7\UHDQG9HKLFOH
'\QDPLFV
+DQV%3DFHMND
3URIHVVRU(PHULWXV
'HOIW8QLYHUVLW\RI7HFKQRORJ\
&RQVXOWDQW712$XWRPRWLYH'HOIW
7KH1HWKHUODQGV
&KDSWHU
7<5(&+$5$&7(5,67,&6$1'
9(+,&/(+$1'/,1*$1'67$%,/,7<
,QWURGXFWLRQ
7KLV FKDSWHU LV PHDQW WR VHUYH DV DQ LQWURGXFWLRQ WR YHKLFOH G\QDPLFV ZLWK
HPSKDVLVRQWKHLQIOXHQFHRIW\UHSURSHUWLHV6WHDG\VWDWHFRUQHULQJEHKDYLRXU
RI VLPSOH DXWRPRELOH PRGHOV DQG WKH WUDQVLHQW PRWLRQ DIWHU VPDOO DQG ODUJH
VWHHULQJLQSXWVDQGRWKHUGLVWXUEDQFHVZLOOEHGLVFXVVHG7KHHIIHFWVRIYDULRXV
VKDSHIDFWRUVRIW\UHFKDUDFWHULVWLFVFI)LJRQYHKLFOHKDQGOLQJSURSHUWLHV
ZLOOEHDQDO\VHG7KHVORSHRIWKHVLGHIRUFH)\YVVOLSDQJOH QHDUWKHRULJLQWKH
FRUQHULQJRUVLGHVOLSVWLIIQHVVLVWKHGHWHUPLQLQJSDUDPHWHUIRUWKHEDVLFOLQHDU
KDQGOLQJDQGVWDELOLW\EHKDYLRXURIDXWRPRELOHV7KHSRVVLEOHRIIVHWRIWKHW\UH
FKDUDFWHULVWLFVZLWKUHVSHFWWRWKHLURULJLQVPD\EHUHVSRQVLEOHIRUWKHRFFXUUHQFH
RIWKHVRFDOOHGW\UHSXOOSKHQRPHQRQ7KHIXUWKHUQRQOLQHDUVKDSHRIWKHVLGH
RUFRUQHULQJIRUFHFKDUDFWHULVWLFJRYHUQVWKHKDQGOLQJDQGVWDELOLW\SURSHUWLHV
RIWKHYHKLFOHDWKLJKHUODWHUDODFFHOHUDWLRQV7KHORDGGHSHQGHQF\RIWKHFXUYHV
QRWDEO\WKHQRQOLQHDUUHODWLRQVKLSRIFRUQHULQJVWLIIQHVVZLWKW\UHQRUPDOORDG
KDVDFRQVLGHUDEOHHIIHFWRQWKHKDQGOLQJFKDUDFWHULVWLFRIWKHFDU)RUWKHTXDVL
)
\
&
)
)
\SHDN
0
]
0
]
)
]
]
9
)
[
)
\
[
\
)LJ &KDUDFWHULVWLFVKDSHIDFWRUVLQGLFDWHG E\ SRLQWVDQGVKDGHGDUHDVRIW\UHRUD[OH
FKDFWHULVWLFVWKDWPD\LQIOXHQFHYHKLFOHKDQGOLQJDQGVWDELOLW\SURSHUWLHV6OLSDQJOHDQG
IRUFHDQGPRPHQWSRVLWLYHGLUHFWLRQVFI$SS
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
VWHDG\VWDWHKDQGOLQJDQDO\VLVVLPSOHVLQJOHWUDFNWZRZKHHOYHKLFOHPRGHOV
ZLOO EH XVHG )URQW DQG UHDU D[OH HIIHFWLYH VLGH IRUFH FKDUDFWHULVWLFV DUH
LQWURGXFHGWRUHSUHVHQWHIIHFWVWKDWUHVXOWIURPVXVSHQVLRQDQGVWHHULQJV\VWHP
GHVLJQIDFWRUVVXFKDVVWHHULQJFRPSOLDQFHUROOVWHHUDQGODWHUDOORDGWUDQVIHU
$OVRWKHHIIHFWRISRVVLEO\DSSOLHGPRGHUDWHEUDNLQJDQGGULYLQJIRUFHVPD\EH
LQFRUSRUDWHGLQWKHHIIHFWLYHFKDUDFWHULVWLFV/DUJHEUDNLQJIRUFHVPD\UHVXOWLQ
ZKHHOORFNDQGSRVVLEO\ODUJHGHYLDWLRQVIURPWKHXQGLVWXUEHGSDWK7KHPRWLRQ
UHVXOWLQJIURPZKHHOORFNZLOOEHGHDOWZLWKLQDQDSSOLFDWLRQRIWKHWKHRU\RID
VLPSOHSK\VLFDOW\UHPRGHOLQ&KDSWHUWKHEUXVKPRGHO7KHDSSOLFDWLRQRI
WKHKDQGOLQJDQGVWDELOLW\WKHRU\WRWKHG\QDPLFVRIKHDY\WUXFNVZLOODOVREH
EULHIO\GHDOWZLWKLQWKHSUHVHQWFKDSWHU6SHFLDODWWHQWLRQZLOOEHJLYHQWRWKH
SKHQRPHQRQ RI RVFLOODWRU\ LQVWDELOLW\ WKDW PD\ VKRZ XS ZLWK WKH FDU WUDLOHU
FRPELQDWLRQ
:KHQWKHZDYHOHQJWKRIDQRVFLOODWRU\PRWLRQRIWKHYHKLFOHWKDWPD\DULVH
IURPURDGXQHYHQQHVVEUDNHWRUTXHIOXFWXDWLRQVZKHHOXQEDODQFHRULQVWDELOLW\
VKLPP\LVVPDOOHUWKDQVD\PDQRQVWHDG\VWDWHRUWUDQVLHQWGHVFULSWLRQRI
W\UHUHVSRQVHLVQHHGHGWRSURSHUO\DQDO\VHWKHSKHQRPHQRQ,Q&KDSWHUV
WKHVHPDWWHUVZLOOEHDGGUHVVHG$SSOLFDWLRQVGHPRQVWUDWHWKHXVHRIWUDQVLHQW
DQG RVFLOODWRU\ W\UH PRGHOV DQG SURYLGH LQVLJKW LQWR WKH YHKLFOH G\QDPLFV
LQYROYHG&KDSWHULVVSHFLDOO\GHYRWHGWRWKHDQDO\VLVRIPRWRUF\FOHFRUQHULQJ
EHKDYLRXUDQGVWDELOLW\
7\UHDQG$[OH&KDUDFWHULVWLFV
7\UHFKDUDFWHULVWLFVDUHRIFUXFLDOLPSRUWDQFHIRUWKHG\QDPLFEHKDYLRXURIWKH
URDGYHKLFOH,QWKLVVHFWLRQDQLQWURGXFWLRQLVJLYHQWRWKHEDVLFDVSHFWVRIWKH
IRUFHDQGPRPHQWJHQHUDWLQJSURSHUWLHVRIWKHSQHXPDWLFW\UH%RWKWKHSXUHDQG
FRPELQHG VOLS FKDUDFWHULVWLFV RI WKH W\UH DUH GLVFXVVHG DQG W\SLFDO IHDWXUHV
SUHVHQWHG)LQDOO\WKHVRFDOOHGHIIHFWLYHD[OHFKDUDFWHULVWLFVDUHGHULYHGIURP
WKHLQGLYLGXDOW\UHFKDUDFWHULVWLFVDQGWKHUHOHYDQWSURSHUWLHVRIWKHVXVSHQVLRQ
DQGVWHHULQJV\VWHP
,QWURGXFWLRQWR7\UH&KDUDFWHULVWLFV
7KHXSULJKWZKHHOUROOLQJIUHHO\WKDWLVZLWKRXWDSSO\LQJDGULYLQJWRUTXHRYHU
DIODWOHYHOURDGVXUIDFHDORQJDVWUDLJKWOLQHDW]HURVLGHVOLSPD\EHGHILQHGDV
WKHVWDUWLQJVLWXDWLRQZLWKDOOFRPSRQHQWVRIVOLSHTXDOWR]HUR$UHODWLYHO\VPDOO
SXOOLQJIRUFHLVQHHGHGWRRYHUFRPHWKHW\UHUROOLQJUHVLVWDQFHDQGDVLGHIRUFH
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
DQGVHOIDOLJQLQJWRUTXHPD\RFFXUDVDUHVXOWRIWKHQRWFRPSOHWHO\V\PPHWULF
VWUXFWXUHRIWKHW\UH:KHQWKHZKHHOPRWLRQGHYLDWHVIURPWKLVE\GHILQLWLRQ]HUR
VOLSFRQGLWLRQZKHHOVOLSRFFXUVWKDWLVDFFRPSDQLHGE\DEXLOGXSRIDGGLWLRQDO
W\UHGHIRUPDWLRQDQGSRVVLEO\SDUWLDOVOLGLQJLQWKHFRQWDFWSDWFK$VDUHVXOW
DGGLWLRQDO KRUL]RQWDO IRUFHV DQG WKH DOLJQLQJ WRUTXH DUH JHQHUDWHG 7KH
PHFKDQLVPUHVSRQVLEOHIRUWKLVLVWUHDWHGLQGHWDLOLQWKHVXEVHTXHQWFKDSWHUV)RU
QRZZHZLOOVXIILFHZLWKVRPHLPSRUWDQWH[SHULPHQWDOREVHUYDWLRQVDQGGHILQH
WKH YDULRXV VOLS TXDQWLWLHV WKDW VHUYH DV LQSXWV LQWR WKH W\UH V\VWHP DQG WKH
PRPHQWDQGIRUFHVWKDWDUHWKHRXWSXWTXDQWLWLHVSRVLWLYHGLUHFWLRQVDFFRUGLQJ
WR)LJ6HYHUDODOWHUQDWLYHGHILQLWLRQV DUH LQXVHDVZHOO,Q$SSHQGL[
YDULRXVVLJQFRQYHQWLRQVRIVOLSFDPEHUDQGRXWSXWIRUFHVDQGPRPHQWVWRJHWKHU
ZLWKUHOHYDQWFKDUDFWHULVWLFVKDYHEHHQSUHVHQWHG
)RUWKHIUHHO\UROOLQJZKHHOWKHIRUZDUGVSHHG9[ORQJLWXGLQDOFRPSRQHQWRI
WKH WRWDO YHORFLW\ YHFWRU 9 RI WKH ZKHHO FHQWUH DQG WKH DQJXODU VSHHG RI
RFDQEHWDNHQIURPPHDVXUHPHQWV%\GLYLGLQJWKHVHWZRTXDQWLWLHV
UHYROXWLRQ
WKHVRFDOOHGHIIHFWLYHUROOLQJUDGLXVUHLVREWDLQHG
'
U
H
9
[
R
$OWKRXJKWKHHIIHFWLYHUDGLXVPD\EHGHILQHGDOVRIRUDEUDNHGRUGULYHQZKHHO
ZHUHVWULFWWKHGHILQLWLRQWRWKHFDVHRIIUHHUROOLQJ:KHQDWRUTXHLVDSSOLHG
DERXWWKHZKHHOVSLQD[LVDORQJLWXGLQDOVOLSDULVHVWKDWLVGHILQHGDVIROORZV
9
[
& U
H
' &
' & R
9
[
&
R
RDQGFRQVHTXHQWO\
7KHVLJQLVWDNHQVXFKWKDWIRUDSRVLWLYH DSRVLWLYHORQJLWXGLQDOIRUFH)[DULVHV
LVLQFUHDVHG
WKDWLVDGULYLQJIRUFH,QWKDWFDVHWKHZKHHODQJXODUYHORFLW\
! R 9[UH'XULQJEUDNLQJWKHIRUHDQG
ZLWKUHVSHFWWR
$W GULYLQJ RQ
DIW VOLS EHFRPHV QHJDWLYH $W ZKHHO ORFN REYLRXVO\
VOLSSHU\URDGV PD\DWWDLQYHU\ODUJHYDOXHV7ROLPLWWKHVOLSWRDPD[LPXP
HTXDO WR RQH LQ VRPH WH[WV WKH ORQJLWXGLQDO VOLS LV GHILQHG GLIIHUHQWO\ LQ WKH
7KLVZLOO
GULYLQJUDQJHRIVOLSLQWKHGHQRPLQDWRURI
QRWEHGRQHLQWKHSUHVHQWWH[W
RLVUHSODFHGE\
/DWHUDO ZKHHO VOLS LV GHILQHG DV WKH UDWLR RI WKH ODWHUDO DQG WKH IRUZDUG
YHORFLW\RIWKHZKHHO7KLVFRUUHVSRQGVWRPLQXVWKHWDQJHQWRIWKHVOLSDQJOH
)LJ$JDLQWKHVLJQRI KDVEHHQFKRVHQVXFKWKDWWKHVLGHIRUFHEHFRPHV
SRVLWLYHDWSRVLWLYHVOLSDQJOH
WDQ ' &
9
9
\
[
)
[
brake slip
0
5%
10%
20%
100%
0, 2
4
16O
slip
angle
)
\
16
O8
4
2
0
slip
angle
0
brake force )
[
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
side force
)
\
brake force
side force
O
O
0 8 16
α
slip angle
0 50 100 %
brake slip
κ
)LJ&RPELQHGVLGHIRUFHDQGEUDNHIRUFHFKDUDFWHULVWLFV
7KHWKLUGDQGODVWVOLSTXDQWLW\LVWKHVRFDOOHGVSLQZKLFKLVGXHWRURWDWLRQRI
WKHZKHHODERXWDQD[LVQRUPDOWRWKHURDG%RWKWKH\DZUDWHUHVXOWLQJLQSDWK
UHPDLQV]HURDQGWKHZKHHOFDPEHURULQFOLQDWLRQDQJOH RI
FXUYDWXUHZKHQ
WKHZKHHOSODQHDERXWWKH[D[LVFRQWULEXWHWRWKHVSLQ7KHFDPEHUDQJOHLV
GHILQHGSRVLWLYHZKHQORRNLQJIURPEHKLQGWKHZKHHOLVWLOWHGWRWKHULJKW,Q
&KDSWHUPRUHSUHFLVHGHILQLWLRQVRIWKHWKUHHFRPSRQHQWVRIZKHHOVOLSZLOOEH
JLYHQ7KHIRUFHV)[DQG)\DQGWKHDOLJQLQJWRUTXH0]DUHUHVXOWVRIWKHLQSXW
VOLS7KH\DUHIXQFWLRQVRIWKHVOLSFRPSRQHQWVDQGWKHZKHHOORDG)RUVWHDG\
VWDWHUHFWLOLQHDUPRWLRQVZHKDYHLQJHQHUDO
' )
)
[
[
)
)
' )
\
\
]
)
0
' 0
]
]
]
)
]
7KHYHUWLFDOORDG)]PD\EHFRQVLGHUHGDVDJLYHQTXDQWLW\WKDWUHVXOWVIURPWKH
QRUPDOGHIOHFWLRQRIWKHW\UH7KHIXQFWLRQVFDQEHREWDLQHGIURPPHDVXUHPHQWV
IRUDJLYHQVSHHGRIWUDYHODQGURDGDQGHQYLURQPHQWDOFRQGLWLRQV
)LJXUHVKRZVWKHDGRSWHGV\VWHPRID[HV[\]ZLWKDVVRFLDWHGSRVLWLYH
GLUHFWLRQVRIYHORFLWLHVDQGIRUFHVDQGPRPHQWV7KHH[FHSWLRQLVWKHYHUWLFDO
IRUFH)]DFWLQJIURPURDGWRW\UH)RUSUDFWLFDOUHDVRQVWKLVIRUFHLVGHILQHGWR
EHSRVLWLYHLQWKHXSZDUGGLUHFWLRQDQGWKXVHTXDOWRWKHQRUPDOORDGRIWKHW\UH
QRWSURYLGHGZLWKD\VXEVFULSWLVGHILQHGSRVLWLYHZLWKUHVSHFWWRWKH
$OVR
QHJDWLYH\D[LV1RWHWKDWWKHD[HVV\VWHPLVLQDFFRUGDQFHZLWK6$(VWDQGDUGV
6$(-H7KHVLJQRIWKHVOLSDQJOHKRZHYHULVFKRVHQRSSRVLWHZLWK
UHVSHFWWRWKH6$(GHILQLWLRQFI$SSHQGL[
,Q)LJW\SLFDOSXUHODWHUDO DQGORQJLWXGLQDO VOLSFKDUDFWHU
LVWLFVKDYHEHHQGHSLFWHGWRJHWKHUZLWKDQXPEHURIFRPELQHGVOLSFXUYHV7KH
FDPEHUDQJOH ZDVNHSWHTXDOWR]HUR:HGHILQHSXUHVOLSWREHWKHVLWXDWLRQ
ZKHQHLWKHUORQJLWXGLQDORUODWHUDOVOLSRFFXUVLQLVRODWLRQ7KHILJXUHLQGLFDWHV
WKDWDGURSLQIRUFHDULVHVZKHQWKHRWKHUVOLSFRPSRQHQWLVDGGHG7KHUHVXOWLQJ
VLWXDWLRQLVGHVLJQDWHGDVFRPELQHGVOLS7KHGHFUHDVHLQIRUFHFDQEHVLPSO\
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
&
)
)]R
[1/rad]
10
0
)\SHDN
)]R
1
0.6
0
car tyre
truck tyre
1
)] )]R
)] )]R
car tyre
truck tyre
)] )]R
1
)\
)]
1
0.5
dry
wet
truck tyre
32
64
88
32
64
88
km/h
0
o
10
)LJ 7\SLFDOFKDUDFWHULVWLFVIRUWKHQRUPDOLVHGFRUQHULQJVWLIIQHVVSHDNVLGHIRUFHDQGVLGH
IRUFHYVQRUPDOLVHGYHUWLFDOORDGDQGVOLSDQJOHUHVSHFWLYHO\)
]RLVWKHUDWHGORDG
H[SODLQHGE\UHDOLVLQJWKDWWKHWRWDOKRUL]RQWDOIULFWLRQDOIRUFH)FDQQRWH[FHHG
WKHPD[LPXPYDOXHUDGLXVRIµIULFWLRQFLUFOH¶ZKLFKLVGLFWDWHGE\WKHFXUUHQW
IULFWLRQFRHIILFLHQWDQGQRUPDOORDG/DWHULQ&KDSWHUWKLVEHFRPHVFOHDUZKHQ
FRQVLGHULQJWKHEHKDYLRXURIDVLPSOHSK\VLFDOW\UHPRGHO7KHGLDJUDPVLQFOXGH
WKHVLWXDWLRQZKHQWKHEUDNHVOLSUDWLRKDVILQDOO\DWWDLQHGWKHYDOXH
ZKLFKFRUUHVSRQGVWRZKHHOORFN
7KH VORSHV RI WKH SXUH VOLS FXUYHV DW YDQLVKLQJ VOLS DUH GHILQHG DV WKH
ORQJLWXGLQDO DQG ODWHUDO VOLS VWLIIQHVVHV UHVSHFWLYHO\ 7KH ORQJLWXGLQDO VOLS
VWLIIQHVVLVGHVLJQDWHGDV&) 7KHODWHUDOVOLSRUFRUQHULQJVWLIIQHVVRIWKHW\UH
GHQRWHGZLWK&) LVRQHRIWKHPRVWLPSRUWDQWSURSHUW\SDUDPHWHUVRIWKHW\UH
DQGLVFUXFLDOIRUWKHYHKLFOH¶VKDQGOLQJDQGVWDELOLW\SHUIRUPDQFH7KHVORSHRI
PLQXVWKHDOLJQLQJWRUTXHYHUVXVVOLSDQJOHFXUYH)LJDW]HURVOLSDQJOHLV
WHUPHGDVWKHDOLJQLQJVWLIIQHVVDQGLVGHQRWHGZLWK&0 7KHUDWLRRIPLQXVWKH
DOLJQLQJWRUTXHDQGWKHVLGHIRUFHLVWKHSQHXPDWLFWUDLOWLIZHQHJOHFWWKHVR
FDOOHGUHVLGXDOWRUTXHWREHGHDOWZLWKLQ&KDSWHU7KLVOHQJWKLVWKHGLVWDQFH
EHKLQGWKHFRQWDFWFHQWUHSURMHFWLRQRIZKHHOFHQWUHRQWRWKHJURXQGLQZKHHO
SODQHGLUHFWLRQWRWKHSRLQWZKHUHWKHUHVXOWLQJODWHUDOIRUFHDFWV7KHOLQHDULVHG
IRUFHDQGPRPHQWFKDUDFWHULVWLFVYDOLGDWVPDOOOHYHOVRIVOLSFDQEHUHSUHVHQWHG
E\WKHIROORZLQJH[SUHVVLRQVLQZKLFKWKHHIIHFWRIFDPEHUKDVEHHQLQFOXGHG
[
)
)
\
0
]
)
' &
' &
)
' &&
% &
)
% &
0
0
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
7KHVHHTXDWLRQVKDYHEHHQDUUDQJHGLQVXFKDZD\WKDWDOOWKHFRHIILFLHQWVWKH
IRUFH DQG PRPHQW VOLS DQG FDPEHU VWLIIQHVVHV EHFRPH SRVLWLYH TXDQWLWLHV
H[FOXGLQJVRPHH[FHSWLRQDOFDVHV
,WLV RI LQWHUHVWWRQRWHWKDWWKHRUGHURIPDJQLWXGHRIWKHW\UHFRUQHULQJ
VWLIIQHVVUDQJHVIURPDERXWWRDERXWWLPHVWKHYHUWLFDOZKHHOORDGZKHQWKH
FRUQHULQJVWLIIQHVVLVH[SUHVVHGDVIRUFHSHUUDGLDQ7KHORZHUYDOXHKROGVIRUWKH
ROGHUELDVSO\W\UHFRQVWUXFWLRQDQGWKHODUJHUYDOXHIRUPRGHUQUDFLQJW\UHV7KH
ORQJLWXGLQDOVOLSVWLIIQHVVKDVEHHQW\SLFDOO\IRXQGWREHDERXWODUJHUWKDQ
WKHFRUQHULQJVWLIIQHVV7KHSQHXPDWLFWUDLOLVRIDQRUGHURIPDJQLWXGHHTXDOWR
DTXDUWHURIWKHFRQWDFWSDWFKOHQJWK
)RUWKHVLGHIRUFHZKLFKLVWKHPRUHLPSRUWDQWTXDQWLW\LQFRQQHFWLRQZLWK
DXWRPRELOHKDQGOLQJSURSHUWLHVDQXPEHURILQWHUHVWLQJGLDJUDPVKDYHEHHQ
SUHVHQWHGLQ)LJ7KHVHFKDUDFWHULVWLFVDUHW\SLFDOIRUWUXFNDQGFDUW\UHVDQG
DUHEDVHGRQH[SHULPHQWVFRQGXFWHGDWWKH8QLYHUVLW\RI0LFKLJDQ7UDQVSRUWD
WLRQ5HVHDUFK,QVWLWXWH8075,IRUPHUO\+65,FI5HI6HJHOHWDO
7KHFDUW\UHFRUQHULQJVWLIIQHVVGDWDVWHPVIURPQHZHUILQGLQJV,WLVVHHQWKDW
WKH FRUQHULQJ VWLIIQHVV FKDQJHV LQ D OHVV WKDQ SURSRUWLRQDO IDVKLRQ ZLWK WKH
QRUPDOZKHHOORDG7KHPD[LPXPQRUPDOLVHGVLGHIRUFH)\SHDN)]DSSHDUVWR
GHFUHDVHZLWKLQFUHDVLQJZKHHOORDG0DUNHGGLIIHUHQFHVLQOHYHODQGVORSHRFFXU
IRUWKHFDUDQGWUXFNW\UHFXUYHVDOVRZKHQQRUPDOLVHGZLWKUHVSHFWWRWKHUDWHG
RU QRPLQDO ORDG 7KH FRUQHULQJ IRUFH YV VOLS DQJOH FKDUDFWHULVWLF VKRZQ DW
GLIIHUHQWVSHHGVDQGURDGFRQGLWLRQVLQGLFDWHWKDWWKHVORSHDW]HURVOLSDQJOHLV
QRWRUKDUGO\DIIHFWHGE\WKHOHYHORIVSHHGDQGE\WKHFRQGLWLRQZHWRUGU\7KH
SHDNIRUFHOHYHOVKRZVRQO\OLWWOHYDULDWLRQLIWKHURDGLVGU\2QDZHWURDGD
PRUH SURQRXQFHG SHDN RFFXUV DQG WKH SHDN OHYHO GURSV VLJQLILFDQWO\ ZLWK
LQFUHDVLQJVSHHG
&XUYHVZKLFKH[KLELWDVKDSHOLNHWKHVLGHIRUFHFKDUDFWHULVWLFVRI)LJ
FDQEHUHSUHVHQWHGE\DPDWKHPDWLFDOIRUPXODWKDWKDVEHFRPHNQRZQE\WKH
QDPHµ0DJLF)RUPXOD¶$IXOOWUHDWPHQWRIWKHHPSLULFDOW\UHPRGHODVVRFLDWHG
ZLWKWKLVIRUPXODLVJLYHQLQ&KDSWHU)RUQRZZHFDQVXIILFHZLWKVKRZLQJWKH
EDVLFH[SUHVVLRQVIRUWKHVLGHIRUFHDQGWKHFRUQHULQJVWLIIQHVV
)\ 'VLQ>&DUFWDQ^% (% DUFWDQ% `@
ZLWKVWLIIQHVVIDFWRU
% &) &'
SHDNIDFWRU
' )] )\SHDN
DQGFRUQHULQJVWLIIQHVV
&) %&' FVLQ^DUFWDQ)]F`