h e ca r and th e d i s com fo r t i t

w o u l d cau s e . [ 3]

T h e sp eed bu mp i s f o rm ed f rom a s i gna l gen e r a t o r and t wo s t ep s i gna l s

a s s ho wn b e lo w in F i gu r e 7 :

F i g u r e 8 : F o r m a t i o n o f s p e e d b u m p .

A s i gna l gen e r a to r em i t s a s i n e w av e w i t h an amp l i t ud e o f 1 5 cm. T wo

s t ep s i gn a l s a r e t h en in t r od u ced . Th e f i r s t s t ep s i gna l i s s en t t o

co r r es po nd to t h e s t a r t o f t h e p os i t i v e s id e o f t h e s i gn wav e . Th e nex t

1 4

s t ep h as an in i t i a l v a lu e equ a l t o t h a t t h e f i n a l v a l u e o f t h e f i r s t s t ep

i np u t . [4] I t i s d e l a yed t o o n e h a l f o f t h e p e r i od o f t he o sc i l l a t i o n o f

t h e w av e to c r ea t e t h e sp eed b um p. [ 5]

Wh en i s o l a t in g the p os i t i ve h a l f o f a s i n e w ave to c r ea t e t h e sp eed

b um p, t h e d i f fe r en t v a lu e s t o b e ap p l i ed to t h e s t ep in pu t s n eed e d t o b e

r e s o lv ed . As t h e s peed chan ges , t h e f req u en c y ch an ges s o n ew p os i t i ons

a l on g t h e w av e n eed t o b e ca l cu l a t ed . [ 6]

2 . P ot ho l e :

A 2 0cm d eep po tho l e w i th a wid th o f 7 5 cm w as mo d e l l ed w i th a c a r

t r av e l l i n g a t 5k m/ hr .

3 . R ep ea t in g Sp eed Bu m ps :

In p u t s a r e c r ea t ed t o s i mu la t e a s e r i es o f s peed b um ps c l os e t o ge t h er .

T h e c i r cu i t t o c r ea t e t h i s r oad co nd i t i o n i s sh ow n b e l o w. I t co ns i s t s o f

a s i gn a l gen er a to r c r ea t in g a s i n e wav e wi t h a 1 5cm am pl i tu d e . T wo

p u l s e gen e ra to r s a r e ad d ed w i th a p h as e d i f f e r ence b e t w een th em, as

s ho w n b e l ow in F i gu r e 8 , wh i ch r e su l t s i n t h e e l im in a t i on o f t h e b o t t om

h a lv es o f t he s in e w av e . Th i s c r ea t es a s e r i es o f sp eed bu mp s . [ 23] [24]

F i g u r e 9 : F o r m a t i o n o f r e p e a t i n g s p e e d b u m p s . [ 2 2 ]

1 5

Results

1 . S p eed B u mp :

R esu l t s f o r 2 . 5km /h r . [7] [8] [9]

R esu l t s f o r 5km /h r . [1 0] [11]

R esu l t s f o r 10 km/ hr . [ 12] [13]

R esu l t s f o r 20 km/ hr . [ 14] [15]

R esu l t s f o r 30 km/ hr . [ 16] [17]

R esu l t s f o r 40 km/ hr . [ 18] [19]

2 . P ot ho l e :

R esu l t s f o r 5km /h r . [2 0] [21]

3 . R ep ea t in g Sp eed Bu m ps :

R esu l t s f o r 2 . 5km /h r . [2 5] [26]

Problems wi th Creat ing Speed Bumps

Fo r s peeds o f 1 0 km /h r an d ov e r , t h e s im ul a t i on t im e h ad t o b e r edu ced

t o 4 s eco nd s t o ens u r e t h e s i n e w av e th a t c r ea t es t he sp eed bu mp h as

en ou gh samp le po in t s t o c r ea t e an accu r a t e s in e w av e . I f t h e s im ul a t io n

t im e w as l e f t a t 1 0 s econ ds a j agged f o rm at i on in t h e in pu t wav es

f o rm ed in s t ead o f a sm oo th s in e wav e . [ 27] T hi s i l l u s t r a t e s t h a t an

i ns u f f i c i en t nu mb er o f s am pl e p o i n t s c an r es u l t i n t he wr o n g r es u l t an t

s i gn a l o r i n t h i s c as e n o s i gn a l a t a l l . [ 2 8] As su ch th e re i s n o t eno u gh

t im e f o r t h e r es po ns e o f t h e ca r t o b e d i s p l ayed . [ 29] [30]

A n ot he r wa y t o c r ea t e mo r e s amp le po in t s o n t h e s i n e w av e was

n eed ed . Us i n g th e s i gn a l bu i ld e r a s in e w av e w i th mo r e s amp le p o in t s

p e r p e r i od w as mad e . Ho w eve r w hen t he s imu l a t i on w as r u n , n o t

e n ou gh s am pl e po in t s w e r e s to r ed i n m emo r y s o o n l y t w o s eco nd s o f

t h e r e ac t i on w as d i s p l a yed . [ 31]

H o w ev e r , t h e s e r esu l t s w e r e s t i l l t oo i n accu r a t e as mo re s i gn a l s w e r e

b e in g mis in t e r p r e t ed an d a s im pl e r w a y t o i n pu t t h e sp eed b ump was

d ev i sed . A t ab l e t op sp eed b um p w as c r ea t ed i ns t ead us in g f ou r r amp s

b u t t h e f r equ en c i es n eed ed t o mo d e l t h e t im e t aken t o go o v e r t h e s p eed

b um p w er e r e t a i ned . Th e wid th o f t he s p eed b um p w as b r ok en in to on e

q u a r t e r o f i t s l en gth a s t he in c l in e , a h a l f l en g th a s t h e f l a t t o p an d th e

1 6

l a s t qu a r t e r a s t he d ec l in e . Th e s ys t em u sed i s sh ow n be l ow in F i gu re

9 .

F i g u r e 1 0 : F o r m a t i o n o f t a b l e t o p s p e e d b u m p . [ 3 2 ]

T h e t ab l e was ap p l i ed to 10 km /h r aga i n t o g i v e a l e s s e r r a t i c r es po ns e .

[ 12] [13] Th i s i n pu t w as u s ed to o b t a i n th e 1 0 , 2 0 , 30 an d 40 km/ hr

r e s u l t s fo r t he s p eed bum p.

Veri f icat ion

Fr o m manu f ac t u re ‟ s d a t a an d f r om o th e r an a l ys i s p e r f o rm ed e l s ew h e re ,

t h e s im ul a t io ns a r e a ccu r a t e . I t c an be s een th a t t h e s im ul a t ion

r e s po ns es i n S i mul i nk m a tch c l os e l y w i t h t h e r es po ns e s f o un d b y A d am

K ad e r e t a l . r e s pon s e c lo s e l y m a t ch es t h e re sp on s e in t h e s im ul a t i on s .

T h e s p eed b um ps m od e l l ed i n t h e s i mul a t io ns w e re lo n ge r t h an t he

s p eed b ump mo d e l l ed b y K ad e r w h i ch acco un t s fo r t h e t wo s t age

r e s po ns e . [ 33]

1 7

Conclusion In co n c lu s i on , t h e h a l f - ca r mo de l l ed i s a ccu r a t e wi th i n th e p a r am et e r s

t h a t w e re s e t ou t e a r l i e r i n t h e p ap e r . D r awb ack s i n c l ud e t h e

r e s t r i c t i o ns i n mo d i f i c a t io n o f t h e mod e l wh i ch o th e r m et ho ds , s u ch a s

t h e S i mM ech an i cs m et ho d , d o n o t su f f e r f r om a s w e l l a s a d i f f i cu l t y i n

i mm ed ia t e u nd e rs t and i n g o f t he m od e l w h en ex amin ed . Th e

S i mM ech an i cs m eth o d i s b as ed mo re i n t he v i su a l e l em en t s o i t i s

e a s i e r t o s ee th e wo r k i n g p a r t s o f t h e m od e l .

H o w ev e r , t h e ma them at i ca l m od e l do es h av e th e ad v an t age i n t h a t i t i s

b ack ed up b y a l r ead y e s t ab l i s h ed fo rm ul ae wh e r eas t h e S imM echan i c s

ap p ro ach h a s a l e s s s t ab l e ma th em a t i ca l b as e .

I t w o u ld b e p os s i b l e t o ex p and th e mo d e l t o a fu l l c a r , h ow ev er wi t h

t h e add ed d egr ees o f f r eed om, new fo r m ul ae w ou l d n eed t o b e d e r i ved

i n o r d er t o e s t ab l i s h t h e r e l a t i on sh ip b e t w een each mo vi n g p a r t , a s

s u ch , a S im M ech an i c s app r o ach wo u l d b e mo r e i d ea l i f ex p an s i on w e re

r eq u i r ed .

Bibliography

h t t p : / / w w w. scc s . swa r th mo r e . edu /u se r s / 06 /ad em/ en g in / e12 / l ab4 /

h t t p : / / e sp ace . l i b r a ry . u q . edu . au /e s e rv /U Q :1 32 21 7 / B5 . 1 . pd f

1 8

Appendix

[ 1] Der i va t i on s :

W e kn o w t h a t :

1 .

2 .

3 .

w h e r e

4 .

A p p l yi n g th i s t o t he s ys t em s ho wn i n F i gu r e 1 :

5 .

6 .

M ul t ip l yi n g ac r os s b y L w e ge t :

7 . f o r t h e

u ns p ru n g m as s .

8 . f o r t h e s p ru n g

m ass .

T h e d i sp l acem en t o f t he f r on t and r ea r s us p en s i on u n i t s c an b e f ou nd

u s i n g th e s e eq u a t i on s wi th t h e f r on t hav in g “1 ” i n t h e s ub s c r ip t an d th e

r ea r h av i n g “2 ” in t h e s ub s cr ip t .

T h e tw o t r a ces r equ i red a r e t h e po s i t i o n o f t h e cen t r e o f g r av i t y o f t h e

s p ru n g m as s and t he an g l e o f t h e sp run g m as s wi t h t h e ho r i zon t a l . W e

can f in d t h e po s i t i o n o f t h e sp ru n g m ass a t t h e c en t r e o f g r av i t y b y

ad d i n g th e t o t a l v e r t i c a l f o r ces a t t h e r e a r w i t h t h e to t a l v e r t i c a l f o r ces

a t t h e f r on t su ch t ha t :

F i g u r e 5 : S p r i n g - M a s s - D a m p e r

S y s t e m

1 9

9 .

T h e d i sp l acem en t o f t h e un sp r un g m ass e s ( a r e r e l a t ed to t h e

r o ad d i s p l acem en t an d a r e d e r i v ed t h ro u gh eq u a t i on s 7 an d 8 f ro m

ab ov e .

T h e p i t ch an g l e o f t h e s p r un g m ass can b e fo un d s i mi l a r l y b y f i n d i n g

t ak in g t he mom ent s c r ea t ed a t t h e f ro n t an d rea r o f t he ca r ab ou t t h e

cen t r e o f g r av i t y, s u ch th a t :

1 0 .

In t h e ab ov e eq u a t io n w h en th e s p ru n g m as s ro t a t es abo u t t h e cen t r e o f

g r av i t y t h e f o r ce s gen e r a t ed b y t h e f r o n t and r ea r su spen s i on s ac t i n

o p pos i t e d i r ec t i on s . F i gu r e 2 b e l o w s ho w s t h e f u l l h a l f - ca r m od e l .

F i g u r e 6 : H a l f - c a r m o d e l .

2 0

[ 2] Calcu l a t i on s o f s t i f f n es s an d da m pin g co e f f i c i en t s f o r s us p en s io n :

I t i s k no w n f rom co n t r o l t h eo r y t h a t :

A s s uch :

U s i n g a d am pin g r a t io , a n a t u r a l f r equ en c y, and a m ass

o f 1 00 0k g , t h e c and k va lu e s a r e fo und t o b e :

[ 3] Tab l e sh owi ng th e va r i ous s p eed s a n d r e su l t i ng f r equen c ie s an d

t im e d e l a ys r eq u i r ed .

s p eed

k m/ hr

s p eed

m /h r

s p eed

m /s p e r i od /2

F r eq u en c y

H z

t im e d e l ay

s ec

2 . 5 2 5 00 0 . 69 4 1 . 08 0 . 92 6 2 . 88 0

5 5 0 00 1 . 38 9 0 . 54 1 . 85 2 1 . 44 0

1 0 1 0 00 0 2 . 77 8 0 . 27 3 . 70 4 0 . 72 0

1 5 1 5 00 0 4 . 16 7 0 . 18 5 . 55 6 0 . 48 0

2 0 2 0 00 0 5 . 55 6 0 . 13 5 7 . 40 7 0 . 36 0

2 5 2 5 00 0 6 . 94 4 0 . 10 8 9 . 25 9 0 . 28 8

3 0 3 0 00 0 8 . 33 3 0 . 09 1 1 .1 11 0 . 24 0

3 5 3 5 00 0 9 . 72 2 0 . 08 1 2 .9 63 0 . 20 6

4 0 4 0 00 0 1 1 .1 11 0 . 06 75 1 4 .8 15 0 . 18 0

2 1

[ 4] In pu t s us ed t o c r ea t e sp eed bu mp .

[5] R es u l t i n g sp eed bu mp .

[ 6] Va l u es f o r t h e tw o i n pu t s t eps .

s a m p l e t i m e

s p e e d k m / h r s t e p 1 s t e p 2

5 1 . 0 8 1 . 3 5

2 2

1 0 0 . 5 4 0 . 6 8

1 5 0 . 3 6 0 . 4 5

2 0 0 . 2 7 0 . 3 4

2 5 0 . 2 1 6 0 . 2 7

3 0 0 . 1 8 0 . 2 3

3 5 0 . 1 5 0 . 1 9

4 0 0 . 1 3 5 0 . 1 7

[ 7] Di s p l acem en t a t 2 .5 km/ hr

[ 8] Ang l e a t 2 . 5 km/h r

[ 9] Disp la cemen t a t f r on t an d r ear a t 2 . 5 km/h r

2 3

[ 10] Di sp la cemen t a t 5 km /h r .

[ 11] An g l e a t 5 km/h r .

2 4

[ 12] Di sp la cemen t a t 10 km/ hr .

[ 13] An g l e a t 1 0 km/ hr .

2 5

[ 14] Di sp la cemen t a t 20 km/ hr .

[ 15] An g l e a t 2 0 km/ hr .

2 6

[ 16] Di sp la cemen t a t 30 km/ hr .

[ 17] An g l e a t 3 0 km/ hr .

2 7

[ 18] Di sp la cemen t a t 40 km/ hr .

[1 9] Ang l e a t 40 km / hr .

2 8

[ 20] Di sp la cemen t f o r a p o t ho l e a t 5km / hr .

[ 21] An g l e f o r a po t ho le a t 5 km /h r .

2 9

[ 22] Ca l cu l a t ed va l u es t o b e i np u t t ed i n t o t h e pu l s e g en er a t or s .

P u l s e G e n e r a t o r 1

P u l s e G e n e r a t o r 2

s p e e d k m / h r P e r i o d

P u l s e w i d t h

p h a s e d e l a y P e r i o d

P u l s e w i d t h

p h a s e d e l a y

5 1 . 0 8 5 0 1 . 0 8 2 . 1 6 5 0 0

1 0 0 . 5 4 5 0 0 . 5 4 1 . 0 8 5 0 0

1 5 0 . 3 6 5 0 0 . 3 6 0 . 7 2 5 0 0

2 0 0 . 2 7 5 0 0 . 2 7 0 . 5 4 5 0 0

2 5 0 . 2 1 6 5 0 0 . 2 1 6 0 . 4 3 2 5 0 0

3 0 0 . 1 8 5 0 0 . 1 8 0 . 3 6 5 0 0

3 5 0 . 1 5 5 0 0 . 1 5 0 . 3 1 5 0 0

4 0 0 . 1 3 5 5 0 0 . 1 3 5 0 . 2 7 5 0 0

[ 23] I np u t s u s ed to c r ea t e s e r i es o f s peed b u mps .

[ 24] R es u l t i n g sp eed bu mp s .

3 0

[ 25] Di sp la cemen t f o r r epea t i n g s p eed b um ps a t 2 .5 km/ hr .

[ 26] An g l e f o r r ep ea t i ng s peed bu m ps a t 2 . 5 km /h r .

3 1

[ 27] I np u t s i gn a l s f o r a s am pl e t i m e o f 1 0 s eco nds a t 10 km / hr .

[ 28] R es u l t i n g sp eed bu mp .

3 2

[ 29] Disp l acem en t w i th i n s u f f i c i en t s am p l in g .

[ 30] An g l e w i t h i n su f f i c i en t s am pl i ng .

3 3

[ 31] S ig na l Bu i ld er .

[ 32] I np u t s i gn a l s a n d r e su l t i ng b um p f o r t h e ta b l e t o p bu m p c rea t io n .

3 4

[ 33] R es po ns es f oun d b y A . K ad er e t a l .