BifurcationMecanique ChauffeEauSolaire DepotLaiton

DS SCIENCES PHYSIQUES MATHSPÉ
calculatrice: autorisée
durée: 4 heures
Sujet
! "
!!#$%
&'(%
)* #$
!!!!* +
,-./
!0 /
!!, /
!!!,-.1
&&- 2
)2
,3- .42
!(5.62
!!'# 2"
!!!*#2"
27%

Un exemple simple de servomécanisme à
bifurcation mécanique
8*#$ 
,9#(9*(:
;2//
8$#$
Sd
(!*(
O A2B A
 3
l=9"m
9      ( 
OB
0*
O
<
B
9 
ROxyz
9    (9      *  
B
*
Oz
0*((9-49 
A2
A
m=9kg
=#$((
m
>
>
8(
Ozx '
R'Ox ' y ' z 
(9
g
7%
.

-9
g=19/2m s−
9 *(
Oz , OA2
(
OA2
Oz
*(
Ox , Ox ' 
9 
/
I. Solide tournant
8($?*
B
  *
Oz
( (
OA2
BA2
0#$@
A@(A8 
S
(
˙
*
Oz
8
R
2 59
R'
?
ex ' , ey ' , ez
9
˙
9
¨
l , 
•
A2
R
•
A2
R
 59
R'
?
ex ' , ey ' , ez
9
˙
9
¨
m ,l ,
• 
P
S
R
• 
LO
S
R
O
" 8(
R
O
O
 *
B
S
•&*-$  
O
9 
S
9
R
O
˙
9
¨
m , g , l , 
ex ' , ey ' , ez
•B
S
C,B
 D
% !*#  (
O A2B A
3 
O A2B
•
R
O
(
O A2B A
@
"7%

AD
•,
 =
•BE
Oz
O
?
O
B
S
•0.(9 .
˙
9- 
D
•**( 
Ek
.(
˙
m ,l ,
DF9E99
˙
9**(
II. Système déformable astreint à tourner uniformément
0*
B
(*
9
Ozx '
(
O A2B A
(
= ez
= ˙
R
9    G      
Mm=Mm ez
@(A0
R'
(
(
R
=
A. Énergie potentielle
+ B.
Sd
D
 B9
Mm
9#$
Sd
R
H...C( 
Sd
R
(D
 D
%7%

/ B
R'
(:H...C(
Sd
R'
        (  B  
.
R'
CC(D
9
R'
1 5
R'
?
ex ' , ey ' , ez
9
A2
R'
˙
$
2B**( 
Ek'
Sd
9
R'
9
˙
$
22B**(
Ep , g 
#$8
*((
=/
2F          (?    $        
R'Ox ' y ' z 
M
m
9    
v
R'
,  
R'
(
R
(
= ez
C  
Oz
  8  (  
H
  E  -(  
M
Oz
  8  
HM =x '  ex '
9
OM =x '  ex' z ez
9
v=dx '
dt 
ex 'dz
dt 
ez
•FC,
M
R'
B
R'
0,.
C(D
•F C C CI
fi , e
M
R'
m
9 9
HM
  :              @
(A  5              
R'
  5  
CC(
x '
2"8$J 9
R'
(9*(9*?
E ' p ,c=− m l
9(
=
9
*
2%0*(    
E ' p
R'
          ?
E ' p=E −− u
K
E
u
  *
m , g , l , 
=g/l2/
2,
E
!
B. Positions d’équilibre relatif du système astreint à tourner uniformément
2+B      *  
Sd
R'
@
/
AD  &
#...C D
2'* 
u2
u2
2/0  
u=2
 L 0 
E ' p
9( 
7%