• Aucun résultat trouvé

A partitioning methodology of complex uncertain structural-acoustic systems in the low and medium frequency ranges

N/A
N/A
Protected

Academic year: 2021

Partager "A partitioning methodology of complex uncertain structural-acoustic systems in the low and medium frequency ranges"

Copied!
12
0
0

Texte intégral

(1)

HAL Id: hal-00692961

https://hal-upec-upem.archives-ouvertes.fr/hal-00692961

Submitted on 1 May 2012

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.

A partitioning methodology of complex uncertain structural-acoustic systems in the low and medium

frequency ranges

L. Gagliardini, M. Kassem, Christian Soize

To cite this version:

L. Gagliardini, M. Kassem, Christian Soize. A partitioning methodology of complex uncertain

structural-acoustic systems in the low and medium frequency ranges. International Conference on

Noise and Vibration Engineering, ISMA 2010, Katholieke Universiteit Leuven, Sep 2010, Leuven,

Belgium. pp.1-11. �hal-00692961�

(2)

A partitioning methodology of complex uncertain structural-acoustic systems in the low and medium frequency ranges

*DJOLDUGLQL/

(1)

.DVVHP0

(1,2)

, 6RL]H&

(2)

36$3HXJHRW&LWURsQ,QGXVWULDODQG7HFKQLFDO'LYLVLRQ 5RXWHGH*LV\9(/,=<9,//$&28%/$<

8QLYHUVLWp3DULV(VW/DERUDWRLUHGH0RGpOLVDWLRQHW6LPXODWLRQ0XOWL(FKHOOH 060(805&156%G'HVFDUWHV0DUQHOD9DOOHH)UDQFH

Abstract

,QWKLVSDSHUDVWUXFWXUDO]RQHSDUWLWLRQLQJPHWKRGRORJ\RIFRPSOH[XQFHUWDLQVWUXFWXUDODFRXVWLFV\VWHPV LVSUHVHQWHG7KLVPHWKRGRORJ\LVEDVHGRQDQHQHUJ\ILHOGIRUPXODWLRQRIWKHYLEURDFRXVWLFSUREOHP,Q WKLV HQHUJ\ILHOG IRUPXODWLRQ WKH YLEURDFRXVWLF V\VWHP LV PRGHOOHG XVLQJ VWDWLVWLFDO DYHUDJLQJ RYHU LQGHSHQGHQW UHDOL]DWLRQV RI D SUREDELOLVWLF YLEURDFRXVWLF PRGHO ,Q WKH YLEURDFRXVWLF HQHUJ\ PRGHO H[FLWDWLRQV DUH UHSUHVHQWHG LQ WHUPV RI LQSXW SRZHU REVHUYDWLRQV DUH UHSUHVHQWHG LQ WHUPV RI UHVSRQVH SRZHU DQG WKH SURSDJDWLRQ EHWZHHQ H[FLWDWLRQ DQG REVHUYDWLRQ SRLQWV LV UHSUHVHQWHG E\ D VHW RI 3RZHU )UHTXHQF\5HVSRQVH)XQFWLRQV3)5):KHQVWDWLVWLFDOO\DYHUDJHGWKHVHSDUDPHWHUVDUHOLQHDUO\UHODWHG WR IRUP WKH HTXDWLRQ RI WKH YLEURDFRXVWLF HQHUJ\ PRGHO ,W LV VKRZQ WKDW DERYH D JLYHQ IUHTXHQF\ WKH SRZHUIUHTXHQF\UHVSRQVHIXQFWLRQVDUHPXFKOHVVGHSHQGDQWRQWKHGLUHFWLRQRURQWKHSUHFLVHORFDWLRQRI WKH H[FLWDWLRQ RU WKH REVHUYDWLRQ WKDQ WKH FODVVLFDO IUHTXHQF\ UHVSRQVH IXQFWLRQV &RQVHTXHQWO\ D VLPSOLILHG PRGHO FDQ EH FRQVWUXFWHG EDVHG RQ D VFDODU IUHTXHQF\ UHVSRQVH IXQFWLRQ $ PHWKRGRORJ\

ZKLFKFRQVLVWVLQILQGLQJDVWUXFWXUDOSDUWLWLRQLQJRIWKHFRPSOH[V\VWHPIRUZKLFKWKLVVLPSOLILHGPRGHOLV YDOLGLVGHYHORSHG3DUWLWLRQLQJFULWHULDDUHWKXVLQWURGXFHGLQRUGHUWRHYDOXDWHVLPXOWDQHRXVO\WKHHUURU DVVRFLDWHGWRWKHPRGHOVLPSOLILFDWLRQDQGWKHSHUWLQHQFHRIDJLYHQ]RQHSDUWLWLRQLQJ$QDSSOLFDWLRQRI WKLVSDUWLWLRQLQJPHWKRGRORJ\WRDQDXWRPRWLYHYHKLFOHVWUXFWXUHLVSUHVHQWHG

1 Introduction

8QFHUWDLQW\ LVVXHV DUH RI PDMRU LQWHUHVW LQ LQGXVWULDO YLEURDFRXVWLF VWXGLHV GXH WR WKH KLJK G\QDPLF VHQVLWLYLW\ WRVPDOOV\VWHPV FKDQJHV $VDQ H[DPSOH ILJ VKRZV PHDVXUHPHQWV RIVWUXFWXUDO )UHTXHQF\

5HVSRQVH )XQFWLRQV )5) SHUIRUPHG RQ SURGXFWLRQ YHKLFOHV FRQVLGHUHG LGHQWLFDO UHJDUGLQJ YLEURDFRXVWLFGHVLJQ

&KDQJHV WKDW FDXVH WKH REVHUYHG VSUHDG PD\ UHVXOW IURP SURFHVV XQFHUWDLQWLHV PDLQO\ JHRPHWULFDO XQFHUWDLQWLHV DW YDULRXV VFDOHV DQG WKHLU FRQVHTXHQFHV LQ WHUPV RI VWLIIQHVV LQWHUQDO VWUHVV RU PDWHULDO EHKDYLRXU«RUIURPSURGXFWGLYHUVLW\SURGXFWFRQWHQWV:KHQFRQVLGHULQJWKHRYHUDOOGHVLJQSURFHVVLW VKRXOG EH PHQWLRQHG WKDW WKH FKDQJHV LQ WKH GHVLJQ LWVHOI WKDW RFFXUV FRQWLQXRXVO\ GXULQJ WKH GHVLJQ SKDVH PD\DOVRJHQHUDWHVDYDULDELOLW\RIDWOHDVWWKHVDPHRUGHURIPDJQLWXGH$OWKRXJKWKHQDWXUHRI WKRVHFKDQJHVLVYHU\GLIIHUHQW±WKHILUVWUHODWHGWRPHFKDQLFDOIDFWVWKHVHFRQGWRPRUHFRQFHSWXDOIDFWV WKH\FDXVHDVSUHDGRIDFWXDOUHVSRQVHVZKHQFRPSDUHGWRDXQLTXHGHVLJQPRGHO%\WKHZD\WKHXVHRID PRGHOLVE\LWVHOIDIDFWRURIXQFHUWDLQW\

5HFHQW ZRUN >@ VKRZHG WKDW LW LV SRVVLEOH WR JOREDOO\ PRGHO HYHU\ NLQG RI XQFHUWDLQWLHV E\ XVLQJ D

SUREDELOLVWLFPRGHOOLQJLQYROYLQJUDQGRPPDWULFHVDQGEDVHGXSRQDPD[LPXPRIHQWURS\SULQFLSOH6XFK

DPRGHOOLQJHQVXUHVDQ\SK\VLFDOFKDQJHZLWKLQDJLYHQUDQJHWREHDFFRXQWHGIRUZKDWHYHULWVQDWXUH

7KHUDQJHRIYDULDWLRQLVFRQWUROOHGE\WKHVRFDOOHGGLVSHUVLRQSDUDPHWHUV7KHVHSDUDPHWHUVDUHUHODWHGWR

(3)

WKHYDULDQFHVRIWKHPHFKDQLFDORSHUDWRUVRIWKHVWUXFWXUDODFRXVWLFSUREOHPPDVVVWLIIQHVVDQGGDPSLQJ 7KLV WKHRU\ ZDV VXFFHVVIXOO\ DSSOLHG WR ODUJH VL]H VWUXFWXUDODFRXVWLF SUREOHPV VXFK DV WKH RQHV HQFRXQWHUHGLQWKHDXWRPRWLYHLQGXVWU\>@SURYLGLQJQHZWRROVWRLQYHVWLJDWHXQFHUWDLQG\QDPLFV\VWHPV DQGPRUHHVSHFLDOO\WKHRQHVZLWKDKLJKG\QDPLFVHQVLWLYLW\

)LJXUH9LEURDFRXVWLFIUHTXHQF\UHVSRQVHIXQFWLRQVPHDVXUHGEHWZHHQWKHGULYHU¶VHDUSRVLWLRQDQGD SRLQWRQWKHERG\IUDPHZRUNIRUSURGXFWLRQYHKLFOHJUD\6WDWLVWLFDODYHUDJHEOXH

WK

FHQWLOHUHG

DQG

WK

FHQWLOHJUHHQ

$IWHUDVKRUWRYHUYLHZRIWKHSUREDELOLVWLFPRGHOOLQJSURFHVVDQHQHUJ\ILHOGDSSURDFKLVSURSRVHGZKLFK H[KLELW LQWHUHVWLQJ DYHUDJHG SURSHUWLHV IRU XQFHUWDLQ V\VWHPV $SSOLFDWLRQV DUH SHUIRUPHG RQ FXUUHQW LQGXVWULDOPRGHOVRIYHKLFOHVLQYROYLQJDERXW

'HJUHHV2I)UHHGRP'2)

2 Stochastic modelling for vibro-acoustic problems

3UREDELOLVWLF PRGHOV GHVFULEHG LQ WKLV VHFWLRQ DUH SUHFLVHO\ GHVFULEHG LQ >@ 7KH PHWKRG XVHG LV D QRQ SDUDPHWULF SUREDELOLVWLF PHWKRG VLQFH LWV SUREDELOLVWLF DVSHFWV DUH QRW UHODWHG WR DQ\ RI WKH V\VWHP SDUDPHWHUV

)RUDXWRPRELOHSUREOHPVDOWKRXJKVRXUFHWHUPVPD\DOVREHXQFHUWDLQWKH\DUHJHQHUDOO\OHVVGLVSHUVHG WKDQWKHFDUERG\)5)V,QWKHIROORZLQJVRXUFHWHUPVZLOOEHNHSWFRQVWDQW

2.1 Stochastic structural-acoustic problem setting and solving :H QDPH KHUH D ³PHDQ PRGHO´ WKH GHWHUPLQLVWLF PRGHO XSRQ ZKLFK WKH VWRFKDVWLF PRGHO LV EDVHG 7KH PHDQ PRGHO LV VXFK DV LWV PHFKDQLFDO RSHUDWRUV DUH HTXDO WR WKH PHDQ PHFKDQLFDO RSHUDWRUV RI WKH VWRFKDVWLFPRGHO

&ODVVLFDOO\ >@ VWUXFWXUDODFRXVWLF SUREOHPV DUH VROYHG XVLQJ )LQLWH (OHPHQWV 0HWKRGV DQG D IUHTXHQF\

GRPDLQVROXWLRQ0HFKDQLFDOHTXDWLRQVLQFRQWLQXRXVPHGLDWKHQOHDGWRDPDWUL[IRUPVXFKDV

» »

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

ZKHUH X

V

UHVS S

D

LV WKH YHFWRU RI FRPSOH[ DPSOLWXGHV RI WKH VWUXFWXUDO UHVS DFRXVWLF UHVSRQVH DW WKH QRGHV ORFDWLRQV SK\VLFDO FRRUGLQDWHV WR WKH H[FLWDWLRQV UHSUHVHQWHG E\ YHFWRUV

ω

I

V

DQG 4

D

ω

VQ

ω

$

V

LV WKH LQ YDFXR G\QDPLF VWLIIQHVV PDWUL[ VXFK WKDW

(4)

V Q V Q V

Q V

QV

0

V

'

V

.

V

$ ω = − ω

+ M ω + ZKHUH

VQ VQ VQ

V V

V

' .

0 DUH UHVSHFWLYHO\ WKH VWUXFWXUDO PDVV GDPSLQJ DQGVWLIIQHVVPDWULFHV

DQ

ω

$

D

LVWKHFDYLW\±ZLWKEORFNHGERXQGDU\DGPLWWDQFHPDWUL[VXFKWKDW

D D D

D

D D D

D Q

M

Q Q

Q

0 ' .

$ ω = − ω

+ ω + ZKHUH

D D D

D D

D Q Q

Q

' .

0 DUH ±DEXVLYHO\ QDPHG DFRXVWLF PDVV GDPSLQJ DQG VWLIIQHVV PDWULFHV DQG

D VQ

&

Q

LV WKH VWUXFWXUDODFRXVWLF FRXSOLQJ PDWUL[ 7KH XQGHUOLQLQJKHUHLQGLFDWHVPHDQYDOXHV

)RU ODUJH VL]H SUREOHPV ±PDQ\ PLOOLRQV '2)6 D PRGDO UHGXFWLRQ LV XVXDOO\ SHUIRUPHG 9LEUDWLRQ DQG DFRXVWLFUHVSRQVHILHOGVDUHUHVSHFWLYHO\H[SUHVVHGXVLQJWKHLQYDFXRVWUXFWXUDOPRGHVDQGWKHEORFNHG ERXQGDU\DFRXVWLFPRGHV$VVRFLDWHGVRFDOOHGJHQHUDOL]HGFRRUGLQDWHV T

V

ω T

ΦD

ω

Ψ

DUHVXFKWKDW

ω ω

ω ω

D D

V V

T ĭ S

T Ȍ X

Φ Ψ

=

=

*HQHUDOL]HG FRRUGLQDWHV YHULI\ WKH VWUXFWXUDODFRXVWLF V\VWHP RI HTXDWLRQ VLPLODU WR EXW DIWHU SURMHFWLRQ RQWKH PRGDO EDVLV RI WKH VWUXFWXUHIRU WKHILUVWOLQH DQG SURMHFWLRQ RQ WKH PRGDO EDVLV RI WKH FDYLW\IRUWKHVHFRQGOLQH

» »

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

$

&

&

$

:KHUHWKHUHGXFHGG\QDPLFVWLIIQHVVPDWUL[RIWKHVWUXFWXUH $

VΨ

ω DQGWKHUHGXFHGDGPLWWDQFH PDWUL[RIWKHFDYLW\ $

ΦD

ω NHHSVLPLODUH[SUHVVLRQVDVDERYH

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

0RGHV SURSHUWLHV HQVXUH UHGXFHG PDVV DQG VWLIIQHVVPDWULFHV DUHSRVLWLYHV ±RUHYHQWXDOO\ VHPLSRVLWLYH GLDJRQDOPDWULFHV5HGXFHGGDPSLQJPDWULFHVDUHPRVWRIWKHWLPHIXOOPDWULFHV

7KH SURSRVHG VWRFKDVWLF PRGHOOLQJ LV EDVHG RQ D PDWKHPDWLFDO WKHRU\ RI UDQGRP PDWULFHV VHH >@ IRU IXUWKHU GHWDLO 5DQGRP UHDOL]DWLRQV RI WKH XQFHUWDLQ V\VWHP FDQ EH FRQVWUXFWHG E\ PHDQ RI UDQGRP PDWULFHV FRQVWUXFWLRQ 7KXV PHDQ PDWULFHV 0

VΨ

'

VΨ

.

VΨ

0

DΦ

'

DΦ

.

DΦ

&

Ȍĭ

DUH UHSODFHG E\ UDQGRP PDWULFHV 0

VȌ

'

VȌ

.

VȌ

0

Dĭ

'

Dĭ

.

Dĭ

&

Ȍĭ

ZKLFKDUHFRQVWUXFWHGXVLQJDQDOJHEUDLFPHWKRGIURPWKHPHDQ PDWULFHVIRUJLYHQGLVSHUVLRQSDUDPHWHUV

(TXDWLRQWKHQEHFRPHVDVWRFKDVWLFHTXDWLRQ

» »

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

$

&

&

$

0DWULFHV LQYROYHG LQ HTXDWLRQ DUH EXLOW XVLQJ HOHPHQWDU\ UDQGRP PDVV VWLIIQHVV GDPSLQJ DQG FRXSOLQJPDWULFHVLQWKHVDPHPDQQHUDV,WLVWREHQRWHGWKDWZKHQUDQGRPL]HGQRQHRIWKHPDWULFHV LV GLDJRQDO DQ\PRUH ZKHQ H[SUHVVHG LQ WKH IXQFWLRQDO EDVLV RI WKH HLJHQPRGHV RI WKH PHDQ PRGHO 1HYHUWKHOHVV HOHPHQWDU\ PDWULFHV UHPDLQ GHILQLWH SRVLWLYH RU GHILQLWH VHPLSRVLWLYH GHSHQGLQJ RQ WKH FDVH

'LVSHUVLRQSDUDPHWHUVKDYHEHHQLGHQWLILHGVXFKDVWKHVWRFKDVWLFPRGHOWRWDOO\LQFOXGHVDFWXDOPHDVXUHG YDOXHV7KHLGHQWLILFDWLRQPD\EHSHUIRUPHGWKURXJKDFRQYHQWLRQDOPLQLPL]DWLRQPHWKRGRUPRUHOLNHO\

LQYRNLQJWKHPD[LPXPRIOLNHOLKRRGSULQFLSOH,QGHHGWKHVWRFKDVWLFPRGHOSURYLGHVSUREDELOLW\GHQVLW\

IXQFWLRQVDOORZLQJWKHFRPSXWLQJRIWKHOLNHOLKRRGRIDQ\PHDVXUHG)5)

6ROXWLRQVRIWKHVWRFKDVWLFSUREOHPDUHREWDLQHGE\WKHPHDQVRID0RQWH&DUORVLPXODWLRQWKDWLVWRVD\

E\VROYLQJDODUJHQXPEHURILQGHSHQGHQWUHDOL]DWLRQVRIV\VWHP7KHQXPEHURIUHDOL]DWLRQVUHTXLUHG

GHSHQGV RQ WKH H[SHFWHG OHYHO RIVWDWLVWLFDO DFKLHYHPHQW7KH )5)¶V PHDQ YDOXHDQG XSSHUERXQG ZLWK

RUSUREDELOLW\FRQYHUJHZLWKLQDIHZKXQGUHGVRIUHDOL]DWLRQV/RZHUERXQGUHTXLUHVPRUHHIIRUW

(5)

,WLVJHQHUDOO\FRQYHQLHQWWRSHUIRUPDTXDQWLOHDQDO\VLV>@VLQFHLWLVWKHPRUHJHQHUDOZD\WRHVWLPDWH SUREDELOLWLHV ZKHQ WKH GHQVLW\ IXQFWLRQ LV XQNQRZQ 8VLQJ D KLJKHU QXPEHU RI UHDOL]DWLRQV PDQ\

WKRXVDQGVRQHPD\HVWLPDWHWKHSUREDELOLW\GHQVLW\IXQFWLRQUHTXLUHGIRUOLNHOLKRRGHVWLPDWLRQV

2.2 Uncertain system modal analysis

,QWKHSUHYLRXVVHFWLRQZHH[SODLQHGKRZWREXLOGDVWRFKDVWLFPRGHOIURPD)(PRGHO:HZLOOGHVFULEH KHUHWKHSK\VLFDOFRQWHQWRIWKLVPRGHO$VIDUDVDQ\UHVSRQVHRIDOLQHDUG\QDPLFDOV\VWHPPD\EHEXLOW IURPLWVHLJHQPRGHVWKHDQDO\VLVRIWKHPRGHVRIWKHVWRFKDVWLFV\VWHPSURYLGHVDFRPSOHWHYLHZRQLWV EHKDYLRXU

)RUHDFKUHDOL]DWLRQRIWKHVWRFKDVWLFPRGHOIHDWXULQJIXOOPDWULFHVDVH[SODLQHGSUHYLRXVO\RQHFDQVHW XSWKHIROORZLQJVWRFKDVWLFHLJHQYDOXHSUREOHP

[ Λ

Ψ

+

Ψ

] Ψ =

V

V

.

0

6ROXWLRQV RI IRU LQGLYLGXDO UHDOL]DWLRQV RI VWLIIQHVV DQG PDVV PDWULFHV SURYLGH WKH VWRFKDVWLF PRGDO EDVLV { } Ȍ

θM

ZKHUH θ

M

LQGLFDWHVWKHUHDOL]DWLRQLQGH[

$VIRUH[SHULPHQWDOPRGDODQDO\VLVDPRGDOVKDSHVLPLODULW\FULWHULRQLVSURSRVHG7KH6WRFKDVWLF0RGDO

$VVXUDQFH&ULWHULRQ60$&LVLQWURGXFHGEHWZHHQWKHPHDQPRGHLDQGVWRFKDVWLFPRGHM

M

( 60$&

LM

θ

° ¿

° ¾

½

° ¯

° ®

­

=

M M M

ș M ș M L L

ș M L

Ȍ Ȍ Ȍ Ȍ

Ȍ

Ȍ

ZKHUH ( { } LVWKHPDWKHPDWLFDOH[SHFWDWLRQ

)LJXUHVKRZVWKH60$&PDWUL[EHWZHHQVWRFKDVWLFPRGHVDQGPHDQPRGHVRIDFDUERG\IRUWKHORZHU RUGHU PRGHV 7KH PDWUL[LV PRVWO\ GLDJRQDO VKRZLQJ PLQRU HIIHFWV RIWKH XQFHUWDLQW\ PRGHOOLQJ RQ WKH PRGDO VKDSHV 2QH FDQ WKHQ GUDZ WKH IUHTXHQF\ GLVWULEXWLRQ RI VHOHFWHG PRGHV LGHQWLILHG E\ WKHLU VLPLODULW\ WR D PHDQ PRGH )LJXUH DQG VKRZ WKH GLVWULEXWLRQ RI XQFHUWDLQ IUHTXHQFLHV RI D FDU ERG\

JOREDOPRGHUHODWHGWRYDULRXVXQFHUWDLQLWLHV7KHGLVWULEXWLRQDVVRFLDWHGWRDGLVSHUVHGPDVVPDWUL[ZLWK LQFUHDVLQJ GLVSHUVLRQ SDUDPHWHU YDOXH LQFUHDVLQJ GLVSHUVLRQ DPRXQW LV SORWWHG RQ ILJXUH )LJXUH VKRZVDVLPLODUSORWIRUGLVSHUVHGVWLIIQHVVPDWULFHVZLWKLQFUHDVLQJGLVSHUVLRQDPRXQW$VH[SHFWHGWKH GLVWULEXWLRQJRHVIODWZKHQWKHGLVSHUVLRQDPRXQWLQFUHDVHV,WLVUHPDUNDEOHWKDWVWLIIQHVVXQFHUWDLQW\OHDG WRDORZHUVKLIWRIWKHPRGDOIUHTXHQF\GLVWULEXWLRQ¶VFHQWUHZKHQLQFUHDVLQJWKHGLVSHUVLRQDPRXQW7KLV UHVXOWPD\EHUHODWHGWRDZHOONQRZQSUDFWLFDOIDFWVWLIIQHVVGHIHFWVORZHUHLJHQIUHTXHQFLHV

)LJXUH60$&PDWUL[EHWZHHQVWRFKDVWLFPRGHV[D[LVDQGPHDQPRGHV\D[LVUHVXOWLQJRIWKH

SURSRVHGQRQSDUDPHWULFPHWKRGIRUPRGHRUGHUVWRRIDQXQFHUWDLQFDUERG\

(6)

)LJXUH7RUVLRQFDUERG\PRGDOIUHTXHQF\GLVWULEXWLRQGXHWRPDVVXQFHUWDLQWLHV2UGHUHGUHGJUHHQ EOXHEODFNF\DQDQG\HOORZFRORXUVLQGLFDWHLQFUHDVLQJGLVSHUVLRQDPRXQW

)LJXUH7RUVLRQFDUERG\PRGDOIUHTXHQF\GLVWULEXWLRQGXHWRPDVVXQFHUWDLQWLHV2UGHUHGUHGJUHHQ EOXHEODFNF\DQDQG\HOORZFRORXUVLQGLFDWHLQFUHDVLQJGLVSHUVLRQDPRXQW

)LJXUH VKRZV WKH 60$& PDWUL[ IRU KLJKHU RUGHU PRGHV XQFHUWDLQ PRGDO VKDSHV DSSHDU WR EH D FRPELQDWLRQ RI DUDQJH RIPHDQ PRGH VKDSHV FUHDWLQJ EOXUVDURXQG WKH GLDJRQDO ,Q WKLVODWHU FDVHLW LV GLIILFXOW WR UHFRJQL]H PRGDO VKDSHV DQG FRQVHTXHQWO\ WR EXLOG VWDWLVWLFV RQ WKH PRGDO IUHTXHQFLHV 7KLV NLQG RI EHKDYLRXU FRXOG W\SLFDOO\ EH UHODWHG WR PLG IUHTXHQFLHV ZKHUH QRW RQO\ PRGDO IUHTXHQF\

GLVWXUEDQFHRFFXUVEXWDOVRPRGDOVKDSH(IIHFWRIWKLVSHFXOLDUEHKDYLRXUZLOOODWHUEHPHQWLRQHG

)LJXUH60$&PDWUL[EHWZHHQVWRFKDVWLFPRGHV[D[LVDQGPHDQPRGHV\D[LVUHVXOWLQJRIWKH SURSRVHGQRQSDUDPHWULFPHWKRGIRUPRGHRUGHUVWRRIDQXQFHUWDLQFDUERG\

(7)

3 Energy density field model

5HIHUHQFH>@GHWDLOVWKHFRQVWUXFWLRQRIWKHSURSRVHGHQHUJ\PRGHOIURPWKHUHDOLVDWLRQVRIDVWRFKDVWLF PRGHO ZLWKRXW DQ\ VLPSOLILFDWLRQ LQ DYHUDJH $ VLPLODU DSSURDFK ZDV SUHYLRXVO\ SURSRVHG LQ >@ LQ D GHWHUPLQLVWLF DQG KLJKHU IUHTXHQFLHV FRQWH[W EXW DOUHDG\ LQYROYLQJ VWDWLVWLFDO LVVXHV VLQFH LW ZDV LQ WKH )UDPHZRUNRIWKH6WDWLVWLFDO(QHUJ\$QDO\VLV

7KLVSURSRVHGPRGHOLVGHULYHGIURPWKHJHQHUDOH[SUHVVLRQRIWKHKDUPRQLFUHVSRQVHRIDYLEURDFRXVWLF V\VWHPWRUDQGRPH[FLWDWLRQV

= 76 7

+

6

55 ((

:KHUH 6

55

LV WKH FURVVVSHFWUDO YLEURDFRXVWLF UHVSRQVH PDWUL[ 6

((

LV WKH FURVVVSHFWUDO H[FLWDWLRQ PDWUL[ 7 LV WKH )5) PDWUL[ EHWZHHQ REVHUYDWLRQ DQG H[FLWDWLRQ SRLQWV REWDLQHG E\

VROYLQJ HTXDWLRQ IRU HYHU\ XQLWDU\ H[FLWDWLRQV DQG

LQGLFDWH D FRQMXJDWH DQG WUDQVSRVHG PDWUL[

(YHU\ TXDQWLW\ XVHG LQ WKLV VHFWLRQ DQG WKH IROORZLQJ LV ±LPSOLFLWO\ ZKHQ QRW PHQWLRQHG D IXQFWLRQ RI IUHTXHQF\&RPSXWDWLRQVKDYHWREHUHSHDWHGDVPDQ\WLPHVDVUHTXLUHGWRIROORZIUHTXHQF\UHVSRQVHV ,QRUGHUWRVLPSOLI\H[SODQDWLRQVRQO\WKHVWUXFWXUDOFDVHLVSUHVHQWHGKHUHUHVSRQVHWHUPVDUHVWUXFWXUDO YHORFLWLHVZKLOHH[FLWDWLRQWHUPVDUHIRUFHV%RWKDUHUHODWHGE\PRELOLW\PDWULFHV,QWKLVFDVHH[SUHVVLRQ LVUHZULWWHQDV

= 76 7

+

6

99 ))

ELV

ZKHUH 6

99

LV WKH YHORFLW\ UHVSRQVH FURVVVSHFWUDO PDWUL[ DQG 6

))

LV WKH IRUFH FURVVVSHFWUDO PDWUL[

,QWKHFDVHRIDQDFRXVWLFSUREOHPUHVSRQVHWHUPVZRXOGEHDFRXVWLFSUHVVXUHVDQGH[FLWDWLRQWHUPVZRXOG EHDFRXVWLFIORZYHORFLW\ERWKEHLQJUHODWHGE\DGPLWWDQFHPDWULFHV

3.1 Energy model construction

7KH SURSRVHG HQHUJ\ PRGHO VLPSO\ UHODWHV WKH YHFWRU RI LQSXW SRZHU RQ LQGLYLGXDO '2)V ʌ

LQ

WR D UHVSRQVHSRZHUDORQJHYHU\REVHUYDWLRQ'2) ʌ

RXW

LQ 2,

RXW

ȟ ʌ

ʌ =

ZKHUH ȟ

2,

LVDSRVLWLYHQRQGLPHQVLRQDOUDQGRP3RZHU)UHTXHQF\5HVSRQVH)XQFWLRQ3)5) PDWUL[

/HWXVLQWURGXFH <

LQ

UHVS <

RXW

WKHVWRFKDVWLFDYHUDJHRIWKHUHDOSDUWRIWKHFRPSOH[PRELOLW\PDWUL[

EHWZHHQDOOLQSXWSRLQWVUHVSDOOREVHUYDWLRQSRLQWV7KHVHPDWULFHVVLQFHWKH\DUHUHDOV\PPHWULFPD\

EHGHFRPSRVHGDV <

LQ

= ;

LQ

<

LQGLDJ

;

+LQ

UHVS <

RXW

= ;

RXW

<

RXWGLDJ

;

RXW+

LQYROYLQJDGLDJRQDOSRVLWLYHPDWUL[

7KHGHFRPSRVLWLRQUHYHDOVWKHSULQFLSDOPRELOLW\SDWWHUQVWKRXJKWWKHXQLWDU\PDWULFHV ;

LQ

UHVS ;

RXW

WKDWGHILQHDQHZYHFWRUEDVLVZKLFKFRPSRQHQWVZLOOEHQDPHGSDWWHUQV3ULQFLSDOPRELOLW\SDWWHUQVVKRZ WKHGLUHFWLRQLQZKLFKWKHYLEURDFRXVWLFUHVSRQVHLVFROOLQHDUZLWKWKHH[FLWDWLRQSDWWHUQWKDWLVWRVD\QR FRXSOLQJH[LVWVEHWZHHQSULQFLSDOPRELOLW\SDWWHUQV

:KDWHYHUWKHH[FLWDWLRQLWFDQEHVKRZQWKDWRQHFDQZULWH

)LQ

GLDJ LQ LQ

< V

ʌ =

(8)

ZKHUH V

)

= ( ;

LQ

6

))

;

+LQ

)

LQ

GLDJ LV WKH YHFWRU RI WKH 3RZHU 6SHFWUDO 'HQVLW\ 36' RI WKH ORDGV H[SUHVVHGLQWKHEDVLVRIWKHLQSXWSULQFLSDOPRELOLW\

2QHWKHQLQWURGXFHVWKHQRQGLPHQVLRQQDOV\PPHWULFDOH[SUHVVLRQRIWKH3)5)PDWUL[

GLDJ LQ 2, GLDJ RXW

2,

< + <

ȟ =

ZKHUH +

2,

LVWKHWHUPWRWHUPVTXDUHGUDQGRPWUDQVIHUPDWUL[H[SUHVVHGLQWKHQHZLQSXWDQG RXWSXWEDVLV

LM 2,LM

2,

7

+ = HW 7

2,

= ;

RXW

7 ;

LQ+

)LQDOO\WKHUHVSRQVHSRZHUDSSHDUVWREH

YRXW

GLDJ RXW

RXW

< V

ʌ =

ZKHUH V

Y

= ( ;

RXW

6

99

;

+RXW

)

RXW

GLDJ LV WKH YHFWRU RI WKH UHVSRQVH YHORFLW\ 36' H[SUHVVHG LQ WKH EDVLVRIRXWSXWSULQFLSDOPRELOLW\

3UDFWLFDOO\ WKLV PHDQV WKDW WKH 36' RI YLEURDFRXVWLF UHVSRQVHV PD\ EH REWDLQHG E\ FRPSXWLQJ VXFFHVVLYHO\

• 7KHLQSXWDQGRXWSXWSULQFLSDOPRELOLW\EDVLV

• 7KHLQSXWSRZHUYHFWRUXVLQJ

• 7KHUHVSRQVHSRZHUYHFWRUXVLQJ

• 7KHUHVSRQVHYHORFLW\36'YHFWRUE\LQYHUVLRQRIH[SUHVVLRQ

$VGHPRQVWUDWHGLQUHIHUHQFHV>@WKHVWRFKDVWLFDYHUDJHRIWKHUHVSRQVH36'¶VH[SUHVVLRQGLDJRQDO WHUPVRIWKHUHVSRQVHFURVVVSHFWUDOPDWUL[DUHFRPSXWHGH[DFWO\ZKHQXVLQJ H[SUHVVLRQV LQVWHDGRI,WVKRXOGEHQRWHGKHUHWKDWFRUUHODWLRQVEHWZHHQUHVSRQVHSDWWHUQVDUHORVWZKHQXVLQJ WKHSURSRVHGDSSURDFKZKLFKPD\FDXVHVRPHWURXEOHLQWKHFDVHRIUDGLDWLRQFRPSXWDWLRQV1HYHUWKHOHVV WKH SURSRVHG HQHUJ\ILHOG PRGHO LV YDOLG IRU VWUXFWXUDODFRXVWLF SUREOHPV QDWXUDOO\ LQFOXGLQJ VRXQG UDGLDWLRQLVVXHVLQWKHVWUXFWXUDODFRXVWLFFRXSOHGIRUPXODWLRQ

3.2 About the principal mobility basis

)LJXUHSULQFLSDOPRELOLW\YDOXHVYVIUHTXHQF\DWRQHSRLQWRQWKHURRIFXUYHGWKLQVKHOO

7KH XVH RI WKH SULQFLSDO PRELOLW\ EDVLV LV D FULWLFDO SRLQW LQ WKH SURSRVHG HQHUJ\ DSSURDFK )LUVW LW

YDOLGDWHVWKHSURSRVHGH[SUHVVLRQVLQDQ\ORDGFDVHLQGHHGH[SUHVVLRQKROGVZKDWHYHUWKHFRUUHODWLRQ

(9)

EHWZHHQ IRUFHV FRRUGLQDWHV LQ WKH LQSXW SULQFLSDO PRELOLW\ EDVLV 5HOHYDQW FRUUHODWLRQV EHWZHHQ IRUFH H[FLWDWLRQVDUHLPSOLFLWO\DFFRXQWHGIRUZKHQFKDQJLQJWKHEDVLV

6HFRQGLWVHSDUDWHVZKHQQHHGHGLQSODQHDQGQRUPDOGLVSODFHPHQWVWKDWDUHNQRZQWRH[KLELWGLIIHUHQW G\QDPLFSURSHUWLHV$VDQH[DPSOHILJXUHDQGVKRZUHVSHFWLYHO\WKHSULQFLSDOPRELOLWLHVFRPSXWHGIRU DSRLQW'2)6RQWKHURRI)LJDQGWKHULJKWHQJLQHPRXQWSRVLWLRQ'2)VRQWKHULJKWIURQWVLGH PHPEHU)LJ2QWKHURRI)LJVOLJKWO\FXUYHGVKHOOWKHQRUPDOGLUHFWLRQVKRZVDPRELOLW\WKDWLV PDQ\ RUGHU RI PDJQLWXGH KLJKHU WKDQ LQSODQH PRELOLW\ 2Q WKH IURQW VLGH PHPEHU FXUYHG EHDP ZLWK YDU\LQJVHFWLRQWKHSULQFLSDOPRELOLWLHVKDYHDERXWWKHVDPHRUGHURIPDJQLWXGH

)LJXUHSULQFLSDOPRELOLW\YDOXHVYVIUHTXHQF\DWRQHSRLQWRQWKHIURQWVLGHPHPEHUFXUYHGEHDP 7KHPLQLPXPH[WHQGRIDQLQSXWDUHDLVDVLQJOHSRLQWZLWKLWVIRUFHV'2)6DVVKRZQIRUILJXUHVDQG ,Q WKLV FDVH WKH SULQFLSDO PRELOLW\ EDVLV LV REWDLQHG E\ D URWDWLRQ RI WKH D[HV SURYLGLQJ D ORFDO D[HV V\VWHPWKDWLVUDWKHUHDV\WRZRUNRXW

:KHQ FRUUHODWHG IRUFHV DUH DSSOLHG WR PXOWLSOH SRLQWV LQ D IUHTXHQF\ UDQJH ZKHUH VWURQJ GLVWDQW LQWHUDFWLRQ H[LVW WKH VL]H RI WKH UHOHYDQW PRELOLW\ PDWUL[ LQFUHDVHV 3DWWHUQV DVVRFLDWHG WR SULQFLSDO PRELOLWLHV JHQHUDOO\ LQYROYH PRUH WKDQ RQH LQSXW SRLQW OHDGLQJ WR D PRUH FRPSOH[ H[FLWDWLRQ VFKHPH 6XFKDSK\VLFDOVFKHPHRQO\RFFXUVDWORZIUHTXHQFLHV:KHQWKHIUHTXHQF\LQFUHDVHVERWKSKHQRPHQD WHQGWRGLVDSSHDUFURVVWHUPVRILQSXWPRELOLW\PDWULFHVDUHYDQLVKLQJZKLOHGLVWDQWIRUFHVWHQG±DWOHDVW LQDYHUDJHRYHUDIUHTXHQF\EDQGWREHXQFRUUHODWHG

,QWKHIROORZLQJZHZLOORQO\FRQVLGHUWKHSULQFLSDOPRELOLW\D[HVDVVRFLDWHGWRRQHVLQJOHSRLQW

4 Simplified energy model and system partition

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

WKHH[SUHVVLRQRIWKH3RZHUWUDQVIHUEHWZHHQDQH[FLWDWLRQDUHDDQGDQREVHUYDWLRQDUHDE\UHSODFLQJ'2)

WR'2)35)5E\DVLQJOHDYHUDJHG3)5)6XFKDVLPSOLILFDWLRQZKHQLWLVUHOHYDQWGHILQHVDSDUWLWLRQRI

WKHVWXGLHGV\VWHPH[FLWDWLRQDUHDVODWHUGHQRWHG,PD\WKHQEHFKDUDFWHUL]HGE\WKHRYHUDOOSRZHULQSXW

ZKLOH REVHUYDWLRQ DUHDV ODWHU GHQRWHG 2 DUH FKDUDFWHUL]HG E\ D XQLIRUP UHVSRQVH SRZHU 2QO\ GLVWDQW

SURSDJDWLRQV DUH FRYHUHG E\ WKH SURSRVHG VLPSOLILFDWLRQ ORFDO HIIHFWV H[FLWDWLRQ QHDUILHOG DUH RXW RI

VFRSHRIWKHSURSRVHGWKHRU\

(10)

)LJXUH$YHUDJHGYHORFLW\)5)VEHWZHHQRQHH[FLWDWLRQ'2)HQJLQHPRXQWDQGREVHUYDWLRQ'2) PD[LPXPPRELOLW\GLUHFWLRQRQWKHURRI

)LJXUH$YHUDJHGYHORFLW\3)5)VEHWZHHQRQHH[FLWDWLRQ'2)HQJLQHPRXQWDQGREVHUYDWLRQ'2) PD[LPXPPRELOLW\GLUHFWLRQRQWKHURRI

2QH WKHQ LQWURGXFHV WKH VFDODU IXQFWLRQ H

2,

DYHUDJH 3)5) ZKLFK FKDUDFWHUL]HG WKH VLPSOLILHG GLVWDQW SRZHUWUDQVPLVVLRQVXFKDV

LQ 2,

RXW

= H Π

Π

ZKHUH Π

LQ

LV WKH RYHUDOO LQSXW SRZHU LQ WKH H[FLWDWLRQ DUHD ¦

=

= Π

1,

O LQN

LQ

ʌ

DQG Π

RXW

LV WKH DYHUDJHGUHVSRQVHSRZHULQWKHREVHUYDWLRQDUHD

,W FDQ EH VKRZQ WKDW WKH EHVW DSSUR[LPDWLRQ ±LQ D OHDVW VTXDUH VHQVH LV SURYLGHG E\ WKH IROORZLQJ H[SUHVVLRQ

¦

¦

¦¦

¦

¦

¦¦

=

=

= =

=

=

= =

=

=

, 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

7KHSRLQWWRSRLQWHUURUWKDWLVDVVRFLDWHGWRWKHSURSRVHGVLPSOLILFDWLRQPD\EHSORWWHGDVWKHPDWUL[RI GLIIHUHQFHVEHWZHHQWKHWUXHPRGHODQGWKHVLPSOLILHGPRGHO

GLDJ O LQ 2, GLDJ

N RXW GLDJ

O LQ O N GLDJ

N RXW NO

2,

= <

<

<

H <

∆ ξ

7KHURRWPHDQVTXDUHGLIIHUHQFHLVDQLQGLFDWRURIWKHVLPSOLILFDWLRQSDUWLWLRQUHOHYDQFH

¦¦

= =

=

, 2

1

O 1

N

NO 2, ,

2

1 FPTH 1

(11)

)LJXUHVKRZVDVDQH[DPSOHWKHPDWUL[ ǻ

2,

RIWKHGLIIHUHQFHEHWZHHQWKHVLPSOLILHGPRGHODQGWKHWUXH PRGHO IRU IRXU VHOHFWHG IUHTXHQFLHV LQ WKH FDVH RI 3)5) EHWZHHQ WKH FDU ERG\ IURQWHQG H[FLWDWLRQ '2)V DQG WKH LQWHULRU FDYLW\ '2)6 )LJXUH VKRZV WKH DVVRFLDWHG URRW PHDQ VTXDUH GLIIHUHQFH

±SORWWHGLQG%DVDIXQFWLRQRIIUHTXHQF\$ERYH+]WKHURRWPHDQVTXDUHGLIIHUHQFHUHPDLQV ORZ ZKLFK LQGLFDWHV WKDW WKH FDU ERG\ IURQWHQG DQG WKH FDYLW\ EHKDYH ±ZLWKLQ D OLPLWHG HUURU DV VHSDUDWHGHQWLWLHVUHODWHGE\DVLQJOH3)5)

,WLVQRWLFHDEOHWKDWDERYH+]RQHPD\EHVXUHWKDWDQ\GHFUHDVHRIWKHLQSXWSRZHURQWKHFDUIURQW HQG ZKDWHYHU WKH ZD\ LW FRPHV OHDGV WR D GHFUHDVH RI WKH UHVSRQVH SRZHU WKDW LV WR VD\ WKH DFRXVWLF SUHVVXUH ,Q VXFK D FDVH QRLVH FRQWURO LQVLGH WKH FDYLW\ MXVW UHOLHV RQ WKH FRQWURO RI WZR IUHTXHQF\

IXQFWLRQV WKH RYHUDOO LQSXW SRZHU DQG WKH DYHUDJH 3)5) 7KH VHQVLWLYLW\ RI ERWK TXDQWLWLHV WR GHVLJQ SDUDPHWHUVLVGLVFXVVHGLQDUHODWHGSDSHU>@

)LJXUH0DWUL[RIGLIIHUHQFHVEHWZHHQWKHSRLQWWRSRLQW3)5)VDQGWKHVLQJOHDYHUDJHG3)5) EHWZHHQWKHERG\IURQWHQG'2)VDQGWKHFDULQWHULRUFDYLW\'2)V+]D+]E

+]FDQG+]G

)LJXUH5RRWPHDQVTXDUHGLIIHUHQFHDVVRFLDWHGWRWKHVLPSOLILFDWLRQRIWKH3)5)EHWZHHQWKH ERG\IURQWHQG'2)VDQGWKHFDULQWHULRUFDYLW\'2)V

5 Conclusion

$IWHU D TXLFN SUHVHQWDWLRQ RI WKH QRQSDUDPHWULF PHWKRG XVHG WR LQYHVWLJDWH XQFHUWDLQ YLEURDFRXVWLF V\VWHPVDQDSSOLFDWLRQWRDFDUERG\PRGDODQDO\VLVVKRZVWKHPDLQIHDWXUHVRIWKHSURSRVHGVWRFKDVWLF PRGHOOLQJ LQ WHUPV RI PRGDOVKDSH DQG HYHQWXDOO\ PRGDO IUHTXHQFLHV 0RGDO VKDSHV DSSHDU WR EH VWDEOH HQRXJKWRSHUIRUPIUHTXHQFLHVVWDWLVWLFVRQO\DWORZIUHTXHQFLHV

D E

F G

(12)

$Q HQHUJ\ ILHOG GHQVLW\ DSSURDFK LV WKHQ LQWURGXFHG ,Q WKH FDVH RI D FRPSOH[ KHWHURJHQHRXV VWUXFWXUH VXFK DV D FDU ERG\ WKLV VWRFKDVWLF DSSURDFK UHYHDOV KRPRJHQHRXV EHKDYLRXUV ZLWK UHVSHFW WR VSDFH DQG IUHTXHQF\6XFKEHKDYLRXUDSSHDUVIRUDQ\ORDGFDVHDIWHUDSURMHFWLRQRQWKHSULQFLSDOPRELOLW\EDVLVRID JLYHQDUHDH[FLWDWLRQRUREVHUYDWLRQ

$ERYHDJLYHQIUHTXHQF\3RZHU)UHTXHQF\5HVSRQVH)XQFWLRQVEHWZHHQH[FLWDWLRQ'2)VZLWKLQDJLYHQ DUHD DQG REVHUYDWLRQ '2)V LQ D GLVWDQW DUHD WHQG WR PHUJH WR D XQLTXH VFDODU 3)5) $ SDUWLWLRQ RI WKH VWXGLHGVWUXFWXUHFDQWKDQEHFUHDWHGLWLVVXFKWKDWRQHVLQJOHDYHUDJHG3)5)LVHQRXJKWRGHVFULEHWKH SRZHUWUDQVIHUEHWZHHQWKHSDUWVFRQVWLWXWHGE\WKHH[FLWDWLRQDQGREVHUYDWLRQDUHDV

$VSHFLILFPHDVXUHLVSURSRVHGWRPHDVXUHWKHUHOHYDQFHRIWKHSURSRVHGVLPSOLILFDWLRQ,WVHHPVWKDWLQ WKHFDVHRIDQDXWRPRWLYHVWUXFWXUHDERYH+]VRPHSDUWVDOUHDG\VDWLVI\DSDUWLWLRQFRQGLWLRQ,WLVLQ SDUWLFXODUWKHFDVHRIWKHFDUIURQWHQGDQGWKHFDYLW\

6XFKDVFKHPHFRXOGEHRIJUHDWKHOSWRLPSURYHWKHQRLVHFRQWUROLQDXWRPRWLYHYHKLFOH,WLVUHPLQGHG WKDWWKHSURSRVHGVFKHPHRQO\KROGVLQDYHUDJHLQDVWRFKDVWLFFRQWH[W7KHH[SHFWHGSURSHUWLHVPD\QRW EHIXOILOOHGIRULQGLYLGXDOUHDOL]DWLRQV

6 References

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

&RPS0HWK$SSO0HFK(QJ

>@ 6RL]H & $ FRPSUHKHQVLYH RYHUYLHZ RI D QRQSDUDPHWULF SUREDELOLVWLF DSSURDFK RI PRGHO XQFHUWDLQWLHVIRUSUHGLFWLYHPRGHOVLQVWUXFWXUDOG\QDPLFV-RXUQDORI6RXQGDQG9LEUDWLRQ

>@ 'XUDQG-)6RL]H&*DJOLDUGLQL/6WUXFWXUDODFRXVWLFPRGHOOLQJRIDXWRPRWLYHYHKLFOHVLQ SUHVHQFHRIXQFHUWDLQWLHVDQGH[SHULPHQWDOLGHQWLILFDWLRQDQGYDOLGDWLRQ-RXUQDORIWKH$FRXVWLFDO 6RFLHW\RI$PHULFD±

>@ 2KD\RQ5DQG6RL]H&6WUXFWXUDO$FRXVWLFVDQG9LEUDWLRQ$FDGHPLF3UHVV6DQ'LHJR

>@ 6HUIOLQJ5-$SSUR[LPDWLRQ7KHRUHPVRI0DWKHPDWLFDO6WDWLVWLFV-RKQ:LOH\ 6RQV

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

YLEURDFRXVWLF DQDO\VLV RI FRPSOH[ VWUXFWXUHV XVLQJ D VWDWLVWLFDO FRPSXWDWLRQDO PRGHO -RXUQDO RI 6RXQGDQG9LEUDWLRQ

>@ 0 .DVVHP & 6RL]H / *DJOLDUGLQL 6WUXFWXUDO SDUWLWLRQLQJ RI FRPSOH[

VWUXFWXUHV LQ WKH PHGLXPIUHTXHQF\ UDQJH $Q DSSOLFDWLRQ WR DQ DXWRPRWLYH YHKLFOH -RXUQDO RI 6RXQG DQG 9LEUDWLRQ 6XEPLWWHG RQ )HEUXDU\

>@ /*DJOLDUGLQL*%RUHOOR9LUWXDO6($7RZDUGVDQLQGXVWULDOSURFHVV6$(,QWHUQDWLRQDO

>@ $-XQG/*DJOLDUGLQL1RLVHFRQWUROXVLQJLQSXWSRZHUDWORZDQGPLGIUHTXHQFLHVVHQVLWLYLW\WR

VWUXFWXUDOGHVLJQFKDQJHV,60$FRQIHUHQFH/HXYHQ

Références

Documents relatifs

A general and efficient method for constructing a reduced-order computational model, which is adapted to predicting the global displacements of complex structures for which there

The main objectives of this paper are (1) to present experimental validation of the nonparametric probabilistic approach of model uncer- tainties and to propose methods to perform

The orthogonal base of the local coordinates system defined by the principal directions of the mean local mobilities is represented by the eigenvectors of the mean mobility matrix

In this paper, we present an energy density field approach for low- and medium-frequency vibroacoustic analysis of complex structures using a statistical computational model..

Gagliardini, Energy-density field approach for low- and medium- frequency vibroacoustic analysis of complex structures using a statistical computational model. Journal of Sound

A general method has been presented for constructing a small-dimension ROM of global displacements, which is adapted for analyzing the dynamical responses of the stiff master part

Steady-state RMS velocity from numerical simulations (crosses) of the friction-damped beam for an input force amplitude of F 1 ¼1.25 N compared with: (a) global upper and lower

Also, n(f ) is slightly but consistently different at each of the measured locations of the soundboard. In this frequency domain, the FEM and the experimental estimations of the