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)22)2)+0 1+ 222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 @)22)2,( , 2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 @A)22)2@,( ,(*=(5*22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 @B
)22 &<(,,+,0 1+ 222222222222222222222222222222222222222222222222222222222 @B)222)&<((*,5 ,+-01+( ,+ 222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 @B)222&<((*,5 ,+-00*1( +0 2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 @)222@7 3C 2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 @)222A&<(,01, ,+-0+ 0+0 222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 @?
)22@ &<(,+-00 1+ 22222222222222222222222222222222222222222222222222222222 A))22A +1 ( +D070, 22222222222222222222222222222222222222222222222222222222222222222222222222222222 AA
)22A2)6+( ,2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 AA)22A2 +* ,,*0+0 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 AA)22A2@*1++ ,+ +* ,,*0+08, 222222222222222222222222222222222222222222222222222222222222222222222 AB
-"# .&* """"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""" 87
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2)2) (,+D 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 A?2)2)2)5 ,,/,<*, 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 A?
E/,<*+*4 E/,<*+*4 E/,<*F-( *1(( ,0< 2222222222222222222222222 A?2)2)2+0,=+,, , ++0 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 AG
E,*=( -+ +4 E,*=(++0+4 E+04 -E%+,, ++0 2222222222222222222222222222222222222222 AG2)2)2@0 0+ 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 AG
2)2 ,0*1*,+,/*1 H, 22222222222222222222222222222222222222222222222222222222222222222222222222222 B2)22)(,1 ,5 D, 2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 B2)22,0/0(,( * ,,( ,1 H,22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 B2)22@,,( ,H+, 1 H,22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 B)2)22A,,( ,07+ H, 2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 B2)22B"0+1 (+ 5,++0,22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 B
2)2@ += ( 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 B@2)2@2)'(H,5 , 2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 B@
E+= ( +,,(++ 2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 B@ E+= ( +,, 0+222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 B@ E+= ( +,,/+1- 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 B@ -E+= ( +,/*1 H222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 B@-E+= ( ,0(( 2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 B@
2)2@2+= ( ,1 ,5 D, 2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 BA E + ,+ ,/,<*1+ 222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 BA E/1, +, H ( =,/,<*( + , 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 B E,,,,1+0,11,(,-+ , 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 B
2)2@2@+= ( ,0/0(,( * ,2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 B? E + ,+ ,/,<*0 222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 B? E7 (H222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 BG E+, ,/,<*( + +*2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 -E+, ,/,<*( + +*222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222
2)2@2AD1,+,/+1-22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 ) E+, +11( 0+ *, (( 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 ) E+, +11( 0+ *( *, ((222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222
2)2A % 50+ , 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 @2)2A2)% 50+ ,(0+(, 1 5 D22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 @
E6+( , 2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 @ E% 50+ ,,+ H,22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 A E% 50+ ,315 2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 A -E"0+1 (+ 5222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 B
2)2A2% 50+ ,(0+(, 0/0(( * 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 E6+( , 2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 E% 50+ I0( 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 E% 50+ 1+=(*1 222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 -E% 50+ *+J= 50+ 315,0+ 222222222222222222222222222222222222222222222222222222222222222222222222222 -E% 50+ ,+,+,222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 - E"0+1 (+ 5222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222
2)2A2@+, ,-,(07+,2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 E+,0+=(*1 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 ? E* 0, 2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 ? E+, ,H+, 1 H, 222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 G
2"2 $.36*'.*$"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""" 0;22) +*7 222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222
22)2)"0707 ,( 1 H222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 22)20K"+17, 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 )22)2@07 D1 +(22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222
E7 D+= + 222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 E&7 0*+( 222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 E&7 *( 1( 222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222
22 -,,07 H,0 + 222222222222222222222222222222222222222222222222222222222222222222222222222 @222)1 0 1(+07 H0 + 2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 A222 + ,H (( 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 A222@ + F1++* ,+ ,(0 - 222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 B222A + /1L+0(7ML1,+0(7M 2222222222222222222222222222222222222222222222222222222222222222222222222
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@2)2) ,0 1 ,/,<*17/, H22222222222222222222222222222222222222222222222222222222222222222222222222222222 ?@2)2 ,( * + ,*<(&2222222222222222222222222222222222222222222222222222222222222222222222222222 ?@
#"2 4$$!( '*!'(&""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""" :8@22) 3/17<,,*( ,+ 2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 ?A@22 1 (++ ( +=F0+*, 222222222222222222222222222222222222222222222222222222222222222222 ?A@22@ 01(, ,+(+1*1F- 222222222222222222222222222222222222222222222222222222222222222222222222 ?B@22A +*( ,+ 0+0 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 ?
@22A2)++*+,,( ,22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 ?@22A2+(0(H+, ,+ H(++ 0+022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 ?@22A2@7 D( +* ,,*0+0222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 ??
@22B 07*++( 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 ??#"# $/ &* !$$&96*""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""" :+
@2@2) & ,H+ 222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 ?G@2@2)2)*(+ +( 222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 ?G@2@2)2+17+,-(( =2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 G@2@2)2@0+0*<(+-0070 2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 G
@2@2 ",( ,07+H17+,*-*222222222222222222222222222222222222222222222222222222222222222 G)@2@22)(+17+,-(( =22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 G)@2@22+(0+022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 G
EH+ ,+,,0**= 2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 G EH+ +-0,0**=,( 1+ 0( <2222222222222222222222222222222222222222222222222222222222222222222222222222222222222 G E( +( 2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 GA
@2@2@ ( ,*1((, 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 G@2@2@2)&<(+-0( , 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 G@2@2@2&<(+-0+ 0+0 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 GG
@2@2A :+( + *<(,* ++(/ H2222222222222222222222222222222222222222222222222222222222222222222222 ))@2@2A2)*1++ ,,*<(, 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 ))
E+, ,00,, =, 2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 )) E+,0((+22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 )@ E7 D*<(222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 )A
@2@2A2:+( + *<(07 , 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 )A E-0& 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 )A E-0( + * H 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 )
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#"8 $.36*'.*$*& /""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""" --;@2A2) += ( ,0/0(,( * ,=, 222222222222222222222222222222222222222222222222222222222222222222222222222 ))
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A2A2 ,0 1 0++0 , H,1 0 1+(,=+0 22222222222222222222222222222222222222222222222222 )A?A2A22)++0 , H,+(, 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 )A?A2A22++ +0/0( ,*22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 )AGA2A22@,0+ ,( ,+.+D070,22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 )BA2A22A++ 01(5 +22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 )B)
A2A2@ ,0 1 15*+0,,/,<**, 2222222222222222222222222222222222222222222222222 )B)A2A2@2)/,<*,*,,- ,,, ,++,+ ,+=, 222222222222222222222222222222222222222222222222222222222 )B)A2A2@2&,( +00(+ +(+ 2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 )BA2A2@2@&,01(5 +222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 )BA2A2@2A/,<* +0H , ,*,,2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 )B
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+( , +, ( 0+ (+ 7<, ,+ 0 1, 0 +1<, + +-, H+
07+1 ,21* 07+1 1,(0D(+7<,40 ,F 1+4(-+,07 D,/,<*,/+* H,+-0, 5+0,0+01,,+,(6& +1+4(-+,07 D/1,*( ,+ ,1,,+,(+( +1+ 0,,.,2,007+1 D1, H+ F ( (, (, (, *7( , 0,,+ , + + * ,0*1*, ,/,<*, ( + ,2 (, +( , =+,4 ( , + 5 (,0*1*,+,/*1 H, (,+= ( 25 ( 5(0,1++*<,- ,F- ,(+,+= ( ,,( ,=,4 (, *1+1,(+07 H0 + =+,,( + *1((40FH , *1( +(*007+1 2+,( , <*07+1 40 ,(+,( ,* ++(/ H 0+,00H ,+1,4(07 D,/,<*F*( ,, +1,( +>*1*1F- 21<,1,+ ,/,<*,+* ,H+ ,, 5* F ( = +-0 D0 + *( 7+* H4 0( 0 , ,( *+ < ,* ++(/ H2 1++* H , 5+ 1+ 07 H 0 +
0 +( @
,10 5 H (, C, 07+ H, , * ,, - 02 5 4 ( H+ <* 07+1 4 (*<( F 0+,*( ( + (, - =+ , , ( += F 0+*,2, ,(+, ,1,,0*1+,+*<(F( =2 ,+(,05F,,(+,D1 *+D =, F 1+ =+0 ,,+ 1 0 1 1 ,+ , *1+0, + 2&+(,+07+* H H - 5*1=(,/,<*4(,0+(0(,* H, 1* 1 4 +, 0+ *,4 0, 17*<, *1+0 +-0 (,=,,+2
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Attracteur
Plan de phase
Remarques et / ou
Section de Poincaré
Exemple
(si possible)
Systèmes non
périodiques
Nœud (stable)
C'est un attracteur point.
On le rencontre en statique. La section de Poincaré n’a donc pas vraiment d’intérêt
ici pour nous.
(Nous ne citerons pas
d'exemple ici.)
Systèmes pseudo-
périodiques
Foyer (stable)
Attracteur point. La solution est dite pseudo-périodique.
La section de Poincaré = (x, t) / t = 0 [ T ] donne un ensemble de points qui convergent
vers le point fixe (foyer).
Oscillateur
linéaire amorti
en régime libre.
Systèmes périodiques
Cycle limite (stable)
La section de Poincaré = (x, t) / t = 0 [ T ]
donne un point P.
Oscillateur
linéaire amorti
en régime forcé
(de période T).
Systèmes quasi-
périodiques
Attracteur quasi-
périodique
La section de Poincaré = (x, t) / t = 0 [ T ] est
une courbe fermée cyclique
Oscillateur linéaire
paramétrique
Systèmes chaotiques
Attracteur étrange
Pas d’attracteur « apparent », on peut avoir toutes
sortes de trajectoires complexes.
Oscillateur non linéaire
dans une configuration particulière
x
D
x
D
x
D
D
x
P
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```
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Raideurs de torsion (Nm/rad)
Inertie / axe rotation (*10-6 kg.m²)
Raideurs de torsion (Nm/rad)
Inertie / axe rotation (*10-6 kg.m²)
k1 99 097 I1 2,81 k15 29 044 I15 6,57 k2 33 743 I2 2,38 k16 29 008 I16 10,12 k3 25 261 I3 4,55 k17 168 308 I17 1,30 k4 43 167 I4 10,12 k18 47 585 I18 2,38 k5 168 308 I5 1,30 k19 47 585 I19 1,30 k6 47 585 I6 10,31 k20 169 182 I20 10,31 k7 47 585 I7 6,57 k21 29 044 I21 6,57 k8 169 182 I8 10,12 k22 29 008 I22 10,12 k9 29 044 I9 1,30 k23 168 308 I23 1,30
k10 29 008 I10 2,38 k24 47 585 I24 2,38 k11 168 308 I11 1,30 k25 49 946 I25 1,13 k12 47 585 I12 2,38 k26 196 763 I26 10,31 k13 47 585 I13 1,30 k27 57 964 I27 3,47 k14 169 182 I14 10,31
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Raideurs de torsion (Nm/rad)
Inertie / axe rotation (*10-6 kg.m²)
Raideurs de torsion (Nm/rad)
Inertie / axe rotation (*10-6 kg.m²)
k1 318 388 I1 1,35 k14 132 653 I14 14,79 k2 207 241 I2 14,79 k15 65 814 I15 4,14 k3 65 814 I3 4,14 k16 65 814 I16 14,79 k4 65 814 I4 14,79 k17 138 593 I17 2,04 k5 137 850 I5 2,05 k18 40 969 I18 2,77 k6 40 951 I6 2,76 K19 40 969 I19 2,04 k7 41 064 I7 2,03 k20 138 593 I20 14,79 k8 138 848 I8 14,79 k21 65 814 I21 4,14 k9 65 814 I9 4,14 k22 65 814 I22 14,79
k10 65 814 I10 14,79 k23 169 817 I23 1,66 k11 131 735 I11 2,14 k24 36 189 I24 3,61 k12 41 906 I12 2,60 k25 26 954 I25 3,91 k13 42 023 I13 2,13
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j"6k "6$ 24 % 92 o ""$ 92 1107 6(=+( 1 (P+(/, (+ "1,: =+ P+,* ,, 1++,2:7 " "-4G4@)B@@42
j"GBk "X 22 o 63 "2 /+* 0 +(/, , 5 + "-,( 6+ + K 7 0(++0,2] -4G4)?QE4@@@4)GGB2
j%G?k % 92 o ""$ 92: =+0, H ,+,2 7*<, : = *1+0,,260 \:DG4-,4)GG?2
j?Bk 2+(0(+ 5( +$=+(+0",1,53 C+(9550",11=/%+(( %+ , K 7 "+ +( (++0,2 45 ] - G? ? 4? : D4)4A)A4)G?B2
j"Gk "X" 24 "2 o &" 22 %+0J(+,7 0 6+ &07+ ,*, &( 4 *(+ + D1 *+ 25 4 4 14 )) Q@E4 @G)A?4)GG2
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j3?@=k 3U2U2o3&292 0+((/0*1+0,0 ((+K 7+ ,, 1+ 245 ] - 54B4?AG?B4)G?@2
j3?@0k 3U 2U2 o 3& 292 0+((/ 0 + ,0 ((+ K 7 *1+0, 7+, + & ,21 :784B)4@4)G?@2
j3?B+k 3U 2U2 /+* 0, 5 3+* 0+((/ D0 /,*, 3+- " *1( ,+ ,2+=7+* 0& ,+0+(% 50+ ,2] - 54B4AB@AB?4)G?B2
j3?B=k 3U 2U2 /+* 0, 5 3+* 0+((/ D0 /,*, 3+- " *1( ,+ ,2+ 7+ 0& , +6(=+(% 50+ ,2] - 54 B4 ABGAA4)G?B2
jk $ 9292 46 $2" 4 G ! 46 27<,0+40(+(/42
j"@k "9220924 2D5 -, /,,4@2
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j;G)k ;XUX 2 &+7*+ 0+( &( + *1 *(+ 5 ( 6+ "+(2 41&+7974G)A)4?))4)GG)2
j3k 3 2 o : 2 ( + /+* 0, 5 6+1+ /,*, K 7 0 55,,+%+0J(+,72] -4G4GQE4?@)42
j3)k 32o:2 0+7+ 0/+* 0,5& -6++ /,*,K 7%+0J(+,720?4 4)4AAA4)2
j3?k 3&92&22o63""27+,+5 =( % 50+ , 7",+05+*1+0,0 ((+21G)4B?4)G?2
j&?Ak & 62"2 o & 92 H0/ ",1, 7++0 , 0, 5 0, K 7 ( +&( 1((++0/1 + /2] -4G4G4))))B4)G?A2
j$Gk $&2DB X27<,0+40(+(+ ,4)GG2
j$G?k $& 2 o 6" 2 0J & , 5 *1+0 ,0 ((+,2 ] -54)QE4@@G@4)GG?2
j$"Ak $"" 922 + : =+ , 5 + %+* K 7 + (-3C + +0 %+/ ,2] -4G4B4))G)4A2
j:)k :3a &2 o 63 "2 +(/, , 5 0+((/ :+/ 6+ &,7 /,*, K 7 (*= 0 $, (H7/2] -4G4A@Q@E4BBBAB4)2
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j:B=k :$U& &224 "a 2"22 o 3":a 62 &( 1( *1+0, 5 + %+(( =K K(+,2+&+7*+ 0+(&(( 245 ] - -X4G4???@B4)GB2
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jU?k U6 22"+ +(: =+ K 7%+0J(+,7 + )245 ] -5 D4)4@@@@4)G?2
jU?)k U6 22"+ +(: =+ K 7%+0J(+,7 + 245 ] -5 D4)@4@?@G4)G?)2
jUGBk U6a2/+* 0,5$,/** 0 0K ,( +8( +/,*,, (*, *2] -4G4)?BQ)E4)BB)4)GGB2
jUGk U6a2 0 5 0",1,5"/+* 0,/,*,K 7( +11,245 ] -G4))G4@A@B@4)GG2
jU)k U622 *, 315% 50+ 5+K5*: =*1+0/,*2] -4G4AQ@E4ABA?B4)2
jU3?+k U3 622 6(=+(/+* 0, 5 +: =*1+0 +,0 ((+2] -4G4))?4@GBAG4)G?2
jU3?=k U3 622 7 : =*1+0 ",1, 5 + 3+* 0+((/ D0 + (+ *, +( +,0 ((+2] -4G4))B4@@@)G4)G?2
jUAk U 324 X:9 92 o %9:$X: 2 550 5 *7 0K , +,0 ((+,7 += ( / 0 ,2] -4G44G)G@4A2
j;X)k ;X3": 2:2 * 0+( &7 5 &( =/ /,* 0 +( *1+0 *(+ 41-D 90'4 ,=74)2
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5*, -+
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1+,,+F,/,<*)QH+ +E(+5*, -+
#+)
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))///
++∆+=
)/// ∆+=
+
Q,07*+ *1( 0 E Q,07*+D1( 0 E+, (+1+ H4 (, ,07*+, *1( 0 , ,1+ 0( <*++1,F (+/+* H (+-0
*+ 0*+,,0, ,+1(+H(((+ *,( + ,,+1(,( 0+2-+074 (,,07*+,D1( 0 ,,1+ 0( <*++1,F (+/+* H+1 Q0+,74 *1+04D1(, E 1 (+H(( ( + * , ( + , Q*+ ((,4 * H,4 0+0 8
HXHH& =++ +-0
>∀≥∀
==
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E *QEH *Q
=
=
=
=
=
=
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HEQH
HEQH
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HEQH
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HEQH
Ql*E 0,
/X/ =+
D, )A
5*E,1(,5+0 (+-0+* ,,( * + ,,(1+,0+(0(+11,1+(+5 ,,0,,+ *+ ((+Q,+= ( 0 ((E2
*+ <+(4+ )
// +=+
b,(+*+ 0 +*1( 5 0+ 2
, ,07*+, + , +00*1+, 0+ *= 11 , H , +((,
- +,0H , 2
"2=
22)E-0 0, < 1+, *1, ∆ 0,+ Q ∆= E4 +(, ( ,07*+ , 0- ,
,(*, 415 D4+ EQ// ∞→→ 2
22E0 , + (+ )
// +=+
- 10** , H -+( F (+ (+ , -+EQ2EQ/EQ/ ) τ∆++=+ b EQ τ , ( 0+ (0+(2 -+( ,
+(*1+,+00,, =(Q, 0+>+ (+,( D+0E2-+074, 10 [ ] J+-0420EQ J >∈∀∆≤τ 4+(,(,07*+,0710 ,F( J2
22@E+= ( 0, <,/,<*)(+*QRE4+(,/S)R/2 + =/ ) ∀+ 4
+(, _ _ ) ( ,07*+ , 0 ,+=(2 *+ < +(4 1 +- ( * + -((,(1+,e40 ,F H +,+= ( 0 ((50 (+-+(e2
,/,<* *+ 0 (4 (, -+(, 11, | QE 2 5
EQ&+DEQ
λ=ρ 4 (+/,10+(H Z 5 F)1H (,07*+, ,+=(2+H, +(+,+= ( (+070+(,(,07*+,50*0-2
22AE* ,,** H17+,+ ( H+ 55 (( F ) ( , -+4
HH+-0HHH =ω+ωξ+ 4 (+ ,( D+0= 0,
( )E, QE0,QEQ ϖϖξω += − b ωξϖ )−= ( ) ϖξω 8 += +,( +( 7* H+4H+F((4(+5*, -+
( )E, Q=E0,QHH
ϖ′′+ϖ′= ω′ξ′−
b ξ′ ,( +* ,,*+( 7* HQ ξ≠ E ω′ ,(+1(,+ 11+( 7* HQ ω≠ E2+0,H4( *10 , ,( +*1( (+1,4(,07*+ +
+* ,,** H+ , H 17+,+* H207 D,07*+* H1,1=(<* 0*10,11 ,2
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D, )B
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#++)
* EQ
α 1+1+ QB2AE
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07 , 1R4 b EQα 4 α 40,1F+00(+ */,1+
#+/)
2@2E&7,"X+,*7,"X+0, ,F ,0 ,07+H1+,∆RS),= - ,
*1((R4s4)sgs4JRS)2 ,0+(0(1+00(,1 , EH4HQ )) ++ F( + ,1 , * + , EH4HQ .4).4) ++ 4.Q)≤.≤JE+-0
[ ])404J.)40 ...4 ∈≤≤∆+=
"A"A"7
=++
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≤≤
=++
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+++
≤≤+
≤≤+
<≤
<≤
HXHH&
H=HH
H=HH
J )
HXHH&
H+HH
H+HH
)))
J.).4.)
J.).4.)
.4.4.4
.).4 ..4
.).4 ..4
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HXHH& =++
1E)1QH
HHHEQH
1E1Q
)1E)1Q
τα+
−τ++τ+τ+≈τ
−−
[ ] HXHH&E2QU
=τ−++τ∆
[ ]
HHXH&
)) =
α∆θ+∆θ++α∆θ++α avec
ττ
τττ=
ττ∆
τττ=θ
∆
∆
∆
∆
2EQU
2EQ2EQU
2EQU2
22EQU
∆α+=
∆α+∆+=
+
+
2HH
22HHH
)
)
D, )
1,0- ((*(+*7"X+1+(+=(+, -+
0) g 0 +) g 2 2 2 g 2 22 2 2 g 2 20J +J) +J g +JJ =) = 222 =J) =J
+-0(,11 ,
.) . 0+ =
<≤
)=J.)
. =≤≤
2
,*7,"X+, +* ,+(FJ2JR)4(,(07 D1,, =(,=)R)40H (,07*+ (4+* ( ,0+
-5+0 (*2JR41+- (+*71 * ( (+*731+D*1(2+*7"X+L0(+,, HM0,1FJRA2
4 , -+( , 1(, + (+ ,( ,* ++(/ H 550 F ( +
,07*+ + * H/11 * ( * 5 Q052@2@2A2 EE2, (, *7, "X+ , +* , , *7, F 1+,4 0+ * F
0((,H,+((,0 +,0H , 22@2@E07*+,F1(, ,1+, [ ] ∆∈τ 40
#+3)
HS) , * 1+ 1+ Q B2A E4 -((* , 1(, , 1+,2 , 1
= 1+D*1(,07*+F@1+,3=(
#+')
+* (,,07*+,F1+,*( 1(,(,1(,0,4 , (+*7/,*QE4
(+*7& (QAE4(,*7, +*,%+,7574(,*7, +*,&((,*7,1 0 00 21(,+ (,,0,*7,4, - ,((0F,1Fj&G)k2
Q,07*+10 ,+ 0 ((*,+=(E
=
−+τ≈τ1
..). HEQEQH bHS),( 0F0707
=++∆
−+−=
∆−+−=
++++
−−++
−−++
))))
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HXHH&
HHGH)?H))H
HHAHBH
H
D, )
2@2AE-(11*,, ,+/(5+ (,+11D *+ ,, -+,
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E1Q)H + ,* 0 -+QB2AEFS) )))) HXHH& ++++ =++
1+ H407 , 1R40H (,07*+K*+J
#+)
-, D1( 0 4+-0& ++(,4 8)4 =γ=β 4= (,07*+, 550,
0,H ,10 ,+,02-, *1( 0 4+-0 A8)=β 8)=γ 4+(,07*+( +00(+ */Q5*(+1<CEH ,10 ,+,0 0 ((*,+=(2
( )[ ]
( )[ ]
( )[ ]
β+β−∆+≈
β+β−−
∆++∆+≈
β+β−∆++∆+≈
+−−
+
+
−
+
++
E1Q)1
E1Q1
E)1Q
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E1Q)
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)
E1Q))
E1Q)
1
)
HH)HH
HH)E)1Q
HHH
HH)1
HHH
( )[ ]( )[ ]
=++
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β+β−∆+∆+=
++++
++
++
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HXHH&
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HH)
HHH
Q γ=ββ=β ) 4 E
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<
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1+-
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EDQD1EDQ, )4DEDQD1EDQ, )4D <+>∀<−>∀
∀D45)QDER), QDE45QDER)S, QDEQDERD1QDE
∀D45) QDER0,QDE45 QDER0,QDE QDERD1QDE
?5<#")#=+,01+ 4,,11,,*((+ Q@2B@E2 ∆RA1+-Xs4
1+-t4Xt4tt4 (5+*H
#+)
#+)
,H∆RA1+-X4 H <∆ 4 b
<
∆
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, 41*QB2)@EQB2)AE4 (,55 *
∆=
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,
#+/)
1,
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(-
#+3)
D, )G
EDQPEDQP5EDQPEDQP54D ) <≤<≤>∀
EQEDQEQ5EDQ5EQEDQEQ5EDQ54D )) −<−−<−>∀
EDQEDQ5EDQEDQ54D ) <<>∀
*+ <- 4+
#+')
*(+(+ QB2)E4 (,5+*+ + ( ,(7<*, -+2
9"6-
4 !- - 7 ? 0 ? -A A "7"
7?C7
4∀∈C?-A#)aA#)? ∀#?)∈
b?c⊂C?-#)J-#)a#)J#)( + 7<*,+00 ,,*,5 ,400 ,,,4+11( HF5)+ , H F54
1+$Rk4S∞j4+R=RD +1<,QB2E4 (-
#+)
, )EDQ)EDQ5)EDQ)EDQ54D ) −<−−<−>∀
0 ,F
#+)
,H(+(+ QB2)EF5 (,(+ ,QB2)@EQB2)AE,*,2
![Total No. of Questions : 8] [Total No. of Pages : 2 [3824 ... · ˘ˇ ˆ ˙ ˝ ˛ ˚ ˜ → ! ˇ ˇ " →˚ ˇ ˇ → ˇ ˝ ˇ ˛ ˝ ( ˇ ˙ % ˙ ˝˝˚ ˝˝ ˛ ˚ ˛ ˙) ˝*ˇ](https://img.pdfslide.fr/doc/110x75/5e6cb5dee2da4d02a34b318b/total-no-of-questions-8-total-no-of-pages-2-3824-.jpg)
