articol dinamica

7/29/2019 articol dinamica

1/6

ANALIZA STABILITATII UNUI AUTOMOBIL

CU AJUTORUL UNUI MODEL SIMPLIFICAT

THE STUDY OF THE CARS STABILITY

USING A SIMPLIFIED MODEL

S.l.drd. ing. L. Simniceanu, Conf. dr. ing. V. Ot, Conf. dr.ing. D-tru Neagoe, Facultatea deMecanica, Universitatea din Craiova

Abstract

In this paper is presented a study of cars stability using a simplified model and a different types of

moving are making obvious. The study is applied for velocitys values witch are content in the

stability field, but also for those values of velocity that surpass the critical value of speed.

1. Theoretical considerationWe will use a simplified model of car for mathematical model construction.

(fig.1)

Fig. 1The velocity of points A1, A2 are:

11

1A

1Y

vi

j

22v

Y

2A

ab

F2

F1

00

0011

a

kji

jvivCAvv yxCA

++=+=rrr

(1)

00

0022

b

kji

jvivCAvv yxCA

++=+=

rrr

(2)

vvv xx == 21


2/6

avv yy +=1 (3)

bvv yy =2

( )v

av

v

vtg

y

x

y

+==

1

1

11 ; (4)

So v

avarctg

y

+

= 11

v

bv

v

vtg

y

x

y

==

2

2

2 ; (5)

So )(2v

b

v

varctg

y =

vdt

vda

rrr

r+= ; = y

xx v

dt

dva ; += v

dt

dva

y

y (6)

The equations for cars moving are:

(7)

+=

++=

+

)sin(cos

)sin(cos

11211

11211

FbYaYI

FYYvvm

z

y

&

The next relation gives the lateral forces:

2,12,12,1 = kY ; Lateral stiffness coefficient of tire;1k

+

+=

+

+

+

=

+

aFbv

bvarctgka

v

avarctgkI

Fv

bv

arctgkv

av

arctgkvvm

yy

z

yy

y

)sin(cos

)sin(cos

112111

112111

&

(8)

For small value of angular lateral deformation we can make the next

approximation:

++

v

a

v

v

v

avarctg

yy,

v

b

v

v

v

bvarctg

yy, 1cos 1 :

The next system is obtained:

+

+

=

+

+

+=

11

2

2

2

121

11

2

2121.

1

zz

y

z

yy

I

ak

vI

bkakv

vI

bkak

mv

k

mv

bkakv

mv

kkv

&

(9)

For a fixed velocity a non-homogenous system is obtained. This has the next

form:

{ } [ ]{ } { }BxAx =+& , (10)

with:


3/6

{ } ;{ } ;, Tx &&& = { } { }Tx ,=

[ ]

+

++

=

vI

bkak

I

bkakmv

bkak

mv

kk

A

zz

2

2

2

121

2

2121 1

{ }

=

11

11

zI

akmv

k

B (11)

2.The stability of systems

The characteristically equation of system is

pmv

kk+

+ 21

2

21

mv

bkak +1 =0 (12)

zI

bkak 21 vI

kbka

z

+ 22

1

2

+p

This can be rewritten:

( )( )0

2

21

2

21

212

2

1

2

2

2121

=

+

+

++

+

++

vIm

bkakmvbkak

mv

kk

vI

kbkap

mv

bkakp

mv

kkp

z

z (13)

If the roots are negative or complex with negative real part, then:

0lim,0lim ==

tt

So the cars moving are stabile.

The previous equation has real negative roots or complex roots with negativereal part if the next conditions are obtained:

( )( )0

0

2

21

2

21212

2

1

2

2

2

1

2

21

>

+

+

+

>

++

+

vIm

bkakmvbkak

mv

kk

vI

kbka

vI

kbka

mv

kk

zz

z(14)

The first relation is content for any car.

The second relation becomes:

(15)( ) 0212

21

2 > bkakmvkkL

The next two cases will be taken into consideration:

I. ( ) 021 bkakthe inequality is content for any cars velocity;


4/6

The time history of , of angular velocity and lateral forces Y1 , Y2 is shownin fig. 2, fig. 3.

The cars moving has a limit point (fig.2) or a limit cycle (fig.3)

II. ( ) 021 > bkak

The inequality is content for smaller velocity then the critical value, witch is givenby:

( )2121

2

bkakm

kkLvcr

= m/s (23)

3. Numerical results

The sinusoidal trajectory

The next numerical values are used:

t( ) 15 sin

2t

:=

v25

3.6:=

5 0 55

0

5

Bj

Sj 2,

4 2 0 2 45

0

5

Aj

Sj 1,

)t(=

)t(=

0 20 40 605

0

5

10

S1

S0

0 20 40 605

0

5

S2

S0

)(t = )(t=

Fig. 2


5/6

The circular trajectory

I 1600:= 10:=m 950:=

v 20:=a 1.2:=

k1 250:=

b 2.4 a:=k2 280:=

D t X,( )

k1 k2+

m v

X0a k1 b k2

m v2

1+

X1

k1

m v+

a k1 b k2

I

X0a

2k1 b

2k2+( )

I vX1

k1 a

I+

:=

)

5 0 5 10 152

0

2

B

S2

150 100 50 0 5010

0

10

A

S1

0 50 1000

2000

4000

Y1

Y2

S0

)t(=

)t(=

(2),(1 tYtY

0 50 100200

100

0

100

S1

S0

0 50 10010

0

10

20

S2

S0

)(t = )(t= Fig. 3


6/6

References

Bolcu, D-tru,

Ot, V.,

Study of variation of steering angle for an autovehicle, The

World Automotive Congress FISITA 2002, Helsinki, paper code:

FO2I240

M.,

Elemente de mecanic tehnic, Editura Universitaria Craiova,

1994.ismaru, L. icri haotice la automobile, Referat nr.2, Bucure]ti, 2000;

. e

5 Gillespie, Th. tals of vehicle dynamics, Harbound, 1992.

.,

L.,

s.a. /2002

. 1.

1

2 Buculei, M.,

Marin,3 C M

4 Ciudakov, E.A Teoria automobilului. Traducere din limba rus. Institutul d

documentare Bucuresti 1958.

Fundamen

6 Ot, V.,

Bolcu, D-tru

Simniceanu,

Dinamica autovehiculelor, Editura Universitaria Craiova, 2005

7 Simniceanu, L., The stability of vehicle-driver system, Analele Universitii din

Craiova, Seria Mecanic, nr.1

8 Untaru, M.,.a Dinamica autovehiculelor pe roi. EDP Bucureti 198

Documents

articol dinamica