PDE control of a rotating shear beam with boundary feedback

3'( &RQWURO RI D 5RWDWLQJ 6KHDU %HDP ZLWK %RXQGDU\ )HHGEDFN

0XVWDID 'R×JDQ

_{ Ž2PHU 0RUJŽXO}

'HSDUWPHQW RI (OHFWULFDO DQG (OHFWURQLFV (QJLQHHULQJ

_{%DVNHQW 8QLYHUVLW\}

_{%LONHQW 8QLYHUVLW\ 7XUNH\}

$EVWUDFW

:H FRQVLGHU D ÀH[LEOH VWUXFWXUH PRGHOHG DV D VKHDU EHDP ZKLFK LV FODPSHG WR D ULJLG ERG\ DW RQH HQG DQG LV IUHH DW WKH RWKHU HQG 7KH ZKROH VWUXFWXUH LV IUHH WR URWDWH RQ WKH KRUL]RQWDO SODQH :H ¿UVW PRGHO WKH V\VWHP E\ XVLQJ 3DUWLDO 'LIIHUHQWLDO (TXDWLRQV 3'( DQG ZH SURSRVH ERXQGDU\ IHHGEDFN ODZV WR DFKLHYH VHW SRLQW UHJXODWLRQ RI WKH URWDWLRQ DQJOH DV ZHOO DV WR VXSSUHVV WKH HODVWLF YLEUDWLRQV 7KH SURSRVHG FRQWURO ODZV DUH EDVHG RQ 3'( PRGHO KHQFH ZH GR QRW UHVRUW WR GLVFUHWL]DWLRQ RI WKH V\VWHP HTXDWLRQV E\ DYDLODEOH PHWKRGV :H XWLOL]H D FRRUGLQDWH WUDQVIRUPDWLRQ EDVHG RQ DQ LQYHUWLEOH LQWHJUDO WUDQVIRUPDWLRQ E\ XVLQJ 9ROWHUUD IRUP DQG EDFNVWHSSLQJ WHFKQLTXHV :H DOVR SUHVHQW VRPH VLPXODWLRQ UHVXOWV

.H\ :RUGV  3DUWLDO 'LIIHUHQWLDO (TXDWLRQV )OH[LEOH 6\V WHPV %RXQGDU\ &RQWURO 6KHDU %HDP .HUQHO )XQFWLRQV 9ROWHUUD 7UDQVIRUPDWLRQ %DFNVWHSSLQJ

, ,1752'8&7,21

7KH SURJUHVV LQ WKH FRQVWUXFWLRQ RI YDULRXV PHFKDQ LFDO VWUXFWXUHV HJ LQ URERWLFV DQG VSDFH VWUXFWXUHV RQ PDFURVFRSLF VFDOH DV ZHOO DV PLFURPDFKLQHV DWRPLF IRUFH PLFURVFRS\ HWF LQ PLFURVFRSLF VFDOH QHFHVVLWDWHV WKH XVH RI OLJKWZHLJKW PDWHULDOV IRU YDULRXV SUDFWLFDO UHDVRQV VHH HJ >@ 6XFK PDWHULDOV XVXDOO\ H[KLELW ÀH[LEOH YLEUDWLRQV DQG WR PRGHO VXFK VWUXFWXUHV RQH KDV WR XVH 3DUWLDO 'LIIHUHQWLDO (TXDWLRQV 3'( ,Q SUDFWLFH ZKHQ GHVLJQLQJ FRQWUROOHUV IRU VXFK V\VWHPV XVXDOO\ WKHVH 3'( PRGHOV DUH UHGXFHG WR 2UGLQDU\ 'LIIHUHQWLDO (TXDWLRQV 2'( E\ XVLQJ YDULRXV PHWKRGV VXFK DV PRGHO UHGXFWLRQ ¿QLWH HOHPHQW DQDO\VLV GLVFUHWL]DWLRQ HWF VHH HJ >@ +RZHYHU VXFK 2'( PRGHOV KDYH VRPH GUDZEDFNV DQG XVXDOO\ WKH GHVLJQHG FRQWUROOHUV EDVHG RQ WKHVH 2'( PRGHOV OLPLW WKH SHUIRUPDQFH RI VXFK V\VWHPV VHH HJ >@

2QH RI WKH PRVW IUHTXHQWO\ HQFRXQWHUHG ÀH[LEOH VWUXF WXUHV LQ PHFKDQLFDO VWUXFWXUHV PHQWLRQHG DERYH LV WKH ÀH[LEOH EHDPV ZKLFK DUH W\SLFDOO\ XVHG WR PRGHO ÀH[LEOH OLQNV RI URERWLF DUPV WLSV RI DWRPLF IRUFH PLFURVFRSHV HWF $PRQJ WKH YDULRXV DGYDQWDJHV RI XVLQJ ÀH[LEOH EHDPV WKH PDLQ RQHV DUH WKHLU OLJKW ZHLJKWV DQG ORZ HQHUJ\ FRQVXPS WLRQV 7KHUH DUH YDULRXV 3'( PRGHOV IRU ÀH[LEOH EHDPV VXFK DV (XOHU%HUQRXOOL 5D\OHLJK 7LPRVKHQNR EHDP HTXDWLRQV $PRQJ WKHVH WKH PRVW DGYDQFHG DQG FRPSUHKHQVLYH RQH LV WKH 7LPRVKHQNR EHDP PRGHO 7KLV PRGHO XQGHU WKH

)LJ  %HDP FRQ¿JXUDWLRQ

´VOHQGHU EHDP´ DVVXPSWLRQ FDQ DOWHUQDWLYHO\ EH UHSUHVHQWHG DV VKHDU EHDP VHH HJ >@ >@

9DULRXV PHWKRGV KDYH EHHQ SURSRVHG IRU FRQWURO RI ÀH[L EOHOLQNV LQ WKH OLWHUDWXUH VHH HJ >@ >@ >@ >@ >@ >@ 5HFHQWO\ >@ >@ >@ SURSRVHG WKH VWUXFWXUDO DSSURDFK WR 3'(V ZLWK EDFNVWHSSLQJ WHFKQLTXHV

,Q WKLV SDSHU ZH FRQVLGHU D VKHDU EHDP FODPSHG WR D ULJLG ERG\ LQ RQH HQG DQG LV IUHH DW WKH RWKHU HQG 7KH ZKROH FRQ¿JXUDWLRQ LV IUHH WR URWDWH RQ WKH KRUL]RQWDO SODQH :H ¿UVW JLYH WKH HTXDWLRQV RI PRWLRQ IRU VXFK V\VWHP :H ZLOO XVH 3'( PRGHO IRU WKH ÀH[LEOH EHDP ZLWKRXW UHVRUWLQJ WR UHGXFLQJ WKH UHVXOWLQJ HTXDWLRQV WR DQ 2'( PRGHO 2XU FRQWURO REMHFWLYH LV WR URWDWH WKH ÀH[LEOH EHDP WR D GHVLUHG DQJOH DQG VXSSUHVV WKH ÀH[XUDO YLEUDWLRQV :H XVH D WHFK QLTXH WKDW LV ¿UVW LQWURGXFHG LQ >@ >@ >@ WR WUDQVIRUP WKH V\VWHP HTXDWLRQV WR DQRWKHU VHW RI HTXDWLRQV ZKLFK KDV ZHOONQRZQ VWDELOLW\ SURSHUWLHV 7KLV WUDQVIRUPDWLRQ LV GRQH E\ XVLQJ DQ LQYHUWLEOH LQWHJUDO RSHUDWRU 9ROWHUUD W\SH ZLWK D VPRRWK NHUQHO :H JLYH WKH DQDO\WLFDO H[SUHVVLRQ RI VXFK D NHUQHO ,Q WKH SURFHVV RI REWDLQLQJ DQ DSSURSULDWH NHUQHO IXQFWLRQ ZH DOVR REWDLQ DSSURSULDWH ERXQGDU\ FRQWURO ODZ WR VWDELOL]H WKH FORVHG ORRS V\VWHP :H DOVR SUHVHQW VRPH VLPXODWLRQ UHVXOWV ZKLFK FRQ¿UPV WKH VWDELOLW\ RI WKH FORVHG ORRS V\VWHP )LQDOO\ ZH JLYH VRPH FRQFOXGLQJ UHPDUNV



ª
$PQZSJHIU
&6$"
 

,, $1$/<7,&$/02'(/

:H FRQVLGHU D ÀH[LEOH VWUXFWXUH FRQVLVWLQJ RI D ÀH[LEOH VKHDU EHDP ZKLFK LV FODPSHG WR D ULJLG ERG\ DW RQH HQG IUHH DW WKH RWKHU DQG DV VKRZQ LQ )LJXUH  5HIHUULQJ WR )LJXUH  WKH YDULRXV V\PEROV UHSUHVHQW WKH IROORZLQJ X_oY_o JOREDO LQHUWLDO V\VWHP RI FRRUGLQDWHV X₁Y₁ ERG\ ¿[HG V\VWHP RI FRRUGLQDWHV DWWDFKHG WR XQGHIRUPHG EHDP θ₁ DQJXODU GLVSODFHPHQW RI EHDP u ÀH[XUDO GLVSODFHPHQW RI EHDP

,Q WKLV VHFWLRQ VWDUWLQJ ZLWK 7LPRVKHQNR EHDP PRGHOWKH SDUWLDO GLIIHUHQWLDO HTXDWLRQV ZLWK ERXQGDU\ FRQGLWLRQV DUH GHULYHG XVLQJ WKH 9ROWHUUD VWDWH WUDQVIRUPDWLRQ VHH HJ >@ >@ 7KH OLQN LV PRGHOOHG LQ FODPSHGIUHH FRQ¿J XUDWLRQ VLQFH QDWXUDO PRGHV RI WKH VHSDUDWHG FODPSHGIUHH OLQNV DJUHH YHU\ ZHOO ZLWK DFWXDO RQHV FRPSDUHG WR SLQQHG IUHH FRQ¿JXUDWLRQ VHH HJ >@ $VVXPLQJ WKH PDQLSXODWRU URWDWHV LQ WKH KRUL]RQWDO SODQH LQ WKH DEVHQFH RI JUDYLW\ WKH SRWHQWLDO HQHUJ\ GHSHQGV RQO\ RQ WKH ÀH[XUDO GHÀHFWLRQV

3DUDPHWHU 'HVFULSWLRQ E <RXQJ¶V 0RGXOXV G 6KHDU 0RGXOXV A FURVVVHFWLRQDO DUHD I FURVVVHFWLRQDO DUHD PRPHQW I_h ,QHUWLD RI WKH KXE L /HQJWK RI WKH EHDP

x &RRUGLQDWH DORQJ WKH D[LDO FHQWHU u(x, t) 7UDQVYHUVH PRYHPHQW

α(x, t) $QJOH RI GLVWRUWLRQ GXH WR VKHDU ˙u(x, t) 7LPH UDWH RI WUDQVYHUVH PRYHPHQW ux(x, t) $[LDO UDWH RI WUDQVYHUVH PRYHPHQW

ρ /LQHDU GHQVLW\

τ₁ ,QSXW WRUTXH DW EDVH PRWRU

θ₁ $QJXODU SRVLWLRQ RI WKH EHDP GXH WR URWDWLRQ

7$%/( ,

3$5$0(7(56 )253'(02'(/

7KH HTXDWLRQV RI PRWLRQ IRU 7LPRVKHQNR EHDP FDQ EH JLYHQ ZLWK WKH QRWDWLRQ LQ 7DEOH , DV IROORZV

¨u = b u_xx− α_x−  x ¨θ₁  μ¨α =  α_xx− α + b u_x  ZKHUH D GRW UHSUHVHQWV WLPH GHULYDWLYH D VXEVFULSW DV LQ u_x GHQRWHV VSDWLDO GHULYDWLYH ZLWK UHVSHFW WR ZUW x IRU 0 ≤ x ≤ L DQG t ≥ 0 +HUH EI GHQRWHV EHQGLQJ VWLIIQHVV b LV GH¿QHG DV b = EI/ρ EA GHQRWHV D[LDO VWLIIQHVV μ LV GH¿QHG DV μ = ρ/EA DQG VKHDU FRHI¿FLHQW  LV D OLQHDU IXQFWLRQ RI EI/GA VHH HJ >@ >@ 6LQFH WKH UDWLR EHWZHHQ WKH OHQJWK RI WKH EHDP DQG LWV WKLFNQHVV LV VXI¿FLHQWO\ ODUJH ZH FDQ WDNH μ= 0 DSSUR[LPDWHO\ 7KHQ WKH VOHQGHU EHDP FDQ HDVLO\ EH PRGHOOHG DV VKHDU EHDP VHH HJ >@ >@ 7KXV WKH JRYHUQLQJ HTXDWLRQV IRU D URWDWLQJ VKHDU EHDP FDQ EH JLYHQ EHORZ

¨u = b u_xx− α_x−  x ¨θ₁  0 =  αxx− α + b ux 

%\ GLIIHUHQWLDWLQJ   ZUW x DQG DGG WR   ZH REWDLQ ¨u_x− b u_xxx+ α_xx+  ¨θ₁+  α_xx− α + b u_x= 0  QRZ LI ZH GLIIHUHQWLDWH   ZUW x DQG VXEWUDFW 1/ WLPHV   WKHQ ZH KDYH ¨u_xx− b u_xxxx+ (1 + )α_xxx+ b u_xx −αx− ¨u + b uxx− 1 αx− x¨θ1= 0.  )LQDOO\ E\ FDOFXODWLRQ RI αxxx IURP   ZH REWDLQ WKH

IROORZLQJ VKHDU EHDP PRGHO DV VLQJOH VHFRQGRUGHULQWLPH IRXUWKRUGHULQVSDFH 3'(

¨u −  ¨uxx+ b uxxxx= −x ¨θ1 

7KH XVXDO FODPSHGIUHH ERXQGDU\ FRQGLWLRQV IRU WKH VKHDU EHDP FDQ EH JLYHQ DV IROORZV VHH HJ >@

u(0, t) = 0 u_x(0, t) = α(0, t)  α(L, t) = 0 u_xx(L, t) = 0  %\ XVLQJ   ZH FDQ REWDLQ α(x, t) LQ WHUPV RI u %\ XVLQJ /DSODFH WUDQVIRUP LQ   ZH JHW WKH IROORZLQJ DIWHU VRPH FDOFXODWLRQV DQG GURSSLQJ WKH t DUJXPHQW LQ WKH UHODWHG IXQFWLRQV VHH HJ >@ α_x(x) = −a2c  _x 0 sinh(cx − cy)u(y)dy − a 2_{u(x) } ZKHUH a2 _{= b/ DQG c}2 _{= 1/  %\ VXEVWLWXWLQJ   LQWR}

  JRYHUQLQJ HTXDWLRQV    FDQ EH VLPSOL¿HG DV VLQJOH HTXDWLRQ EHORZ )RU WKH ULJLG ERG\ URWDWLRQ E\ DSSO\LQJ FRQVHUYDWLRQ RI PRPHQWXP DW WKH EDVH ZH REWDLQ RSHQ ORRS V\VWHP DV IROORZV

¨u = b u_xx+ a2u+ a2c  _x

0 sinh(cx − cy) u(y)dy 

− x ¨θ1

I_h¨θ₁− EI u_xx(0, t) = τ₁  ,,, &21752//(5'(6,*1

7KH FRQWUROOHU IRU WKH ULJLG SDUW FDQ EH GHVLJQHG WR SURYLGH H[SRQHQWLDO GHFD\LQJ RI GHULYDWLYHV DQG WR DFKLHYH WKH GHVLUHG VHW SRLQW VXFK WKDW

τ₁= −EI u_xx(0, t) − k₁ ˙θ₁− k₂(θ₁− θ_d)  ZKHUH k₁, k₂ DUH SRVLWLYH FRQVWDQWV DQG θ_d DUH WKH GHVLUHG SRVLWLRQ 7R VLPSOLI\ WKH ÀH[LEOH HTXDWLRQ   ZH ¿UVW GH¿QH D QHZ VWDWH YDULDEOH w(·) E\ DSSO\LQJ DQ LQYHUWLEOH 9ROWHUUD VWDWH WUDQVIRUPDWLRQ ZLWK VPRRWK NHUQHO k(x, y)

w(x) = u(x) −  _x

0

k(x, y) u(y)dy 

)RU WKH UDWLRQDOH DQG PHWKRGRORJ\ EHKLQG XVLQJ VXFK D WUDQVIRUPDWLRQ VHH HJ >@ >@ 7KH NHUQHO k(x, y) VKRXOG EH FKRVHQ VR WKDW WKH UHVXOWLQJ HTXDWLRQV LQ WHUPV RI WKH WUDQVIRUPHG YDULDEOH w(x) KDV QLFH VWDELOLW\ SURSHUWLHV 6XFK D UHVXOWLQJ V\VWHP FDQ EH JLYHQ DV IROORZV

w¨ = b (w_xx− e w) 

w_x(L) = −c_o˙w(L) 

w_x(0) = c₁(/b) ¨θ₁  ZKHUH e, co, c1 DUH SRVLWLYH FRQWUROOHU JDLQV 1RWH WKDW WKH

ERXQGDU\ FRQGLWLRQ   LV RI FUXFLDO LPSRUWDQFH WR REWDLQ HTXDWLRQV   DQG   VHH DOVR UHPDUN  +HQFH WKH RSHQORRS V\VWHP JLYHQ E\    FDQ EH WUDQVIRUPHG LQWR WKH FORVHGORRS V\VWHP JLYHQ E\    ZLWK WKH IROORZLQJ ERXQGDU\ FRQWURO ODZ

u_x(L, t) =  L 0 k_x(L, y)u(y)dy + k(L, L)u(L, t)  −co˙u(L, t) + co  _L 0 k(L, y) ˙u(y)dy. 2EYLRXVO\ ZH QHHG WR GHVFULEH WKH NHUQHO k(x, y) RU LWV SURSHUWLHV DW WKLV VWDJH ,Q DGGLWLRQ WR HTXDWLRQV    WKH IROORZLQJ HTXDWLRQV DUH DOVR UHTXLUHG WR REWDLQ WKH FRQWURO ODZ   DQG VRPH FRQGLWLRQV IRU WKH NHUQHO

w_x(x) = u_x(x) −  x 0 k_x(x, y)u(y)dy − u(x)k(x, x) w_xx(x) = u_xx(x) − u(x)[k_x(x, x) + k_y(x, x)]  − ux(x) k(x, x) − u(x) kx(x, x) −  _x 0 k_xx(x, y) u(y)dy w¨(x) = ¨u(x) −  _x 0 k(x, y) ¨u(y)dy  %\ IROORZLQJ WKH PHWKRGRORJ\ LQWURGXFHG LQ >@ WKH WHUP ¨u(·) VKRXOG EH VXEVWLWXWHG LQ   E\ XVLQJ   7KXV REWDLQHG w¨(x) DQG w_xx(x) JLYHQ E\   FDQ EH UHSODFHG LQ   $IWHU VRPH FDOFXODWLRQV LW FDQ EH VKRZQ WKDW WKH UHTXLUHG NHUQHO k(x, y) VKRXOG VDWLVI\ WKH QRQKRPRJHQRXV 3'( DV IROORZV k_yy− k_xx+ f k = −1.5 sinh(cx − cy)  k(x, x) = −fx/2  k(0, 0) = 0  k(x, 0) = x −  _x 0 y k(x, y) dy  k_y(x, 0) = 0 

ZKHUH f = a2_{/b+ e 7KH DQDO\WLFDO VROXWLRQ RI  }

 IRU WKH NHUQHO IXQFWLRQ k(x, y) FDQ EH REWDLQHG E\ XVLQJ VWDQGDUG PHWKRGV VHH HJ >@ $IWHU VRPH OHQJWK\

FDOFXODWLRQV ZH REWDLQ WKH IROORZLQJ H[SOLFLW IRUP RI WKH NHUQHO k(x, y) k(x, y) = 0.5[h(0) + h(L)] + 0.5  _L 0 J₀(z)g(ξ)dξ  −0.5yf  _L 0 (J1(z)/z) h(ξ)dξ − m sinh(cx − cy) ZKHUH m = −−1.5_{/f DQG z =} _(y2_{− (x − ξ)}2_)f

J₀(·), J₁(·) DUH %HVVHO IXQFWLRQV h(x) = (m − 0.81) sinh(cx) + 0.62521x DQG g(x) = −cm cosh(cx) 1RWH WKDW WKH IXQFWLRQV h(·) DQG g(·) DUH JHQHUDWHG E\ ERXQGDU\ FRQGLWLRQV RI WKH NHUQHO )RU DQ DOWHUQDWH H[SUHV VLRQ RI WKH NHUQHO VHH WKH $SSHQGL[

'XH WR VSDFH OLPLWDWLRQV ZH GR QRW SUHVHQW DQ\ VWDELOLW\ UHVXOWV DW WKH PRPHQW %XW QRWH WKDW WKH VWDELOLW\ RI FORVHG ORRS V\VWHP FDQ EH DQDO\]HG ZLWK WKH /\DSXQRY VWDELOLW\ WKHRU\ E\ XVLQJ WKH IROORZLQJ /\DSXQRY IXQFWLRQ FDQGLGDWH >@

V = d₁(w_x2+ ew2) +  ˙w2+ d₂ < w, ˙w >  ZKHUH d1, d2 DSSURSULDWH SRVLWLYH FRQVWDQWV wx2 =

_L

0 w2xdxLV WKH QRUP DQG <· > GHQRWHV WKH XVXDO LQQHU

SURGXFW LQ L₂(0, L) VSDFH

5HPDUN  1RWH WKDW WR LPSOHPHQW WKH FRQWURO ODZ

JLYHQ E\   ZH QHHG WKH PHDVXUHPHQWV RI u(x, t), u(L, t) DQG ˙u(L, t) 7KH ODVW WZR FDQ EH PHDVXUHG HDVLO\ +RZHYHU WR PHDVXUH u(x, t) LV QRW SUDFWLFDO EXW LW FDQ EH HVWLPDWHG E\ XVLQJ DQ DSSURSULDWH REVHUYHU )ROORZLQJ WKH LGHDV JLYHQ LQ >@ VXFK DQ REVHUYHU FDQ EH GHVLJQHG ZLWK WKH KHOS RI WKH NHUQHO k(x, y) JLYHQ DERYH DV H[SODLQHG LQ WKH QH[W VHFWLRQ

5HPDUN  7KH ERXQGDU\ FRQGLWLRQ   FDQ EH LPSOH

PHQWHG DV WKH VHFRQGDU\ FRQWUROOHU HDVLO\ GXH WR DFWXDWLRQ DQG PHDVXUHPHQW DW WKH FODPSHGHQG )XUWKHUPRUH WKLV VHFRQGDU\ FRQWURO ODZ LV LQGLVSHQVDEOH WR VLPSOLI\ WKH RSHQ ORRS V\VWHP DQG WR REWDLQ WKH DQDO\WLF NHUQHO VROXWLRQ 7KLV LV DFKLHYHG E\ GHFRXSOLQJ WKH ULJLG DQG ÀH[LEOH FRRUGLQDWHV UHVXOW LQ WKH ERXQGDU\ FRQGLWLRQ   ZLOO GHFD\ H[SRQHQ WLDOO\ IDVW HQRXJK 7KH ZKROH VWUXFWXUH RI WKH SURSRVHG GHVLJQ DQG LWV LPSOHPHQWDWLRQ FDQ EH VWUDLJKWIRUZDUG LQ D SXUH FRQWURO ZD\ FRPSDUH ZLWK WKH PXFK PRUH FRPSOH[ KDQGOLQJ RI WKH VDPH SUREOHP VHH HJ >@

,9 2%6(59(5'(6,*1

6LPLODU WR WKH FRQWUROOHU GHVLJQ WR GH¿QH WKH REVHUYHU HUURU G\QDPLFV ZH QHHG DQ DX[LOLDU\ G\QDPLFV ZLWK ZHOO NQRZQ VWDELOLW\ SURSHUWLHV 6XFK D V\VWHP FDQ EH JLYHQ DV IROORZV  ¨w˜ = b ( ˜w_xx− e₂w˜)  ˜ w_x(L) = −c₂˙˜w(L)  ˜ w_x(0) = 0 



ZKHUH e2, c2DUH SRVLWLYH REVHUYHU JDLQV DQG REVHUYHU HUURU

DVVXPHG WR EH ]HUR DW FODPSHG HQG $IWHU DSSO\LQJ 9ROWHUUD VWDWH WUDQVIRUPDWLRQ REVHUYHU HUURU ˜u(x) FDQ EH H[SUHVVHG DV IROORZV

˜u(x) = ˜w(x) −  x

0

p(x, y) ˜w(y)dy.  $OVR REVHUYHU HUURU FDQ EH GH¿QHG DV ˜u(x) = u(x) − ˆu(x) KHUH DQG REVHUYHU NHUQHO p(x, y) LV WKH GXDO YHUVLRQ RI WKH FRQWUROOHU NHUQHO k(x, y) ZLWK DSSURSULDWH ERXQGDU\ FRQGLWLRQV +HQFH ZH DUH UHDG\ WR LQWURGXFH REVHUYHU G\QDPLFV ZKLFK LV GULYHQ E\ w˜(x) VXFK WKDW

 ¨ˆu = b ˆu_xx+ a2ˆu + a2c  _x

0 sinh(cx − cy) ˆu(y)dy

− x ¨θ1+ py(x, L)[u(L, t) − ˆu(L, t)]

−c2p(x, L)[ ˙u(L, t) − ˙ˆu(L, t)]. 

1RWH WKDW py(x, L), p(x, L) DUH GXDO FRXQWHUSDUWV WR WKH

FRQWUROOHU JDLQV kx(L, y), k(L, y) 7KH REVHUYHU G\QDPLFV

FRPSO\ ZLWK XVXDO REVHUYHU VHWXS ZKLFK FDQ EH H[SODLQHG DV FRS\ RI WKH SODQW SOXV HUURU IHHGEDFN $IWHU VRPH OHQJWK\ FDOFXODWLRQV E\ XVLQJ WKH LQWHJUDO WUDQVIRUPDWLRQ   DQG DX[LOLDU\ G\QDPLFV   ZH JHW WKH IROORZLQJ HTXDWLRQV p_xx− p_yy+ ˜f p= −1.5 sinh(cx − cy)  a2c  x 0 sinh(cx − cy)  y 0 p(y, s) ˜w(s) ds dy = c₂p(x, L) ˙˜u(L, t) − p_y(x, L) ˜u(L, t)  ZKHUH ˜f = a2/b+e₂ 7KH DQDO\WLFDO VROXWLRQ RI   IRU WKH REVHUYHU NHUQHO IXQFWLRQ p(x, y) FDQ EH REWDLQHG HDVLO\ E\ GXDOLW\ RI HTXDWLRQV    )LQDOO\ HTXDWLRQ   ZLOO KHOS XV WR FRQVWUXFW WKH REVHUYHU G\QDPLFV ZLWK DX[LOLDU\ RQH w˜(x) LQ OLHX RI ˜u(x) %HVLGHV WKH QHZ FRQWURO ODZ EDVHG RQ REVHUYHU FDQ EH JLYHQ EHORZ

u_x(L, t) =  L 0 k_x(L, y)ˆu(y)dy + k(L, L)u(L, t)  −co˙u(L, t) + co  L 0 k(L, y) ˙ˆu(y)dy. 9 6,08/$7,215(68/76

)RU WKH VLPXODWLRQV ZH ZLOO XVH WKH IROORZLQJ VHW RI SDUDPHWHUV ZKLFK DUH WDNHQ IURP >@

7KH SURSRVHG FRQWURO VFKHPH LV WHVWHG ZLWK WKH VLPX ODWLRQ SURJUDP LPSOHPHQWHG LQ 0$7/$% 7KH 3'(V DUH GLVFUHWL]HG LQ WKH VSDFH GRPDLQ E\ WKH ¿QLWH GLIIHUHQFH PHWKRG WR REWDLQ 2'(V DW HDFK RI WKH QRGHV 7KHQ 2'(V DUH VROYHG QXPHULFDOO\ ,QVWHDG RI GHDOLQJ ZLWK FRPSOH[LW\ RI WKH IRXUWK RUGHU GHULYDWLYH DSSUR[LPDWLRQ WKH VHFRQG RUGHU GHULYDWLYH DSSUR[LPDWLRQ KDV EHHQ XVHG E\ WKH YLUWXH RI FRRUGLQDWH WUDQVIRUPDWLRQ 7KRVH VWDWHV DUH

3DUDPHWHU 9DOXH /HQJWK RI WKH EHDP L= 0.6 m 7LPH VWHS Δt = 1e − 4 VHF 6SDWLDO VWHS Δx = L/30 <RXQJ¶V 0RGXOXV E 70 GP a 'HQVLW\ 2742 kgm−3 7KLFNQHVV RI WKH EHDP m t₁= 0.003175 +HLJKW RI WKH EHDP b_o= 0.0654 m 6KHDU FRHI¿FLHQW = 1.6801 +XE LQHUWLD kgm2 _I h= 0.0055 θ_dGHVLUHG π/3 rad &RQWUROOHU JDLQV co= 59 k₁= I_h· 600 k₂= I_h· 800 c₁= 1, e = 0.9 2EVHUYHU JDLQV c₂= 82 DQG e₂= 0.3 7$%/( ,, 3$5$0(7(56 2) 7+(%($0

PRUH PHDQLQJIXO LQ D UHDO SUREOHP DV ZHOO VLQFH WKH\ FRU UHVSRQGV WR SK\VLFDO YDULDEOHV VXFK DV GHÀHFWLRQV YHORFLW\ DQG EHQGLQJ PRPHQWV +RZHYHU WKH QXPEHU RI 2'(V WR VROYH DQG WKH FRPSXWDWLRQ WLPH DUH LQFUHDVHG LQ UHWXUQ RI WKH UREXVW VWDELOLW\ RI WKH QXPHULF VFKHPH 7KH H[SOLFLW ¿QLWH GLIIHUHQFH VFKHPH WKDW UHTXLUHV YHU\ VPDOO WLPHVWHSV DQG HDV\ WR LPSOHPHQW HI¿FLHQWO\ LV DGDSWHG IURP >@ &KDRWLF YLEUDWLRQV RI D PRGL¿HG (XOHU%HUQRXOOL EHDP WKDW DUH GLI¿FXOW WR FDWFK KDYH EHHQ VROYHG VXFFHVVIXOO\ E\ WKH VDPH QXPHULF VFKHPH LQ >@ 2Q WKH RWKHU KDQG UREXVW QXPHULFDO VWDELOLW\ LV VXFFHHGHG E\ PDNLQJ WKH UDWLR Δt / Δx2 _{≤ 0.5 DV ORZ DV SRVVLEOH 7KH SDUDPHWHUV XVHG}

LQ WKH PRGHO IRU V\VWHP     DUH OLVWHG LQ 7DEOH ,, 7KH VLPXODWLRQ UHVXOWV DUH SUHVHQWHG LQ )LJXUHV    1RWH WKDW )LJXUH  UHSUHVHQWV WKH EHQGLQJ VWUDLQ RI WKH EHDP QHDU WKH IUHH HQG SRLQW x = L 'HFRXSOLQJ EHWZHHQ WKH ULJLG FRRUGLQDWHV DQG ÀH[LEOH RQHV LV VXFFHHGHG E\ FRQWURO ODZV DQG LV REVHUYHG GXULQJ VLPXODWLRQV )RU θ1DQG ˙θ1 VHH

)LJXUHV    FRQYHUJLQJ SURSHUWLHV WZR H[SRQHQWLDO GH FD\LQJ PRGHV IRU GHULYDWLYHV FDQ EH DGMXVWHG LQGHSHQGHQWO\ ZLWK k1, k2 7KHUHIRUH WKH ¨θ1 FDQ EH XVHG DW ERXQGDU\

FRQGLWLRQ   E\ H[SRQHQWLDO GHFD\LQJ WLPH KLVWRU\ DQG DOVR WR VLPSOLI\ WKH FORVHGORRS V\VWHP )DVW FRQYHUJHQFH WR ]HUR IRU REVHUYHU HUURU UDWH LV TXLWH VDWLVIDFWRU\ LQ )LJXUH  )LQDOO\ VPRRWK WLPH KLVWRULHV RI DOO YDULDEOHV RI LQWHUHVW ZLWKRXW RYHUVKRRW VKRZ WKH HIIHFWLYHQHVV RI WKH FRQWUROOHU SHUIRUPDQFH ZLWK UHODWLYHO\ ORZ FRQWURO HQHUJ\

9, &21&/86,216

7KH PRVW FRPSOH[ EHDP PRGHO LV VLPSOL¿HG DQG EHFDPH HDV\ WR DQDO\]H DQG WR FRQWURO $OVR ZH GR QRW XVH DQ\ GDPSLQJ WHUP LQ WKH V\VWHP PRGHO ,Q WKLV SDSHU ZH FRQVLGHU WKH VHW SRLQW FRQWURO RI D URWDWLQJ VKHDU EHDP :H DVVXPHG WKDW WKH VKHDU EHDP LV IUHH WR URWDWH RQ WKH KRUL]RQWDO SODQH ,Q WKLV UHVHDUFK WKH V\VWHP HTXDWLRQV LV ¿UVW WUDQVIRUPHG LQWR DQRWKHU VHW RI HTXDWLRQV ZKLFK KDV ZHOONQRZQ VWDELOLW\ SURSHUWLHV E\ XVLQJ DQ LQYHUWLEOH 9ROWHUUD VWDWH WUDQVIRUPDWLRQ :H DOVR REWDLQHG WKH NHUQHO

      í í í í í í í í   8W>PVHF@ WLPH>VHF@

)LJ  )OH[XUDO 9HORFLW\ DW WKH HQG RI WKH EHDP

            [ í 8>P@ WLPH>VHF@ )LJ  'HÀHFWLRQ DW WKH HQG RI WKH EHDP       í        8[[>P@ WLPH>VHF@ )LJ  %HQGLQJ VWUDLQ RI WKH EHDP               θ >UDGLDQ@ WLPH>VHF@ )LJ  -RLQW $QJOH RI WKH EHDP               θ >UDGLDQVHF@ WLPH>VHF@

)LJ  -RLQW $QJXODU 9HORFLW\ RI WKH EHDP

      í           τ >1P@ WLPH>VHF@ )LJ  &RQWURO 7RUTXH



              2EVHUYHUHUURUUDWH>PVHF@ WLPH>VHF@ )LJ  2EVHUYHU (UURU 5DWH

IXQFWLRQ RI VXFK D WUDQVIRUPDWLRQ DQDO\WLFDOO\ E\ WKH YLUWXH RI SURSRVHG GHVLJQ 7KLV SURFHVV LV DOVR YHU\ HI¿FLHQW WR LPSURYH WKH REVHUYHU G\QDPLFV WKDW ZLOO JLYH WKH UHTXLUHG ERXQGDU\ FRQWURO ODZV 2XU VLPXODWLRQ UHVXOWV VKRZ WKDW WKH SURSRVHG FRQWURO VFKHPH LV HIIHFWLYH LQ ERWK DFKLHYLQJ FRUUHFW RULHQWDWLRQ DQG LQ VXSSUHVVLRQ RI ÀH[XUDO YLEUDWLRQV

$33(1',;

1RWH WKDW WKH NHUQHO H[SUHVVLRQ JLYHQ E\   LV REWDLQHG E\ FKDQJLQJ YDULDEOHV WR KDYH KRPRJHQRXV K\SHUEROLF NHUQHO 3'( LQ OLHX RI   ZLWK DSSURSULDWH ERXQGDU\ FRQGLWLRQV DV IROORZV r(x, y) = k(x, y) + m sinh(cx − cy)  r_yy− r_xx+ f r = 0  r(x, x) = −fx/2  r(0, 0) = 0  r(x, 0) = k(x, 0) + m sinh(cx)  r_y(x, 0) = −c · m cosh(cx)  ZKHUH m= −−1.5_{/f DQG WKH WHUPV WKDW SURGXFH E\ WKH}

VHFRQG RUGHU VSDWLDO GHULYDWLYHV FDQFHOOHG HDFK RWKHU %\ SHUIRUPLQJ LQWHJUDOV LQ   WKDW FRQWDLQV %HVVHO IXQFWLRQV J₀(·) DQG J₁(·) DQG DIWHU VRPH OHQJWK\ FDOFXODWLRQV ZH REWDLQ WKH IROORZLQJ H[SOLFLW IRUP IRU WKH NHUQHO k(x, y)

k(x, y) = 0.5[h(0) + h(L)] +(c.m/4)(sinh(cx) + cosh(cx)) . 1 c2+ f exp(−y  c2+ f) +1₄(sinh(cx) + cosh(cx)) .(m − 0.81039)[exp(−yc2+ f) − exp(−cy)] −0.62521 x(1 − cos(y √ f)) 2(m − 0.81039) +_{2(m − 0.81039)}0.62521 fy [1 − J0(  (y2_{− x}2_)f)] −m sinh(cx − cy)  5()(5(1&(6

>@ / 0HLURYLWFK )XQGDPHQWDOV RI 9LEUDWLRQV 1HZ <RUN 0F*UDZ+LOO 

>@ + %DUXK $QDO\WLFDO '\QDPLFV %RVWRQ 0F*UDZ+LOO  >@ 0 'R×JDQ < ,VWHIDQRSXORV ¶2SWLPDO 1RQOLQHDU &RQWUROOHU 'HVLJQ

IRU )OH[LEOH 5RERW 0DQLSXODWRUV ZLWK $GDSWLYH ,QWHUQDO 0RGHO¶ ,(7 &RQWURO 7KHRU\ DQG $SSOLFDWLRQV    

>@ -8 .LP DQG < 5HQDUG\ µ%RXQGDU\ FRQWURO RI WKH 7LPRVKHQNR EHDP¶ 6,$0 - RI &RQWURO DQG 2SWLPL]DWLRQ YRO  SS  

>@ )< :DQJ DQG < *DR  $GYDQFHG 6WXGLHV RI )OH[LEOH 5RERWLF

0DQLSXODWRUV 1HZ -HUVH\ :RUOG 6FLHQWL¿F 

>@ = + /XR % = *XR DQG Ž2 0RUJŽXO 6WDELOLW\ DQG 6WDELOL]DWLRQ RI

,Q¿QLWH 'LPHQVLRQDO 6\VWHPV ZLWK $SSOLFDWLRQV /RQGRQ 6SULQJHU

9HUODJ 

>@ Ž2 0RUJŽXO µ2ULHQWDWLRQ DQG 6WDELOL]DWLRQ RI D )OH[LEOH %HDP $W WDFKHG WR D 5LJLG %RG\ 3ODQDU 0RWLRQ¶ ,((( 7UDQVDFWLRQV RQ

$XWRPDWLF &RQWURO YRO 1R SS 

>@ Ž2 0RUJŽXO µ'\QDPLF ERXQGDU\ FRQWURO RI WKH 7LPRVKHQNR EHDP¶

$XWRPDWLFD YRO 1R SS 

>@ % = *XR µ5LHV] %DVLV 3URSHUW\ DQG ([SRQHQWLDO 6WDELOLW\ RI &RQ WUROOHG (XOHU%HUQRXOOL %HDP (TXDWLRQV ZLWK 9DULDEOH &RHI¿FLHQWV¶

6,$0 - RI &RQWURO DQG 2SWLPL]DWLRQ YRO 1R SS



>@ $ 6P\VKO\DHY DQG 0 .UVWLF µ&ORVHGIRUP ERXQGDU\ VWDWH IHHG EDFNV IRU D FODVV RI ' SDUWLDO LQWHJURGLIIHUHQWLDO HTXDWLRQV¶ ,(((

7UDQVDFWLRQV RQ $XWRPDWLF &RQWURO YRO 1R SS



>@ $ 6P\VKO\DHY DQG 0 .UVWLF µ%DFNVWHSSLQJ REVHUYHUV IRU D FODVV RI SDUDEROLF 3'(V¶ 6\VWHPV DQG &RQWURO /HWWHUV YRO SS 

>@ 0 .UVWLF $ $ 6LUDQRVLDQ DQG $ 6P\VKO\DHY  µ%DFNVWHSSLQJ ERXQGDU\ FRQWUROOHUV DQG REVHUYHUV IRU WKH VOHQGHU 7LPRVKHQNR EHDP3DUW ,'HVLJQ¶ 3URFHHGLQJV RI WKH $PHULFDQ &RQWURO &RQIHU

HQFH SS  0LQQHDSROLV 

>@ 0 .UVWLF % = *XR $ %DORJK DQG $ 6P\VKO\DHY  µ&RQWURO RI D WLSIRUFH GHVWDELOL]HG VKHDU EHDP E\ REVHUYHUEDVHG ERXQGDU\ IHHGEDFN¶ 6,$0 - RI &RQWURO DQG 2SWLPL]DWLRQ YRO 1R SS 

>@ ' 3RUWHU DQG '6* 6WLUOLQJ ,QWHJUDO (TXDWLRQV &DPEULGJH &DP EULGJH 8QLYHUVLW\ 3UHVV 

>@ ** +DVWLQJV DQG :-%RRNµ$ /LQHDU '\QDPLF 0RGHO IRU )OH[LEOH 5RERWLF 0DQLSXODWRUV¶,((( &RQWURO 6\VWHP 0DJD]LQH YRO SS  

>@ -1 5HGG\ $Q ,QWURGXFWLRQ WR WKH )LQLWH (OHPHQW 0HWKRG 1HZ <RUN 0F*UDZ+LOO 

>@ / 6LHYHUV 0 - %DODV $ 9 )ORWRZ µ0RGHOOLQJ RI :HE FRQYH\DQFH V\VWHPV IRU PXOWLYDULDEOH FRQWURO¶ ,((( 7UDQVDFWLRQV RQ $XWRPDWLF

&RQWURO YRO 1R SS 

>@ 0HLURYLWFK / $QDO\WLFDO 0HWKRGV LQ 9LEUDWLRQV /RQ GRQ0DF0LOODQ 

>@ / 'HEQDWK 1RQOLQHDU 3'(V IRU VFLHQWLVWV DQG HQJLQHHUV %RVWRQ %LUNKDXVHU 

>@ 0 'R×JDQ µ2SWLPDO 1RQOLQHDU &RQWUROOHU 'HVLJQ IRU )OH[LEOH 5RERW 0DQLSXODWRUV¶ 3K' 7KHVLV %R×JD]LFÜL 8QLYHUVLW\ ,VWDQEXO -DQ 

>@ 16 $EK\DQNDU (. +DOO ,, 69 +DQDJXG µ&KDRWLF 9LEUDWLRQV RI %HDPV1XPHULFDO 6ROXWLRQ RI 3DUWLDO 'LIIHUHQWLDO (TXDWLRQV¶$60(

-RXUQDO RI $SSOLHG 0HFKDQLFV YRO SS