• Aucun résultat trouvé

Modélisation non-paramétrique des incertitudes en vibroacoustique appliquée aux automobiles

N/A
N/A
Protected

Academic year: 2021

Partager "Modélisation non-paramétrique des incertitudes en vibroacoustique appliquée aux automobiles"

Copied!
6
0
0

Texte intégral

(1)

HAL Id: hal-00550535

https://hal.archives-ouvertes.fr/hal-00550535

Submitted on 28 Dec 2010

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Modélisation non-paramétrique des incertitudes en vibroacoustique appliquée aux automobiles

L. Gagliardini, J.-F. Durand, C. Fernandez, M. Kassem, Christian Soize

To cite this version:

L. Gagliardini, J.-F. Durand, C. Fernandez, M. Kassem, Christian Soize. Modélisation non-

paramétrique des incertitudes en vibroacoustique appliquée aux automobiles. 10ème Congrès Français

d’Acoustique, Apr 2010, Lyon, France. pp.1-5. �hal-00550535�

(2)

qPH&RQJUqV)UDQoDLVG$FRXVWLTXH

/\RQ$YULO

0RGpOLVDWLRQQRQSDUDPpWULTXHGHVLQFHUWLWXGHVHQYLEURDFRXVWLTXH DSSOLTXpHDX[DXWRPRELOHV

/DXUHQW*DJOLDUGLQL

-HDQ)UDQoRLV'XUDQG

&KDUOHV)HUQDQGH]

0RUDG.DVVHP

&KULVWLDQ6RL]H

36$±3HXJHRW&LWURsQ5RXWHGH*LV\)9(/,=<9,//$&28%/$<

8QLYHUVLWp3DULV(VW/DERUDWRLUHGH0RGpOLVDWLRQHW6LPXODWLRQ0XOWL(FKHOOH060(805&156

%G'HVFDUWHV0DUQHOD9DOOHH)UDQFH

DGUHVVHSUpVHQWHMHDQIUDQFRLVGXUDQG#DVWULXPHDGVQHWDGUHVVHSUpVHQWHFKDUOHVIHUQDQGH]#JGIVXH]FRP

/DGLVSHUVLRQGHV)RQFWLRQVGH5pSRQVHHQ)UpTXHQFHYLEURDFRXVWLTXHVHVWXQSKpQRPqQHTXHO¶RQSHXWREVHUYHUSRXUOD SOXSDUWGHVREMHWVLQGXVWULHOVHOOHWUDGXLWO¶K\SHUVHQVLELOLWpG\QDPLTXHGHVVWUXFWXUHVFRXSOpHVjFHOOHGHVFDYLWpVTXLDSSDUDLW GpV ORUV TXH OD GHQVLWp PRGDOH GpSDVVH XQ FHUWDLQ VHXLO /D PRGpOLVDWLRQ VH GRLW GH SUHQGUH HQ FRPSWHFH IDLW DYpUp HW GRQW O¶HIIHWQHSHXWrWUHFRQVLGpUpFRPPHQpJOLJHDEOH/DPRGpOLVDWLRQpWDQWLPSDUIDLWHHOOHLQGXLWHOOHPrPHGHVpFDUWVSDUUDSSRUW jODUpDOLWpLQFHUWLWXGHVGHPRGqOHOXLPrPHUpVXOWDWGHODVLPSOLILFDWLRQLQKpUHQWHjWRXWHPRGpOLVDWLRQHWLQFHUWLWXGHVGHV GRQQpHV GX PRGqOH /D PRGpOLVDWLRQ QRQSDUDPpWULTXH GHV LQFHUWLWXGHV V¶DSSX\DQW VXU XQ SULQFLSH GH PD[LPXP G¶HQWURSLH TXLSHUPHWGHPRGpOLVHUJOREDOHPHQWO¶HIIHWGHFHVGLIIpUHQWVW\SHVG¶LQFHUWLWXGHDpWpDSSOLTXpHDYHFVXFFqVjODPRGpOLVDWLRQ GHVYpKLFXOHVDXWRPRELOHV

'DQV XQ SUHPLHU WHPSV RQ UDSSHOOH OH SUREOqPH DFRXVWLTXHVWUXFWXUDO G¶XQ YpKLFXOH FRXSOp j VD FDYLWp LQWpULHXUH HW OD PDQLqUH GRQW RQ HQ GpGXLW XQ PRGqOH VWRFKDVWLTXH OHV SDUDPqWUHV GH GLVSHUVLRQ VRQW HQILQ LGHQWLILpV GH IDoRQ j FH TXH OH PRGqOHUHSUpVHQWHDXPLHX[ODUpDOLWp2QSURSRVHHQVXLWHXQHH[SORLWDWLRQPRGDOHSXLVpQHUJpWLTXHGXPRGqOHVWRFKDVWLTXH TXLIDLWDSSDUDLWUHTXHOTXHVSURSULpWpVLQWpUHVVDQWHV

,QWURGXFWLRQ

/DSULVHHQFRPSWHGHVGLVSHUVLRQVHVWSDUWLFXOLqUHPHQW LPSRUWDQWH HQ YLEURDFRXVWLTXH DSSOLTXpH j GHV SURGXLWV LQGXVWULHOV j FDXVH GH O¶K\SHUVHQVLELOLWp GH OD UpSRQVH G\QDPLTXH GHV V\VWqPHV PpFDQLTXHV j GH IDLEOHV PRGLILFDWLRQV $ WLWUH G¶H[HPSOH OD ILJXUH PRQWUH XQ H[HPSOH GH PHVXUH GH )RQFWLRQ GH 5pSRQVH HQ )UpTXHQFH )5) VXU YpKLFXOHV DXWRPRELOHV FRQVLGpUpV FRPPH LGHQWLTXHVHQERXWGHOLJQHGHSURGXFWLRQ

)LJXUH)RQFWLRQGH5pSRQVHHQ)UpTXHQFH 9LEURDFRXVWLTXHPHVXUpHHQWUHXQSRLQWGDQVO¶KDELWDFOHHW

XQSRLQWVXUODVWUXFWXUHSRXUYpKLFXOHVQHXIVJULV 6XSHUSRVLWLRQGHODPR\HQQHVWDWLVWLTXHEOHXDLQVLTXHGHV

pPH

FHQWLOHURXJHHW

pPH

FHQWLOHVYHUW

/D PRGLILFDWLRQ GX V\VWqPH pWXGLp SHXW UpVXOWHU G¶LPSHUIHFWLRQV ORUV GH OD SURGXFWLRQ YDULDWLRQV JpRPpWULTXHVFRPSRUWHPHQWPpFDQLTXHGHVPDWpULDX[RX ELHQG¶pYROXWLRQVGHODFRQFHSWLRQPLVHDXSRLQWHQWUDLQDQW XQH PRGLILFDWLRQ GHV GHVVLQV GLYHUVLWp OLpHV j XQH GpFOLQDLVRQ FRPPHUFLDOH %LHQ TXH GH QDWXUHV WUqV GLIIpUHQWHV±O¶XQHGHQDWXUHSK\VLTXHO¶DXWUHDWWDFKpHjXQ

SURFHVVXV LQWHOOHFWXHO FHV PRGLILFDWLRQV HQJHQGUHQW XQH GLVSHUVLRQGHODUpSRQVHGHVREMHWVSK\VLTXHVSDUUDSSRUWj FHOOHG¶XQ PRGqOHXQLTXHFHQVpUHSUpVHQWHUO¶HQVHPEOH/H UHFRXUVjXQPRGqOHHVWHQOXLPrPHXQIDFWHXUG¶pFDUWQRQ PDLWULVpHQWUHO¶REMHWG¶pWXGHHWODUpDOLWp

'HVWUDYDX[UpFHQWV>@RQWPRQWUpTX¶LOpWDLWSRVVLEOH GH PRGpOLVHU JOREDOHPHQW OHV LQFHUWLWXGHV DWWDFKpHV j XQ PRGqOH j O¶DLGH GH PRGqOHV SUREDELOLVWHV TXL V¶DSSXLHQW VXUXQSULQFLSHGHPD[LPXPG¶HQWURSLH&HWWHPRGpOLVDWLRQ JDUDQWLW DLQVL TXH WRXWHV OHV PRGLILFDWLRQV G¶XQH DPSOHXU GRQQpH D\DQW XQ VHQV SK\VLTXH VHURQW SULVHV HQ FRPSWH TXHOTXH VRLW OHXU QDWXUH /¶DPSOHXU GHV PRGLILFDWLRQV HVW FRQWU{OpH SDU GHV SDUDPqWUHV GLWV GH GLVSHUVLRQ TXL FRQWU{OHQW OD YDULDQFH GHV RSpUDWHXUV FRQVWLWXWLIV GH WRXW SUREOqPHYLEURDFRXVWLTXHOLQpDLUHRSpUDWHXUGHPDVVHGH UDLGHXU HW G¶DPRUWLVVHPHQW GHV FDYLWpV HW GHV VWUXFWXUHV DLQVL TXH OHV RSpUDWHXUV GH FRXSODJH IOXLGHVWUXFWXUH /¶DSSOLFDWLRQGHFHVWKpRULHVDpWpPHQpHDYHFVXFFqVGDQV OHFDVGHPRGqOHVGHJUDQGHWDLOOHWHOVTXHFHX[XWLOLVpVGDQV O¶LQGXVWULHDXWRPRELOH>@

/¶XWLOLVDWLRQ GH WHOV PRGqOHV SHUPHW G¶DERUGHU VRXV XQ QRXYHO DQJOH OHV TXHVWLRQV TXH SRVHQW OHV V\VWqPHV LQFHUWDLQV HW SOXV SDUWLFXOLqUHPHQW FHX[ VXMHWV j O¶K\SHUVHQVLELOLWpG\QDPLTXH

$SUqV XQ EUHI UDSSHO GH OD WHFKQLTXH GH PRGpOLVDWLRQ XWLOLVpH RQ PRQWUHUD TXHOTXHV UpVXOWDWV W\SLTXHV FRQFHUQDQW OHV FDUDFWpULVWLTXHV PRGDOHV G¶XQ PRGqOH VWRFKDVWLTXH 2Q LQWURGXLUD HQVXLWH XQH DSSURFKH pQHUJpWLTXH TXL SHUPHW GH PHWWUH HQ pYLGHQFH G¶DXWUHV SURSULpWpVLQWpUHVVDQWHVGHVV\VWqPHVLQFHUWDLQV

,OHVWLPSRUWDQWGHQRWHUTXHOHVSURSULpWpVTXLYRQWrWUH PRQWUpHVLFLUpVXOWHQWGXPRGqOHVWRFKDVWLTXHXWLOLVpHWQRQ SDVO¶LQYHUVH

/HV UpVXOWDWV PRQWUpV RQW WRXV pWp REWHQXV j SDUWLU GH

PRGqOHV YLEURDFRXVWLTXH GH YpKLFXOHV WHOV TX¶LOV VRQW

XWLOLVpV GDQV O¶LQGXVWULH &HV PRGqOHV RQW XQH WDLOOH GH

O¶RUGUHGH

'HJUpV'H/LEHUWp''/

(3)

0RGpOLVDWLRQQRQSDUDPpWULTXH GHVLQFHUWLWXGHVHQYLEURDFRXVWLTXH

/HV PRGqOHV XWLOLVpV VRQW GpFULWV GH PDQLqUH GpWDLOOpH GDQVODUpIpUHQFH>@

2Q DSSHOOH PRGqOH PR\HQ OH PRGqOH GpWHUPLQLVWH VXU OHTXHO V¶DSSXLH OH PRGqOH VWRFKDVWLTXH /H PRGqOH PR\HQ HVW WHO TXH OHV RSpUDWHXUV PR\HQV GX PRGqOH VWRFKDVWLTXH VRQW pJDX[ DX[ RSpUDWHXUV GX PRGqOH PR\HQ 3RXU XQ SUREOqPH YLEURDFRXVWLTXH OH PRGqOH V¶DSSXLH FODVVLTXHPHQW >@ VXU XQH IRUPXODWLRQ PDWULFLHOOH LVVXH G¶XQH UpVROXWLRQ HQ UpJLPH KDUPRQLTXH GHV pTXDWLRQV DX[

GpULYpHV SDUWLHOOHV GHV PLOLHX[ FRQWLQXV SDU XQH WHFKQLTXH G¶pOpPHQWVILQLVGHODIRUPH

» »

¼ º

« «

¬

= ª

» »

¼ º

« «

¬ ª

» »

¼ º

« «

¬ ª

ω

ω ω

ω ω ω

ω

D V D

V D

Q 7 Q Q

Q Q V

Q

4 I S

X

$

&

&

$

D D V

D V

V

R X

V

UHVS S

D

HVWOHYHFWHXUGHVYLWHVVHVFRPSOH[H GH OD VWUXFWXUH UHVS FDYLWp DX[ Q°XGV GX PDLOODJH FRRUGRQQpHV SK\VLTXHV

VQ

ω

$V

HVW OD PDWULFH GH UDLGHXU G\QDPLTXH GH OD VWUXFWXUH LQ YDFXR WHOOH TXH

V Q V Q V Q V

QV 0 V 'V .V

$

ω

=−

ω

+M

ω

+

R

VQ VQ VQ

V V

V ' .

0

VRQW UHVSHFWLYHPHQW OHV PDWULFHV GH PDVVH DPRUWLVVHPHQW HW UDLGHXU GH OD VWUXFWXUH

DQ

ω

$

D

HVW OD PDWULFH G¶DGPLWWDQFH GH OD FDYLWp j SDURLV EORTXpHV WHOOH TXH

D D D D

D D D

D Q M Q Q

Q 0 ' .

$

ω

=−

ω

+

ω

+

R

D D D

D D

D Q Q

Q ' .

0

VRQW ± DEXVLYHPHQWQRPPpHVPDWULFHVGHPDVVHG¶DPRUWLVVHPHQW HWGHUDLGHXUGHODFDYLWpHW

D VQ

&

Q

HVWODPDWULFHGHFRXSODJH IOXLGHVWUXFWXUH

/H VRXOLJQHPHQW LQGLTXH LFL TX¶LO V¶DJLW ELHQ GHV PDWULFHVPR\HQQHV

/HVWHUPHVG¶H[FLWDWLRQ I

V

ω HW 4

D

ω VHURQWWRXMRXUV VXSSRVpVLQYDULDEOHV

3RXU OHV SUREOqPHV GH JUDQGH WDLOOH ±SOXVLHXUV PLOOLRQV GH GHJUpV GH OLEHUWp LO HVW FODVVLTXH GH UHFRXULU j GHV WHFKQLTXHV GH UpGXFWLRQ PRGDOH GHV PRGqOHV j FDXVH GH OHXUJUDQGHHIILFDFLWpQXPpULTXH/HVFKDPSVYLEUDWRLUHVHW DFRXVWLTXHV VRQW DORUV SURMHWpV VXU OD EDVH IRQFWLRQQHOOH WURQTXpH TXH FRQVWLWXHQW SRXU OD VWUXFWXUH OHV PRGHV SURSUHVGHODVWUXFWXUHLQYDFXRHWSRXUODFDYLWpOHVPRGHV SURSUHV GH OD FDYLWp j SDURLV EORTXpHV /HV FRRUGRQQpHV JpQpUDOLVpHV T

ΨV

ω T

ΦD

ω TXL FDUDFWpULVHQW DORUV OHV FKDPSVYLEUDWRLUHVHWDFRXVWLTXHVVRQWWHOOHVTXH

ω ω

ω ω

D D

V V

T ĭ S

T Ȍ X

Φ

=

Ψ

=

&HV FRRUGRQQpHV JpQpUDOLVpH VRQW OHV VROXWLRQV GH O¶pTXDWLRQ YLEURDFRXVWLTXH VLPLODLUH j PDLV H[SULPpH SRXU OD VWUXFWXUH GDQV OD EDVH GHV PRGHV SURSUHV GH OD VWUXFWXUHLQYDFXRHWSRXUODFDYLWpGDQVODEDVHGHVPRGHV SURSUHVGHODFDYLWpjSDURLVEORTXpHV

» »

¼ º

« «

¬

= ª

» »

¼ º

« «

¬ ª

» »

¼ º

« «

¬ ª

Φ Ψ

Φ Ψ Φ Φ

Ψ

Φ Ψ Ψ

ω

ω ω

ω ω

ω ω

D V D

V D

7

V

4 I T

T

$

&

&

$

2ODPDWULFHUpGXLWHGHUDLGHXUG\QDPLTXH $

VΨ

ω HW OD PDWULFH UpGXLWH G¶DGPLWWDQFH JDUGHQW GHV H[SUHVVLRQV VLPLODLUHV j FHOOH GRQQpHV SOXV KDXW HQ FRRUGRQQpHV SK\VLTXHV

V V V

V

0 ' .

$

Ψ

ω = − ω

Ψ

+ M ω

Ψ

+

Ψ

D D D

D Φ

M

Φ Φ

Φ

= − 0 + ' + .

$ ω ω

ω

3DU GpILQLWLRQ GH OD SURMHFWLRQ UpDOLVpH OHV PDWULFHV GH PDVVH HW GH UDLGHXU VRQW GLDJRQDOHV GDQV OHV FRRUGRQQpHV JpQpUDOLVpHV (Q UHYDQFKH OD PDWULFH G¶DPRUWLVVHPHQW ± IOXLGH RX VWUXFWXUH HVW SOHLQH GpV ORUV TX¶HOOH Q¶HVW SOXV SURSRUWLRQQHOOHVRLWjODPDWULFHGHUDLGHXUVRLWjODPDWULFH GHPDVVH

/DPRGpOLVDWLRQVWRFKDVWLTXHSURSRVpHV¶DSSXLHVXUXQH WKpRULHPDWKpPDWLTXHGHPDWULFHVDOpDWRLUHVYRLU>@SRXU FRQVWUXLUH GHV UpDOLVDWLRQV LQGpSHQGDQWHV GH O¶pTXDWLRQ 2Q VXEVWLWXH DORUV DX[ PDWULFHV FRQVWLWXWLYHV GX SUREOqPH PR\HQ 0

VΨ

'

VΨ

.

VΨ

0

DΦ

'

DΦ

.

DΦ

&

Ȍĭ

OHV PDWULFHV DOpDWRLUHV 0

VȌ

'

VȌ

.

VȌ

0

Dĭ

'

Dĭ

.

Dĭ

&

Ȍĭ

FRQVWUXLWHV GH PDQLqUH DOJpEULTXH j SDUWLU GHV PDWULFHV PR\HQQHV HW G¶XQSDUDPqWUHGHGLVSHUVLRQDVVRFLpjFKDTXHPDWULFH

/¶pTXDWLRQGHYLHQWDORUVXQHpTXDWLRQVWRFKDVWLTXH

» »

¼ º

« «

¬

= ª

» ¼

« º

¬

» ª

¼

« º

¬ ª

Φ Ψ Φ

Ψ Φ

Φ Ψ

Φ Ψ Ψ

ω

ω ω

ω ω

ω ω

D V D

V 7 D

V

4 I T

T

$

&

&

$

/HV PDWULFHV TXL FRPSRVHQW OH V\VWqPH VRQW FRQVWUXLWHV GH OD PrPH PDQLqUH TXH SRXU O¶H[SUHVVLRQ PDLVHQXWLOLVDQWOHVPDWULFHVpOpPHQWDLUHVDOpDWRLUHV

,O HVW j QRWHU TX¶DXFXQH GHV PDWULFHV DOpDWRLUHV FRQVWUXLWHV VXLYDQW >@ Q¶HVW GLDJRQDOH ORUVTXH H[SULPpH GDQVODEDVHGHVYHFWHXUVSURSUHVGXV\VWqPHPR\HQHOOHV JDUGHQWFHSHQGDQWOHVPrPHVSURSULpWpVGHGpILQLWLRQHWGH SRVLWLYLWp TXH OHV PDWULFHV PR\HQQHV /HV SDUDPqWUHV GH GLVSHUVLRQRQWpWpLGHQWLILpVGHIDoRQjUHSUpVHQWHUDXPLHX[

OHV pFDUWV REVHUYpV H[SpULPHQWDOHPHQW HQWUHV PHVXUHV HW DYHFOHVPRGqOHV

/HV VROXWLRQV GX SUREOqPH VWRFKDVWLTXH VRQW REWHQXHV SDUXQH PpWKRGHGH0RQWH&DUORF¶HVWjGLUHHQUpVROYDQW XQJUDQGQRPEUHGHUpDOLVDWLRQVGXSUREOqPH2QSHXWDORUV FDOFXOHU GHV LQGLFDWHXUV VWDWLVWLTXHV j SDUWLU GH O¶HQVHPEOH GHV UpVXOWDWV REWHQXV &RPSWH WHQX GX PDQTXH GH FRQQDLVVDQFH GH OD IRUPH GHV GHQVLWpV GH SUREDELOLWp GHV UpVXOWDWV REWHQXV XQH DQDO\VH SDU TXDQWLOHV HVW VRXYHQW OD SOXVIDFLOHPHQWXWLOLVDEOHSRXUGHVDSSOLFDWLRQVG¶LQJpQLHULH

>@

/H QRPEUH GH UpDOLVDWLRQV QpFHVVDLUHV GpSHQG GH O¶REMHFWLI UHFKHUFKp GH TXHOTXHV FHQWDLQHV j TXHOTXHV PLOOLHUV VHORQ TX¶RQ O¶RQ V¶LQWpUHVVH j GHV LQGLFDWHXUV VWDWLVWLTXHV UREXVWHV PR\HQQH RX LQWHUYDOOH GH FRQILDQFH RX TXH O¶RQ VRXKDLWH XQH FRQYHUJHQFH GH OD IRQFWLRQ GH GHQVLWpGHSUREDELOLWp

$QDO\VHVWDWLVWLTXHGHODEDVH PRGDOHG¶XQV\VWqPHLQFHUWDLQ

/¶DQDO\VH VWDWLVWLTXH D MXVTX¶j SUpVHQW PDMRULWDLUHPHQW pWpDSSOLTXpHjGHV)5)VYLEUDWRLUHVDFRXVWLTXHVRXYLEUR DFRXVWLTXH 2Q SHXW DXVVL HIIHFWXHU GHV VWDWLVWLTXHV VXU OH FRPSRUWHPHQW PRGDO GX V\VWqPH 3RXU OD VWUXFWXUH SDU H[HPSOH RQ UpVRXW SRXU FKDTXH UpDOLVDWLRQ GX PRGqOH VWRFKDVWLTXHXQSUREOqPHDX[YDOHXUVSURSUHVGHODIRUPH

[ Λ 0

VΨ

+ .

ΨV

] Ψ =

2QREWLHQWDLQVLXQHFROOHFWLRQGHEDVHVPRGDOHV { } Ȍ

θM

R

θ

M

LQGLTXHOHQXPpURGHUpDOLVDWLRQ

'HPDQLqUHVLPLODLUHjFHTXLHVWIDLWORUVTX¶RQFRPSDUHGHV

EDVHV PRGDOHV H[SpULPHQWDOHV RQ GpILQLW XQ FULWqUH TXL

FDUDFWpULVHODVLPLOLWXGHGHVGpIRUPpHVPRGDOHV/H60$&

(4)

6WRFKDVWLF0RGDO$VVXUDQFH&ULWHULRQHQWUHGHX[PRGHVL HWMHVWGpILQLSDU

M

( 60$&

LM

°¿

θ

° ¾

½

°¯

° ®

­

=

M M

M

ș M ș M L L

ș M L

Ȍ Ȍ Ȍ Ȍ

Ȍ

Ȍ

R ( { } HVWO¶HVSpUDQFHPDWKpPDWLTXH

/H 60$& WUDGXLW OHV VLPLOLWXGHV TXL H[LVWHQW HQ PR\HQQH HQWUH OHV PRGHV GX PRGqOH PR\HQ HW FHX[ GX PRGqOHVWRFKDVWLTXH

/D ILJXUH PRQWUH OD PDWULFH GX 60$& SRXU OHV SUHPLHUV PRGHVG¶XQH VWUXFWXUH2QREVHUYH XQH PDWULFHj GLDJRQDOH IRUWH TXL WUDGXLW OH IDLW TXH HQ PR\HQQH OHV GpIRUPpHV PRGDOHV VRQW SURFKHV GH FHOOHV GX PRGqOH PR\HQ

6XUODILJXUHOHPrPHFULWqUHHVWWUDFpHQWUHGHVPRGHV G¶RUGUH SOXV pOHYp 2Q FRQVWDWH FHWWH IRLVFL XQH GLDJRQDOH IORXH DYHF GHV YDOHXUV SOXV IDLEOHV TXL WUDGXLVHQW OH IDLW TX¶HQ PR\HQQH OHV PRGHV LQFHUWDLQV RQW GHV DOOXUHV GLIIpUHQWHVGHVPRGHVGHODVWUXFWXUHPR\HQQH

)LJXUH&ULWqUH60$&SRXUOHVSUHPLHUVPRGHV>

+]@G¶XQHVWUXFWXUHDXWRPRELOHLQFHUWDLQHPRGpOLVpHj O¶DLGHG¶XQHWHFKQLTXHQRQSDUDPpWULTXH

)LJXUH&ULWqUH60$&SRXUOHVPRGHVHQWUH>+]HW +]@G¶XQHVWUXFWXUHDXWRPRELOHLQFHUWDLQHPRGpOLVpHj

O¶DLGHG¶XQHWHFKQLTXHQRQSDUDPpWULTXH

&HVUpVXOWDWVSHUPHWWHQWGHSUpFLVHUODQRWLRQGHEDVVHV IUpTXHQFHVVRXYHQWDVVRFLpHjXQFRPSRUWHPHQW©PRGDOª HQ\DMRXWDQWODQRWLRQGHVWDELOLWpGHODIRUPHSURSUHSRXU O¶HQVHPEOHpWXGLp&HWWHVWDELOLWpHVWLQGLVSHQVDEOHSRXUTXH OHV LQWHUSUpWDWLRQV TXL VRQW DVVRFLpHV DX[ PRGHV DLHQW XQ VHQV$LQVLF¶HVWVHXOHPHQWSRXUOHVSUHPLHUV PRGHVGHOD VWUXFWXUH TXH O¶RQ SHXW OpJLWLPHPHQW IDLUH OD VWDWLVWLTXH LQGLYLGXHOOHGHVIUpTXHQFHVSURSUHV

/D ILJXUH PRQWUH OD GLVWULEXWLRQ GHV IUpTXHQFHV SURSUHVG¶XQPRGHJOREDOGHVWUXFWXUHFDOFXOpHjO¶DLGHGX

PRGqOH VWRFKDVWLTXH GDQV OH FDV R VHXOH OD PDVVH HVW LQFHUWDLQH SRXU GHV YDOHXUV FURLVVDQWHV GX SDUDPqWUH GH GLVSHUVLRQ GH OD PDVVH VWUXFWXUDOH 2Q FRQVWDWH TX¶HIIHFWLYHPHQW OD IUpTXHQFH SURSUH HVW G¶DXWDQW SOXV GLVSHUVpHTXHOHSDUDPqWUHGHGLVSHUVLRQDXJPHQWH

6XU OD ILJXUH HVW WUDFpH OD GLVWULEXWLRQ GHV IUpTXHQFHV SURSUHVG¶XQ PRGHJOREDOGH VWUXFWXUHFDOFXOpHjO¶DLGHGX PRGqOH VWRFKDVWLTXH GDQV OH FDV R VHXOH OD UDLGHXU HVW LQFHUWDLQH HW SRXU GHV YDOHXUV FURLVVDQWHV GX SDUDPqWUH GH GLVSHUVLRQ GH OD UDLGHXU VWUXFWXUDOH 2Q FRQVWDWH LFL TX¶HQ SOXV GH VH GLVSHUVHU OD IUpTXHQFH SURSUH HVW HQ PR\HQQH G¶DXWDQW SOXV IDLEOH TXH OH SDUDPqWUH GH GLVSHUVLRQ HVW JUDQG /RUVTXH OH QLYHDX GH GLVSHUVLRQ GH OD UDLGHXU HVW LPSRUWDQWODIUpTXHQFHSURSUHHVWHQPR\HQQHLQIHULHXUHj FHOOH GX PRGqOH PR\HQ &¶HVW XQH FRQVWDWDWLRQ TXL SHXW VRXYHQWrWUHIDLWHGDQVODSUDWLTXH

)LJXUH'LVWULEXWLRQGHVIUpTXHQFHVSURSUHVG¶XQPRGH

JOREDOG¶XQHVWUXFWXUHGRQWVHXOHODPDVVHHVWGLVSHUVpH

&RXOHXUVURXJHYHUWEOHXQRLUF\DQMDXQHDVVRFLpHVjGHV YDOHXUVFURLVVDQWHVGXSDUDPqWUHGHGLVSHUVLRQGHODPDVVH

)LJXUH'LVWULEXWLRQGHVIUpTXHQFHVSURSUHVG¶XQPRGH

JOREDOG¶XQHVWUXFWXUHGRQWVHXOHODUDLGHXUHVWGLVSHUVpH

&RXOHXUVURXJHYHUWEOHXQRLUF\DQMDXQHDVVRFLpHVjGHV YDOHXUVFURLVVDQWHVGXSDUDPqWUHGHGLVSHUVLRQGHODUDLGHXU

&RQVWUXFWLRQG¶XQPRGqOH pQHUJpWLTXH

/D UpIpUHQFH >@ GpWDLOOH OD FRQVWUXFWLRQ G¶XQ PRGqOH pQHUJpWLTXH GpGXLW VDQV K\SRWKqVH VLPSOLILFDWULFH GX PRGqOHJpQpUDOGHUpSRQVHOLQpDLUHG¶XQV\VWqPHPpFDQLTXH HQUpJLPHKDUPRQLTXHjGHVH[FLWDWLRQVTXHOFRQTXHV

= 76 7

+

6

99 ))

R 6

99

HVW OD PDWULFH LQWHUVSHFWUDOH GH UpSRQVH HQYLWHVVH 6

))

ODPDWULFHLQWHUVSHFWUDOHG¶H[FLWDWLRQ7HVW OD PDWULFH GHV )5)V HQWUH SRLQWV G¶REVHUYDWLRQ HW SRLQWV

(5)

G¶H[FLWDWLRQ REWHQXH HQ LQYHUVDQW OH V\VWqPH HW

LQGLTXH OD WUDQVSRVLWLRQ HW OD FRQMXJDLVRQ G¶XQH PDWULFH FRPSOH[H

7RXWHV OHV JUDQGHXUV XWLOLVpHV GDQV FH SDUDJUDSKH HW OH VXLYDQW GpSHQGHQW LPSOLFLWHPHQW GH OD IUpTXHQFH SXLVTXH OHVGpULYDWLRQVTXLVXLYHQWV¶DSSXLHQWVXUGHV)5)

/H PRGqOH pQHUJpWLTXH SURSRVp UHOLH OH YHFWHXU GHV SXLVVDQFHV LQMHFWpHV DX[ ''/ G¶H[FLWDWLRQ ʌ

LQ

j XQH SXLVVDQFHGHUpSRQVHDX[''/G¶REVHUYDWLRQ ʌ

RXW

LQ 2,

RXW

ȟ ʌ

ʌ =

R ȟ

2,

HVWXQHPDWULFHDOpDWRLUHGH)RQFWLRQVGH5pSRQVH HQ)UpTXHQFHGHOD3XLVVDQFH)5)3DGLPHQVLRQQHOOH

3RXUFHWWHFRQVWUXFWLRQRQLQWURGXLW <

LQ

UHVS <

RXW

OD PR\HQQHGHODSDUWLHUpHOOHGHODPDWULFHGHPRELOLWpHQWUH OHV ''/ GHV SRLQWV G¶H[FLWDWLRQ UHVS G¶REVHUYDWLRQ TXH O¶RQ GpFRPSRVH VRXV OD IRUPH <

LQ

= ;

LQ

<

LQGLDJ

;

+LQ

UHVS

=

+RXW

GLDJ RXW RXW

RXW

; < ;

< FH TXL SHUPHW G¶LQWURGXLUH OHV GLUHFWLRQVSULQFLSDOHVGHPRELOLWpPR\HQQHVDX[TXHOOHVRQ DFFqGH JUkFH j OD PDWULFH GH SDVVDJH PR\HQQH

;

LQ

UHVS

;

RXW

/¶H[LVWHQFH G¶XQH WHOOH GpFRPSRVLWLRQ HVW JDUDQWLH SDUOHIDLWTXHODSDUWLHUpHOOHG¶XQHPDWULFHGHPRELOLWpHVW SDUFRQVWUXFWLRQGpILQLHSRVLWLYH

2QSHXWDORUVpFULUH

)LQ

GLDJ LQ LQ

< V

ʌ =

R V

)

= ( ;

LQ

6

))

;

+LQ

)

LQ

GLDJ HVW OH YHFWHXU GHV 'HQVLWpV 6SHFWUDOHVGH3XLVVDQFH'63GHVHIIRUWVDSSOLTXpVSURMHWp GDQVODEDVHGHVPRELOLWpVSULQFLSDOHV

RXW

Y GLDJ RXW RXW

< V

ʌ =

R V

Y

= ( ;

RXW

6

99

;

+RXW

)

RXW

GLDJ HVW OH YHFWHXU GHV '63 GHV YLWHVVHV YLEUDWRLUHV RX SUHVVLRQ DFRXVWLTXH SURMHWp GDQVODEDVHGHVPRELOLWpVSULQFLSDOHV

GLDJ LQ 2, GLDJ RXW

2,

< + <

ȟ =

R +

2,

HVWODPDWULFHGHVPRGXOHVFDUUpVWHUPHjWHUPHGH ODPDWULFHGHWUDQVIHUWDOpDWRLUHSURMHWpHGDQVOHVQRXYHOOHV

EDVHV

2,LM

2, LM

7

+ = HW 7

2,

= ;

RXW

7 ;

+LQ

/HV H[SUHVVLRQV JDUDQWLVVHQW O¶pTXLYDOHQFHGHVH[SUHVVLRQVHWFHTXLVLJQLILHTXH OHV '63 GHV UpSRQVHV VRQW FDOFXOpHV H[DFWHPHQW HQ PR\HQQHVWRFKDVWLTXHVLRQXWLOLVHOHVH[SUHVVLRQV DXOLHXGH&HWWHDOWHUQDWLYHSHUPHWGHPHWWUH HQ pYLGHQFH GHV SURSULpWpV VWDWLVWLTXHV LQWpUHVVDQWHV TXH O¶RQYDGpWDLOOHUFLGHVVRXV

8QHDSSURFKHVLPLODLUHDpWpSURSRVpHVDQVMXVWLILFDWLRQ GDQV>@SRXUGHVDSSOLFDWLRQVHQKDXWHVIUpTXHQFHV

0RGqOHpQHUJpWLTXHVLPSOLILp

/¶REVHUYDWLRQ GHV )5)3 PR\HQQHV IDLW DSSDUDLWUH XQH JUDQGH UpJXODULWp j OD IRLV IUpTXHQWLHOOH HW VSDWLDOH j O¶LQWpULHXU GH ]RQHV ELHQ GpILQLHV TXH OHV )5) QH SUpVHQWDLHQW SDV /HV ILJXUHV VXLYDQWHV SHUPHWWHQW GH FRPSDUHU GHV )5) YLEUDWRLUHV PR\HQQHV ILJXUH DX[

)5)3 FRUUHVSRQGDQWHV ILJXUH SRXU XQ PrPH SRLQW

G¶H[FLWDWLRQ HW XQ HQVHPEOH GH SRLQWV G¶REVHUYDWLRQ VLWXpV GDQV XQH PrPH ]RQH 2Q FRQVWDWH TX¶DXGHOj G¶XQH IUpTXHQFH G¶HQYLURQ +] OHV )5)3 WHQGHQW j VH UDSSURFKHUOHVXQHVGHVDXWUHVGHWHOOHVRUWHTX¶RQSRXUUDLW WRXWHVOHVDSSUR[LPHUSDUXQHXQLTXHYDOHXU6DFKDQWTX¶XQ FRPSRUWHPHQW VLPLODLUH H[LVWH SRXU GHV ''/ G¶H[FLWDWLRQ VLWXpVGDQV XQH PrPH]RQHLOGHYLHQWSRVVLEOHG¶H[SULPHU OH WUDQVIHUW GH SXLVVDQFH HQWUH GHX[ ]RQHV SDU XQ VFDODLUH SOXW{W TX¶XQH PDWULFH &HWWH VLPSOLILFDWLRQ ORUVTX¶HOOH HVW SRVVLEOH WUDGXLW O¶H[LVWHQFH G¶XQH SDUWLWLRQ GH OD VWUXFWXUH HQ ]RQHV GRQW OH FRPSRUWHPHQW YLEURDFRXVWLTXH H[SULPp SDUODSXLVVDQFHGHUpSRQVHHVWXQLIRUPH

,O HVW LQWpUHVVDQW GH QRWHU TXH OH PRGqOH VLPSOLILp SURSRVpQ¶DSSRUWHUDDXFXQHVLPSOLILFDWLRQSRXUOD]RQHTXL UHoRLW OHV H[FLWDWLRQV /HV TXHVWLRQV OLpHV DX[ FKDPSV SURFKHVQHVRQWSDVWUDLWpHVSDUODWKpRULHSURSRVpH

)LJXUH)5)VYLEUDWRLUHVPR\HQQHVHQWUHXQSRLQW G¶H[FLWDWLRQGXPRWHXUHWSRLQWVGXSDYLOORQGDQVOD

GLUHFWLRQQRUPDOH

)LJXUH)5)3VPR\HQQHVHQWUHXQSRLQWG¶H[FLWDWLRQ GXPRWHXUHWSRLQWVGXSDYLOORQGDQVODGLUHFWLRQQRUPDOH

2QLQWURGXLWGRQFOHVFDODLUH H

2,

WHOTXH

LQ 2,

RXW

= H Π

Π

R Π

LQ

HVW OD SXLVVDQFH WRWDOH LQMHFWpH GDQV ]RQH G¶H[FLWDWLRQ ¦

=

= Π

1,

O LQN

LQ

ʌ

HW Π

RXW

HVW OD SXLVVDQFH GH UpSRQVHPR\HQQHGDQVOD]RQHG¶REVHUYDWLRQ

/DYDOHXUTXLFRQGXLWDXPRGqOHVLPSOLILpOHSOXVSHUWLQHQW FHOXL TXL SURFXUH OD PHLOOHXUH HVWLPDWLRQ GHV UpSRQVHV TXDGUDWLTXHVPR\HQQHVSDU]RQHHVWWHOOHTXH

¦

¦

¦¦

¦

¦

¦¦

=

=

= =

=

=

= =

=

=

, 2

, 2

2 ,

, 2

1

N GLDJ

N RXW 1

O GLDJ

O LQ 1

O 1

N 2,NO 1

N GLDJ

N RXW 1

O GLDJ

O LQ 1

O 1

N

GLDJ N RXW ON 2, GLDJ

O LQ 2,

<

<

7

<

<

<

<

H

ξ

(6)

&HWWH YDOHXU pWDQW FRQQXH RQ SHXW pWXGLHU OHV HUUHXUV FRPPLVHV ORUV GH OD VLPSOLILFDWLRQ 2Q WUDFH SRXU FHOD OD PDWULFHGHVpFDUWVHQWUHOHPRGqOHpQHUJpWLTXHHWOHPRGqOH pQHUJpWLTXHVLPSOLILp

GLDJ O LQ 2, GLDJ

N RXW GLDJ

O LQ O N GLDJ

N RXW NO

2,

= <

<

<

H <

Δ ξ

8QLQGLFDWHXUJOREDOGHODSHUWLQHQFHGXPRGqOHSURSRVpHVW ODPR\HQQHTXDGUDWLTXHGHVpFDUWV

¦¦

= =

Δ

=

1, 2

O 1

N

NO 2, ,

2

1 FPTH 1

)LJXUH7UDFHUGHODPDWULFHG¶HUUHXUGXPRGqOHVLPSOLILp HQWUHOHEORFDYDQW''/HWO¶KDELWDFOH''/j

+]D+]%+]FHW+]G

)LJXUH9DOHXUTXDGUDWLTXHPR\HQQHGHO¶pFDUWLQWURGXLW SDUOHPRGqOHVLPSOLILpHQWUHOHEORFDYDQWHWO¶KDELWDFOH

/D ILJXUH PRQWUH j WLWUH G¶H[HPSOH OD PDWULFH ǻ

2,

j SOXVLHXUVIUpTXHQFHVSRXUGHVREVHUYDWLRQVGDQVODFDYLWpHW GHV H[FLWDWLRQV j O¶DYDQW GX YpKLFXOH /D ILJXUH PRQWUH O¶LQGLFHFPTHH[SULPpHQG%HWIRQFWLRQGHODIUpTXHQFH,O FRQILUPHOHIDLWTX¶DXGHVVXVGH+]ODFDYLWpHWOHEORF DYDQW SHXYHQW rWUH FRQVLGpUpV FRPPH VpSDUp /RUVTXH OD SDUWLWLRQHVWOLFLWHLOHVWLQWpUHVVDQWGHQRWHUTXHODUHODWLRQ LPSOLTXHTXHODUpSRQVHGDQVOD]RQHG¶REVHUYDWLRQOD FDYLWp HVW SURSRUWLRQQHOOH j OD SXLVVDQFH WRWDOH LQMHFWpH GDQV OD ]RQH VRXUFH OH EORF DYDQW TXHOOHV TXH VRLHQW OHV H[FLWDWLRQVFRQVLGpUpHV

&RQFOXVLRQ

$SUqV DYRLU UDSSHOp OH SULQFLSH GH OD PRGpOLVDWLRQ SUREDELOLVWH QRQSDUDPpWULTXH GHV LQFHUWLWXGHV SRXU GHV SUREOqPHV GH YLEURDFRXVWLTXH VRQ pWHQGXH HW VHV

OLPLWDWLRQVXQPRGqOHSUREDELOLVWHDpWpXWLOLVpSRXUpWXGLHU OHFRPSRUWHPHQWPRGDOG¶XQHFDUURVVHULHG¶DXWRPRELOHTXL FRQVWLWXHXQERQH[HPSOHGHV\VWqPHLQFHUWDLQ,ODSSDUDLW WRXWG¶DERUGTXHODIRUPHGHVPRGHVHVWVWDEOHYLVjYLVGHV LQFHUWLWXGHV VHXOHPHQW GDQV OHV EDVVHV IUpTXHQFHV &¶HVW GRQFVHXOHPHQWGDQVFHWWHEDQGHGHIUpTXHQFHTXHO¶RQSHXW WUDFHUODGLVWULEXWLRQGHVIUpTXHQFHVSURSUHVGHODVWUXFWXUH GLVSHUVpH 2Q REVHUYH DORUV TX¶XQH DXJPHQWDWLRQ GH OD GLVSHUVLRQ GH OD UDLGHXU IDLW HQ PR\HQQH GLPLQXHU OHV IUpTXHQFHVSURSUHVOHVSOXVEDVVHV

2Q LQWURGXLW HQVXLWH XQH DSSURFKH pQHUJpWLTXH TXL SHUPHW GDQV OH FDV G¶XQH VWUXFWXUH WUqV FRPSOH[H HW KpWpURJqQH GH IDLUH DSSDUDLWUH GHV FRPSRUWHPHQWV KRPRJqQHVSDUEDQGHVGHIUpTXHQFHHWSDU]RQH/HPRGqOH SURSRVp LPSRVH GH SURMHWHU OHV H[FLWDWLRQV HW OHV UpSRQVHV GDQV GHV EDVHV GH PRELOLWp SULQFLSDOH TXL GLDJRQDOLVHQW OD SDUWLH UpHOOH GH OD PDWULFH GH PRELOLWp DVVRFLpH j OD ]RQH FRQVLGpUpH$XGHOjG¶XQHFHUWDLQHIUpTXHQFHOHVWUDQVIHUWV GH SXLVVDQFH HQWUH ]RQHV SHXYHQW rWUH PRGpOLVpV JOREDOHPHQW SDU XQ VFDODLUH 8Q FULWqUH SHUPHWWDQW GH PHVXUHU OD SHUWLQHQFH GH FHWWH PRGpOLVDWLRQ VLPSOLILpH HVW SURSRVp 2Q FRQILUPH DLQVL TXH GHV ]RQHV VWUXFWXUDOHV RX DFRXVWLTXHV SUpVHQWHQW HQ PR\HQQH GHV FRPSRUWHPHQWV KRPRJqQHV TXL SHUPHWWHQW G¶HQYLVDJHU OD PLVH HQ SODFH G¶XQVFKpPDGHSDUWLWLRQGHODVWUXFWXUHpWXGLpH&HVFKpPD pYROXH HQ IRQFWLRQ GH OD IUpTXHQFH /HV SUHPLqUHV VpSDUDWLRQV VH SURGXLVHQW j UHODWLYHPHQW EDVVH IUpTXHQFH SRXUXQYpKLFXOHDXWRPRELOH+]

8Q WHO VFKpPD FRQWULEXHUDLW IRUWHPHQW j DPpOLRUHU OD PDLWULVH GH OD FRQFHSWLRQ YLEURDFRXVWLTXH GDQV OHV PR\HQQHVIUpTXHQFHV

5pIpUHQFHV

>@ 6RL]H&´ 5DQGRP PDWUL[ WKHRU\ IRU PRGHOLQJ UDQGRP XQFHUWDLQWLHV LQ FRPSXWDWLRQDO PHFKDQLFV´

&RPS 0HWK $SSO 0HFK (QJ

>@ 6RL]H & ´$ FRPSUHKHQVLYH RYHUYLHZ RI D QRQ SDUDPHWULF SUREDEOLVWLF DSSURDFK RI PRGHO XQFHUWDLQWLHV IRU SUHGLFWLYH PRGHOV LQ VWUXFWXUDO G\QDPLFV´-6RXQG9LE

>@ 'XUDQG-)6RL]H&*DJOLDUGLQL/³6WUXFWXUDO DFRXVWLFPRGHOLQJRIDXWRPRWLYHYHKLFOHVLQ SUHVHQFHRIXQFHUWDLQWLHVDQGH[SHULPHQWDO LGHQWLILFDWLRQDQGYDOLGDWLRQ´-RXUQDORIWKH

$FRXVWLFDO6RFLHW\RI$PHULFD±

>@ 2KD\RQ5DQG6RL]H&´6WUXFWXUDO$FRXVWLFVDQG 9LEUDWLRQ´$FDGHPLF3UHVV6DQ'LHJR

>@ 6HUIOLQJ 5 - ³$SSUR[LPDWLRQ 7KHRUHPV RI 0DWKHPDWLFDO6WDWLVWLFV´-RKQ:LOH\ 6RQV

>@ 0.DVVHP&6RL]H/*DJOLDUGLQL(QHUJ\GHQVLW\

ILHOG DSSURDFK IRU ORZ DQG PHGLXPIUHTXHQF\

YLEURDFRXVWLFDQDO\VLVRIFRPSOH[VWUXFWXUHVXVLQJD VWDWLVWLFDO FRPSXWDWLRQDO PRGHO -RXUQDO RI 6RXQG DQG9LEUDWLRQ

>@ /*DJOLDUGLQL*%RUHOOR9LUWXDO6($7RZDUGVDQ LQGXVWULDOSURFHVV6$(,QWHUQDWLRQDO

D E

F G

Références

Documents relatifs

The bounded solutions o/the equation (1.3) are periodic/unctions whose group o/periods contains the support o/ItA. The/ollowing properties o//a are

&amp;HSHQGDQWLOIDXWHQFRUHDJLUGHIDoRQEHDXFRXSSOXVHIILFDFHIDFHDX[EHVRLQVVDQLWDLUHVGHV

2Q QRWH F OD FpOpULWp GH O¶RQGH VRQRUH ORUVTXH OH IOXLGH HVW DX UHSRV GDQV OH UpIpUHQWLHO G¶pWXGH ( 'RQQHU O¶H[SUHVVLRQ GH OD YLWHVVH Y $% GH O¶RQGH VRQRUH GDQV OH IOXLGH HQ

Por lo tanto, esta versión derivada del CIF para niños y jóvenes amplía la cobertura del volumen principal de la CIF ofreciendo contenido específico y detalles adicionales para

‡ ,O IDXW pYDOXHU O¶LQWpUrW UHODWLI SRXU OD VDQWp GDQV OH PRQGH GHV SURJUqV GH OD JpQRPLTXH DX QLYHDX GH OD SUDWLTXH HW GH OD SUHVWDWLRQ GHV VRLQV SDU UDSSRUW DX FR€W HW j

Construire un cercle connaissant son centre et un point, ou son centre et son rayon.Vocabulaire approprié pour décrire les figures planes usuelles : carré, rectangle,

/D UHFKHUFKH SRXU DPpOLRUHU OD ILDELOLWp GHV V\VWqPHV LQWHUDFWLIV V¶HVW SRXU O¶LQVWDQW SULQFLSDOHPHQW GLULJpH YHUV OD SUpYHQWLRQ G¶LQWURGXFWLRQ GH IDXWHV SDU

Nous montrons dans ce paragraphe comment on peut adapter les coordonn6es actions-angles ~ une situation off la famille de vari&amp;6s lagrangiennes invariantes