1: 3102/ 30/ 22 : - -

$3 $9 : 8765 .03'&% 21! 0!/ -, : +*) (& (! '&% $!#"!

**********************************************************************************

:

1 1# ! 0% /. +*$) '& %$# ! -

( .J) ?7+ > PE %/. > +'= ;. :$) 50 +716 50 ) % : I >H $$$ % /. C F 7 &C %)> C -

( : + !'I )> .O7 %$# >'I : +*$) N( + !'I ) +/. > - .PPE+'= 50 P

$ ) % ( + !'I ) 17 +/. > -

( %Q+ !'I )>

(! (& -7'@' ;?9 ? >= < ;: *'&% !#"!

$3 $9: A; 8765

40

:

:1-

1h 2h

.* ( + $#) ! 2h 1h

:2-

1

+ 1 = 2

*( 0+ $# ) $# 2 1

inagrefseraf/etis/moc.elgoog.setis.www 3102/30/22:

2: 3102/ 30/ 22 : - -

$3 $9 : 8765 .03'&% 21! 0!/ -, : +*) (& (! '&% $!#"!

( .T

S* )17 *$) 5 %Q ( %Q+ !'I ) +/. > - . ePE )$ ) % +'= 50( %Q+ !'I ) 17 +/. > -

( U &H + !'I )>

.17 VH 'U % O$ ( U &H+ !'I ) +/. > - . ePE ) % +'= 50 &$ ( U &H+ !'I ) 17 +/. > -

:543 C 7 != O 7 !N) ?$Z + %

'N )!X &C F$ T0 > -

.1 *$) &C T0 \ *$) %*$) )!X . ON) 17. T0 O7 T0 %

'N T0 )$ T0 V$ 1 ON) X )!O F' -

( . 0 = ) T0 V$ ( 0 = x))$

:

2 :1-

) %X B %+>. X A%7 # X ( %7 )1 %7$ ) 'a ?) m` ( S) !'I TC - ( .+H +cP$) 16 >'I )M$VC

Az )!O F' T0 d

$ &C ;. F

ON) B ' :&C # X ON) - ( .4-/H) 1# ' )!OC M %O7 X $VC Mz 1# ' )!OC A%O7 X

:3- )2(B

)1(A

0 = > = . * (A @=+ $# ) ! 1 2

z :4-

A

B

zA

zM

h

PP z=0,E=0

M

- - : 22 /03 /2013 :3

!"#!$ %&' !( &( )*+ : ,- /!0 !12 %&'30. 5678 : 9$ 3$

1- > N1% 0T VF 7N g$NT% ) I'! + ( X 71 7!M N1 $)F 0TECM \7%EPPA X 71 $d 0T A EPPM 71 $d 0T X M.

2- > 61 0T VF 7N g$NT% ) I'! ( X 71 7!M N1 $)F 0TEC \7% zA

V+ XA CO!) ' #1 zM X V+ M CO!) ' #1 .

3- X 71 $d 0T N1 i #$%' #$+NO 61 1 AX CV$ M. C:

1- N1 $)F 0T MCE \7% PPAE PPME: - 7 > ) : I'! + ( - 7 X!) : C$j .)NO C CFT. - $! 0 7!+ \ . - #$O #$% > 61 0T VF 7N g$NT% A M:

EA + EN' - E7 = EM ECA + EppA = ECM + EppM ECA = 0 (vA = 0 )

mNn ` : EppA = ECM + EppM ECM = EppA - EppM .............................................. (1)

1- N1 $)F 0T MCE \7% Az Mz: - 7 > ) : I'!S ( - 7 X!) : C$j .)NO C CFT. -$! )Po : HdP. - #$O #$% > 61 0T VF 7N g$NT% A M:

EA + EN' - E7 = EM

ECA = 0 (vA = 0 ) m.g.z - m.g.z = )z - (z m.g =m.g.h = )P(W MAMAM-A

-5: z

A

B

Az

Mz

h

0=E,0=z PP

M

P

- - : 22 /03 /2013 :4

!"#!$ %&' !( &( )*+ : ,- /!0 !12 %&'30. 5678 : 9$ 3$

mNn ` : E = m.g.z - m.g.z + 0 CMMA

m.g.z - m.g.z = E MACM .................................................. (2) 3- X 71 $d 0T N1 AX CV$ M:

7 gN : ECM = EppA - EppM ............................................................. (1)

m.g.z - m.g.z = E MACM .................................................. (2) 7> %T%:

EppA = m.g.zA EppM = m.g.zB

=: I'! 71)S ( V+ 61z > p& CO!) ' # ) I'!S + ( $P 05 U+ 0qO% 1 )NO:

Epp = m.g.z

2: ) 05 $d 0T H 05 HF+

- 7On )F I$' ! 61 %P 1)'%AB # U')H/-6(. - : g? #$% 0T qF 7)O q U `% T$F '!g? 61 TAB

U' #.

1-P g? 61 NT Hd 0T $nF Hd IP .O >) g? ( `O #$%

CV$ X C7%w 0T VF O = . 2-0T 7 HN' g? H U C& `OT0 P > AB U' # -T .; ` y70 C '!: C IO% %!w ? C& 0)/(; C '!: C 0T ` - q% %!w ? C&

E)A$!(

-6:

5: 3102/ 30/ 22 : - -

$3 $9 : 8765 .03'&% 21! 0!/ -, : +*) (& (! '&% $!#"!

H +!7 &C U 50 # 5) ?g C # +')!X 17 N dq-3 ?g C \ # ')!O H +!7 &C U 50 O # 5) -4

\ # ) T0 SO(C # )!1)T0 'N T0 : Pq+n w!% 16 ( .)!1

:C :+d$H -1

:(!'I) > Fn$ T0 : !$ oP) 16 > -

. 1 50 'N . AB W(T)>0 ! &C ! F) Ta 0 ++) $ . 1 50 7 . AB W(P) F) ! f 0 \? . : )1 ?g P% P ; \ B 6 A?)$ P% P w # : T0 -

: C ( ?g ) 1$` + Fn$ T0 >

:T0 O FV

)('%$"

r W(P)

W(f)r

r W(T)

:7-

B

AP

f

R

T

- - : 22 /03 /2013 :6

!"#!$ %&' !( &( )*+ : ,- /!0 !12 %&'30. 5678 : 9$ 3$

> 61 0T VF 7N g$NT%)g? ( #$O #$%A B : EA + EN' - E7 = EB

E = )f(W + )P(W ( - )T( W+ E CBB-AB-AB-ACA # \ P %P $)F 0T A 6 B ECA = ECB mNn :

0 = )f(W - )P(W - )T(W B-AB-AB-A 2- XT0 P &'AB )+ 0 HO ' 05 g? HN')T(W r a$F ) ?\ ='%

# ! 0T .; ?w 0 HO ')f(Wr

CN > Hd 0 HO ' )P(W r )5 # 1 g? 7: X N )$Pc+ C&. 3- 7!+ IO N 71 X!)'+ # C g? )5 # 05 C 0T .; q 0T

# g? X&)A 6 B )+ 0 HO ' + C )T(W B-A 71 0T .; !) # N $ ' 0T .; $0 g? : #$% N )$Pc+ #1 >+ 05 H 61 g?

Hd 0 H1)P(W BAr

.

4- IO )5 # O 05 7!+ O!)' # \ C g? 0T C % OS HOV?\

HO ' $0 ?\ 0)f(Wr

C 7> g? id $7 0T C& )$*+ H 61.

3: :

` I'! a%)M `) y%P H5 ) Q% C&)5 7? 6 K = 50 N/m X # a' `) IP A$& $7% 1) 7% X 6? Q% H$T'B Q% ;71 H$T' I'> 1) 7O+ #

7%x) H/-8. (

-8:

- - : 22 /03 /2013 :7

!"#!$ %&' !( &( )*+ : ,- /!0 !12 %&'30. 5678 : 9$ 3$

1- > 61 0T VF 7N g$NT% ) I'! + +Q% ( #$O #$%A B ' N1% X # C&:B 'F O!) > $) 0T yNP $d 0T

) I'! + Q% + ( X 71B 7% \T' Q% # x 0qO% 1 )NO :Epe = m.g.x.

2- V I$0 H! # %)> ) M C& 0 T\ $% ) Hx i 61 nF+ Q% C 7> C& 7:

x

2(m2) Epe = Mgx (J) x(m) M(kg) 0.04 0.100 0.06 0.150 0.08 0.200 0.10 0.250

- 7> H . )NOg = 10 m/s2.

- $N I EPe = f(x2) ; O #$% IP H/ # $NEPe = Ke x2 Z$? Ke =T $N H$ `%'?.

3- y%d #$$OKe H C& `T $% 0 ` C& V '! g$O% U g%' Q% )O% I0 I'> Q% # > + 71 ).

C 7> $1 HnF i Hd: T = M.g (N)

x(m) M(kg) 0.02 0.100 0.03 0.150 0.04 0.200 0.05 0.250

-9:

- - : 22 /03 /2013 :8

!"#!$ %&' !( &( )*+ : ,- /!0 !12 %&'30. 5678 : 9$ 3$

- 0qO% `1 )NO% Q% )++ yNP ) 61 1 :T = M.g H 0qO .; 61 1 g%' 7>.

- $N I T = f(x)$% IP H/ # `O ` #T = Ke' x Z$? Ke' `%'? =T $N ; H$ . 3- C$0 0 Ke Ka' $%K Q% ) y%P . 5- V 61 nF+ V ) Q% O% %)> ) . Q% )++ 0 N1 i

) 0T N1 $. C:

1- NPx.g.m = peE:

- 7 > ) : I'! + +Q%( - 7 X!) : C$j .)NO C CFT. - > 61 )Po $! 0 7!+ \ . - #$O #$% 0T VF 7N g$NT% A B:

EA + EN' - E7 = EB ECA + EppA + EpeA = ECB + EppB + EpeB ECA = 0 (vA = 0 ) EppA = m.g.zA = m.g.x EpeA = 0 (x = 0) ECB = 0 (vB = 0 ) EppB = 0 (CO!) ')

7 mNn: EpeB = m.g.x

2- - 7> :

x2(m2) Epe = Mgx (J) x(m) M(kg)

1.6 . 10-3 0.04 0.04 0.100 3.6 . 10-3 0.09 0.06 0.150 6.4 . 10-3 0.16 0.08 0.200

10-2 0.25 0.10 0.250

-10:

A

BP

Tx

- - : 22 /03 /2013 :9

!"#!$ %&' !( &( )*+ : ,- /!0 !12 %&'30. 5678 : 9$ 3$

- $N )2x(f = peE:

- $N Epe = f(x2) H/ # `O I$' #1 N1 Epe = Ke x2 Z$? Ke $N ; H$ . -$N # :

25 = 0.002 . 3

0.05 . 3 = Ke

3- 0qO% `1 )NO Q% )++ NP g.M= T ? C& :

- 7 > : I'!)S. ( - 7 X!) : C$j .)NO C CFT. - > 61 )Po $! : HdP)++ Q% T. - > + ) :

0=F ext 0 = T + P

F g& $1O/ 0qO H$F%ox HV: F C0/ : P - T = 0 M.g - T = 0 T = M.g

0

0,05

0,1

0,15

0,2

0,25

0,3

0 0,002 0,004 0,006 0,008 0,01 0,012

x2 (m2)

E pe(J)

-10:

P

T

x

- - : 22 /03 /2013 :10

!"#!$ %&' !( &( )*+ : ,- /!0 !12 %&'30. 5678 : 9$ 3$

- 7> :

T = M.g (N)

x(m) M(kg) 1.0 0.02 0.100 1.5 0.03 0.150 2.0 0.04 0.200 2.5 0.05 0.250

- $N )x(f= T :

$NT = f(x) H/ # `O 7N # ) I$' #1 N1 T = Ke' x Z$? Ke' $N ; H$ . - $N # :

50 = 0.01 . 3

0.5 . 3 = 'Ke

3- C$0 eK 'aK $% K: K = 50

Ke = 25 2K

=Ke

Ke' = 50 Ke' = K 4- )+ 0 7 N1 T $) 0T peE:

gN :

xK = E 2ePe

T = Ke x

Z$? :2K

=Ke K='Ke :

0

0,5

1

1,5

2

2,5

3

0 0,01 0,02 0,03 0,04 0,05 0,06

x (m)

T(N)

- - : 22 /03 /2013 :11

!"#!$ %&' !( &( )*+ : ,- /!0 !12 %&'30. 5678 : 9$ 3$

K x21

= E 2Pe

T = K x =:

- `) y%P ) Q% H$T' 71 K 7% a*S x )++ % `% aN+) I'> 61 )Po T +7 0qO% 1 )NO:

T = K x

- 7% a*S Q% H$T' 71 x > p& ) I'! +Q% (1 )NO $) 05 U+ ;71

0qO%:

2pe K x 2

1=E

- Q% ) y%P 7 k % N/m

345: 71 Q% .>+ C& \T' Q% 71 Q% H .>+% + Q% )++ 0 !

#$F C& Q% F 61 gNT ? T*S )H/. (

3: :

- .17 mT' C ? Q% =@ 0) 1 ( `& y%PC = 0.125 N.m/radyNd C& C7 `&)5 # TO Q% O%)2 ( `) y%P )OK = 12.5 N/m HV Q% )F )T 61 gNT )1 (

1 0 61 AO g$NT+ T $ O!) . )H/. (

-12:

T

T

+ ? C& Q%

T ? C& Q%

*S ? C& Q%

- - : 22 /03 /2013 :12

!"#!$ %&' !( &( )*+ : ,- /!0 !12 %&'30. 5678 : 9$ 3$

1- T\ ) H C& '0 NT 7/ C& )$j x Q% )2 ( 7 HV Q% )1 (

;c% $1 HnF i Hd C 7>R = 10 cm:

R.F=)F(M=M o/ F (N) (rd) x(cm) 0.5 0.4 4 1.0 0.8 8 1.5 1.2 12

- 7> H . - $N I M = f() O ' HV ! 1 )$*+ Hd ; F `O #$% IP

H/ #M = Ce Z$? CeN ; H$ `%'? =T $. 2- C& HO' HV Q% C& 0T 'F )/-1 ( HV Q% C& 0T HN)1 (

Q% C& 0T X H C& '+)2 . ( 7> g%' 7> C& 7 i O% C:

2(rd)2 C& 0T Q% )1( EPe = 1/2Kx2(J) (rd) x(cm) 0.4 4 0.8 8 1.2 12

- 0T )$*+ 6F I Epe Q% C& )1 ( X%) \7%2 H/ # `O #$% IP

Epe = Ce' 2 Z$? Kc' `%'? =T $N ; H$ . -#$% 0 C$0 Ce Ce' $0 C U' H0 y%P . 3- $0 0 Ce Q% H& y%P $0 X C . ?q+

4- H/ 61 =+ HV Q% $) 0T N1 i :2pe C21

=E.

5-61 nF+ V ) H& Q% O% %)> ) V . ! 1 N1 i $V 0T N1 H.

-6:

31: 3102/ 30/ 22 : - -

$3 $9 : 8765 .03'&% 21! 0!/ -, : +*) (& (! '&% $!#"!

:C

: >7 - -1

: M =f() N$ -

1N 1# '$I ) # N7 O` eC = M 1N 1# '$I ) # N7 O` )(f = MN$ . $H ; N$ eC ?$Z eC = M

: # N$ -

= 521.0 6.1 . 5.0

5 . 20.0 eC =

: >7 - -2

- - : 22 /03 /2013 :14

!"#!$ %&' !( &( )*+ : ,- /!0 !12 %&'30. 5678 : 9$ 3$

- $N )2(f = pe E:

$NEpe = f() H/ # `O 7N # ) I$' #1 N1 Epe = Ce' 2 Z$? Ce' $N ; H$ . 0.0625 =

0.5 . 1.60.01 . 5

= Ce

!- #$% eC 'eC X C: C = 0.125 Ce = 0.125 Ce = C

Ce = 0.125 Ce = 2C

4- )+ 0 7 N1 T $) 0T peE: gN :

M = Ce EPe = Ce' 2

Z$? :Ce = C 2C

'Ce :

M = C

EPe = 21

C 2

=: - `& y%P H& HV 71 C 1 H& ! % `% aN+) I'> 61 )Po 7% M 1 )NO

0qO%:

M = C

- - : 22 /03 /2013 :15

!"#!$ %&' !( &( )*+ : ,- /!0 !12 %&'30. 5678 : 9$ 3$

- 7% H& HV 71 > p& ) I'! + H& U ( 05 U+ ;71 0qO% 1 )NO $& :

C 21

= E 2pe

% HV y%P 7N.m/rad.

7 $P .; # ' H$F . C O : X0 H:

www.sites.google.com/site/faresfergani

** : : & C0)&** I% y I0 P

)-$T'0 Fares_Fergani@yahoo.Fr

Tel : 0771998109

? # 7 C& H c% C)w 7)N g)5 #1 jq% !).

N' )