7\UHDQG9HKLFOH '\QDPLFV +DQV%3DFHMND
3URIHVVRU(PHULWXV 'HOIW8QLYHUVLW\RI7HFKQRORJ\ &RQVXOWDQW712$XWRPRWLYH...
194 downloads
1592 Views
15MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
7\UHDQG9HKLFOH '\QDPLFV +DQV%3DFHMND
3URIHVVRU(PHULWXV 'HOIW8QLYHUVLW\RI7HFKQRORJ\ &RQVXOWDQW712$XWRPRWLYH'HOIW 7KH1HWKHUODQGV
&KDSWHU 7<5(&+$5$&7(5,67,&6$1' 9(+,&/(+$1'/,1*$1'67$%,/,7<
,QWURGXFWLRQ
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
)
9
\
)
&
)
\SHDN
)
[
0
]
0
)
\
]
]
)
[ \
]
)LJ &KDUDFWHULVWLF VKDSH IDFWRUV LQGLFDWHG E\ SRLQWV DQG VKDGHG DUHDV RI W\UH RUD[OH FKDFWHULVWLFVWKDWPD\LQIOXHQFHYHKLFOHKDQGOLQJDQGVWDELOLW\SURSHUWLHV6OLSDQJOHDQG IRUFHDQGPRPHQWSRVLWLYHGLUHFWLRQVFI$SS
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
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
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<
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
R
FDQEHWDNHQIURPPHDVXUHPHQWV%\GLYLGLQJWKHVHWZRTXDQWLWLHV
WKHVRFDOOHGHIIHFWLYHUROOLQJUDGLXVUHLVREWDLQHG
U '
9
[
H
R
$OWKRXJKWKHHIIHFWLYHUDGLXVPD\EHGHILQHGDOVRIRUDEUDNHGRUGULYHQZKHHO ZHUHVWULFWWKHGHILQLWLRQWRWKHFDVHRIIUHHUROOLQJ :KHQ D WRUTXHLVDSSOLHG DERXWWKHZKHHOVSLQD[LVDORQJLWXGLQDOVOLSDULVHVWKDWLVGHILQHGDVIROORZV
'&
9 &U 9 [
H
'&
R
&
[
R
7KHVLJQLVWDNHQVXFKWKDWIRUDSRVLWLYH
DSRVLWLYHORQJLWXGLQDOIRUFH)[DULVHV
WKDWLVDGULYLQJIRUFH,QWKDWFDVHWKHZKHHODQJXODUYHORFLW\ ZLWKUHVSHFWWR
DQGFRQVHTXHQWO\ R
!
R
LVLQFUHDVHG
9[ UH'XULQJEUDNLQJWKHIRUHDQG
DIW VOLS EHFRPHV QHJDWLYH $W ZKHHO ORFN REYLRXVO\
$W GULYLQJ RQ
VOLSSHU\URDGV PD\DWWDLQYHU\ODUJHYDOXHV7ROLPLWWKHVOLSWRDPD[LPXP HTXDO WR RQH LQ VRPH WH[WV WKH ORQJLWXGLQDO VOLS LV GHILQHG GLIIHUHQWO\ LQ WKH GULYLQJUDQJHRIVOLSLQWKHGHQRPLQDWRURI
R
LVUHSODFHGE\
7KLVZLOO
QRWEHGRQHLQWKHSUHVHQWWH[W /DWHUDO ZKHHO VOLS LV GHILQHG DV WKH UDWLR RI WKH ODWHUDO DQG WKH IRUZDUG YHORFLW\RIWKHZKHHO7KLVFRUUHVSRQGVWRPLQXVWKHWDQJHQWRIWKHVOLSDQJOH )LJ $JDLQWKHVLJQRI KDVEHHQFKRVHQVXFKWKDWWKHVLGHIRUFHEHFRPHV SRVLWLYHDWSRVLWLYHVOLSDQJOH
WDQ
'&
9 9
\
[
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
side force
side force
brake force
)\
) [
brake slip
) \
0, 2
O
8
0 5%
4
10%
16
20%
slip angle
16 O
4 2
slip angle
100%
0 O
0
O
8
16
slip angle
0
α
50
100 %
brake slip κ
0
brake force
) [
)LJ&RPELQHGVLGHIRUFHDQGEUDNHIRUFHFKDUDFWHULVWLFV
7KHWKLUGDQGODVWVOLSTXDQWLW\LVWKHVRFDOOHGVSLQZKLFKLVGXHWRURWDWLRQRI WKHZKHHODERXWDQD[LVQRUPDOWRWKHURDG%RWKWKH\DZUDWHUHVXOWLQJLQSDWK FXUYDWXUHZKHQ
UHPDLQV]HURDQGWKHZKHHOFDPEHURULQFOLQDWLRQDQJOH RI
WKH ZKHHO SODQH DERXW WKH [ D[LV FRQWULEXWH WR WKH VSLQ 7KH FDPEHU DQJOH LV GHILQHG SRVLWLYH ZKHQ ORRNLQJ IURP EHKLQG WKH ZKHHO LV WLOWHG WR WKH ULJKW ,Q &KDSWHUPRUHSUHFLVHGHILQLWLRQVRIWKHWKUHHFRPSRQHQWVRIZKHHOVOLSZLOOEH JLYHQ7KHIRUFHV)[DQG)\DQGWKHDOLJQLQJWRUTXH0]DUHUHVXOWVRIWKHLQSXW VOLS7KH\DUHIXQFWLRQVRIWKHVOLSFRPSRQHQWVDQGWKHZKHHOORDG)RUVWHDG\ VWDWHUHFWLOLQHDUPRWLRQVZHKDYHLQJHQHUDO
) ') [
[
)
]
) ') \
\
) 0 '0 ]
]
]
)
]
7KHYHUWLFDOORDG)]PD\EHFRQVLGHUHGDVDJLYHQTXDQWLW\WKDWUHVXOWVIURPWKH QRUPDOGHIOHFWLRQRIWKHW\UH7KHIXQFWLRQVFDQEHREWDLQHGIURPPHDVXUHPHQWV IRUDJLYHQVSHHGRIWUDYHODQGURDGDQGHQYLURQPHQWDOFRQGLWLRQV )LJXUHVKRZVWKHDGRSWHGV\VWHPRID[HV[\] ZLWKDVVRFLDWHGSRVLWLYH GLUHFWLRQV RIYHORFLWLHVDQGIRUFHVDQGPRPHQWV7KHH[FHSWLRQLVWKHYHUWLFDO IRUFH)]DFWLQJIURPURDGWRW\UH)RUSUDFWLFDOUHDVRQVWKLVIRUFHLVGHILQHGWR EHSRVLWLYHLQWKHXSZDUGGLUHFWLRQDQGWKXVHTXDOWRWKHQRUPDOORDGRIWKHW\UH $OVR
QRWSURYLGHGZLWKD\VXEVFULSW LVGHILQHGSRVLWLYHZLWKUHVSHFWWRWKH
QHJDWLYH\D[LV1RWHWKDWWKHD[HVV\VWHPLVLQDFFRUGDQFHZLWK6$(VWDQGDUGV 6$(-H 7KHVLJQRIWKHVOLSDQJOHKRZHYHULVFKRVHQRSSRVLWHZLWK UHVSHFWWRWKH6$(GHILQLWLRQFI$SSHQGL[ ,Q)LJW\SLFDOSXUHODWHUDO
DQGORQJLWXGLQDO
VOLSFKDUDFWHU
LVWLFVKDYHEHHQGHSLFWHGWRJHWKHUZLWKDQXPEHURIFRPELQHGVOLSFXUYHV7KH FDPEHUDQJOH
ZDVNHSWHTXDOWR]HUR:HGHILQHSXUHVOLSWREHWKHVLWXDWLRQ
ZKHQHLWKHUORQJLWXGLQDORUODWHUDOVOLSRFFXUVLQLVRODWLRQ7KHILJXUHLQGLFDWHV WKDWDGURSLQIRUFHDULVHVZKHQWKHRWKHUVOLSFRPSRQHQWLVDGGHG7KHUHVXOWLQJ VLWXDWLRQ LV GHVLJQDWHGDVFRPELQHGVOLS7KHGHFUHDVHLQIRUFHFDQEHVLPSO\
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
&)
)\
)
car tyre
]R
)
]
[1/rad] 10
1
dry
32 64 88
truck tyre
0
1
32 64 88 km/h
) ) ]
]R
wet
)
0.5
\SHDN
)
]R
truck tyre
1
car tyre
truck tyre
0.6 0
) ) ]
1
0
10
o
]R
)LJ 7\SLFDOFKDUDFWHULVWLFVIRUWKHQRUPDOLVHGFRUQHULQJVWLIIQHVVSHDNVLGHIRUFHDQGVLGH
)]RLVWKHUDWHGORDG
IRUFHYVQRUPDOLVHGYHUWLFDOORDGDQGVOLSDQJOHUHVSHFWLYHO\
H[SODLQHGE\UHDOLVLQJWKDWWKHWRWDOKRUL]RQWDOIULFWLRQDOIRUFH)FDQQRWH[FHHG WKHPD[LPXPYDOXHUDGLXVRIµIULFWLRQFLUFOH¶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¶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
) '& ) ' & %& 0 ' && % & [
)
\
)
]
)
0
0
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
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µ0DJLF)RUPXOD¶$IXOOWUHDWPHQWRIWKHHPSLULFDOW\UHPRGHODVVRFLDWHG ZLWKWKLVIRUPXODLVJLYHQLQ&KDSWHU)RUQRZZHFDQVXIILFHZLWKVKRZLQJWKH EDVLFH[SUHVVLRQVIRUWKHVLGHIRUFHDQGWKHFRUQHULQJVWLIIQHVV )\
'VLQ>&DUFWDQ^%
(%
DUFWDQ%
`@
ZLWKVWLIIQHVVIDFWRU %
&) &'
SHDNIDFWRU '
)]
)\SHDN
DQGFRUQHULQJVWLIIQHVV &)
%&'
FVLQ^DUFWDQ)]F `
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
7KHVKDSHIDFWRUV&DQG(DVZHOODVWKHSDUDPHWHUVFDQGFDQGWKHIULFWLRQ FRHIILFLHQW SRVVLEO\ GHSHQGLQJ RQ WKH YHUWLFDO ORDG DQG VSHHG PD\ EH HVWLPDWHGRUGHWHUPLQHGWKURXJKUHJUHVVLRQWHFKQLTXHV
(IIHFWLYH$[OH&RUQHULQJ&KDUDFWHULVWLFV
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µZRUNLQJFXUYHV¶DVWKHVHZHUHUHIHUUHG WRE\WKHP9n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
$
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
DDQGE
D
,Q D FXUYH WKH YHKLFOH ERG\ UROOV DERXW WKH UROO D[LV 7KH ORFDWLRQ DQG
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
D
roll axis \
0
2 F
,
P
<
2
[] ; K
king-pin 1 2
F
K
2
V
2
\
1 $
F
D
2
1
K1
) ]
) ] E
X
1R
2
2
2L
1
H ) \
V ) \
Y 0
=
U
1R
1L
wheel plane
O D
) \
D
1
) \
1
1 1
) ]
1L
1
D
]
9 [
1
)LJ9HKLFOHPRGHOVKRZLQJWKUHHGHJUHHVRIIUHHGRPODWHUDO\DZDQGUROO
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
ZKLFKUHVXOW
IURPVXVSHQVLRQVSULQJVDQGDQWLUROOEDUV 7KHIRUHDQGDIWSRVLWLRQRIWKHFHQWUHRIJUDYLW\RIWKHERG\LVGHILQHGE\D DQGELWVKHLJKWIROORZVIURPWKHGLVWDQFHK¶WRWKHUROOD[LV7KHERG\PDVVLV GHQRWHGE\PDQGWKHPRPHQWVRILQHUWLDZLWKUHVSHFWWRWKHFHQWUHRIPDVVDQG KRUL]RQWDODQGYHUWLFDOD[HVE\,[,]DQG,[]7KHVHODWWHUTXDQWLWLHVZLOOEHQHHGHG LQDODWHUSKDVHZKHQWKHGLIIHUHQWLDOHTXDWLRQVRIPRWLRQDUHHVWDEOLVKHG7KH XQVSUXQJ PDVVHV ZLOO EH QHJOHFWHG RU WKH\ PD\ EH LQFOXGHG DV SRLQW PDVVHV DWWDFKHGWRWKHUROOD[LVDQGWKXVPDNHWKHPSDUWRIWKHVSUXQJPDVVWKDWLVWKH YHKLFOHERG\ )XUWKHUPRUHWKHPRGHOIHDWXUHVWRUVLRQDOVSULQJVDURXQGWKHVWHHULQJD[HV 7KHNLQJSLQLVSRVLWLRQHG DWDVPDOOFDVWHUDQJOHWKDWJLYHVULVHWRWKHFDVWHU OHQJWKHDVLQGLFDWHGLQWKHGUDZLQJ7KHWRWDOVWHHULQJWRUVLRQDOVWLIIQHVVOHIW SOXVULJKWLVGHQRWHGE\F
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
(IIHFWLYH$[OH&RUQHULQJ6WLIIQHVV
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
& '& % & '& % )
)
)
) )
R
) R
]
]
ZKHUHWKHLQFUHPHQWRIWKHZKHHOYHUWLFDOORDGLVGHQRWHGE\ RIWKHFRHIILFLHQWYHUVXVORDGFXUYHVDW)]
)]DQGWKHVORSHV
)]RDUHUHSUHVHQWHGE\
:KHQ WKH YHKLFOH PRYHV VWHDGLO\ DURXQG D FLUFXODU SDWK D FHQWULSHWDO DFFHOHUDWLRQD\RFFXUVDQGDFHQWULIXJDOIRUFH.
PD\FDQEHVDLGWRDFWRQWKH
YHKLFOHERG\DWWKHFHQWUHRIJUDYLW\LQWKHRSSRVLWHGLUHFWLRQ7KHERG\UROODQJOH WKDWLVDVVXPHGWREHVPDOOLVFDOFXODWHGE\GLYLGLQJWKHPRPHQWDERXWWKH UROOD[LVE\WKHDSSDUHQWUROOVWLIIQHVVZKLFKLVUHGXFHGZLWKWKHWHUPPJK¶GXH WRWKHDGGLWLRQDOPRPHQWPJK¶
'
&PD KN F % F &PJKN
\
7KHWRWDOPRPHQWDERXWWKHUROOD[LVLVGLVWULEXWHGRYHUWKHIURQWDQGUHDUD[OHV LQSURSRUWLRQWRWKHIURQWDQGUHDUUROOVWLIIQHVVHV7KHORDGWUDQVIHU WKHLQQHU WRWKHRXWHUZKHHOVWKDWRFFXUVDWD[OHL
)]LIURP
RU LQDVWHDG\VWDWH
FRUQHULQJPRWLRQZLWKFHQWULSHWDODFFHOHUDWLRQD\IROORZVIURPWKHIRUPXOD
) ' PD ]L
L
\
ZLWKWKHORDGWUDQVIHUFRHIILFLHQWRID[OHL
L
'
F
L
V F %F &PJKN
L
KN %
O&D K O L
L
7KHDWWLWXGHDQJOHRIWKHUROOD[LVZLWKUHVSHFWWRWKHKRUL]RQWDOLVFRQVLGHUHG VPDOO,QWKHIRUPXODVLGHQRWHVKDOIWKHWUDFNZLGWKKLVWKHGLVWDQFHIURPWKH
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
FHQWUHRIJUDYLW\WRWKHUROOD[LVDQGD
DDQGD
E7KHUHVXOWLQJYHUWLFDOORDGV
DWD[OHLIRUWKHOHIW/ DQGULJKW5 ZKHHOVEHFRPHDIWHUFRQVLGHULQJWKHOHIWDQG ULJKWLQFUHPHQWVLQORDG
) ' ) ) ' ò) % ) ]L/
]L
]L/
]L
]L
) '& ) ) ' ò) & ) ]L5
]L
]L5
]L
]L
7KHZKHHOVDWWKHIURQWD[OHDUHVWHHUHGDERXWWKHNLQJSLQVZLWKWKHDQJOH 7KLV DQJOHUHODWHVGLUHFWO\WRWKHLPSRVHGVWHHULQJZKHHODQJOH
VWZ
WKURXJKWKHVWHHULQJ
UDWLRQVW
'
VWZ
Q
VW
,QDGGLWLRQWRWKLVLPSRVHGVWHHUDQJOHWKHZKHHOVPD\VKRZDVWHHUDQJOHDQGD FDPEHU DQJOH LQGXFHG E\ ERG\ UROO WKURXJK VXVSHQVLRQ NLQHPDWLFV 7KH IXQFWLRQDO UHODWLRQVKLSV ZLWK WKH UROO DQJOH PD\ EH OLQHDULVHG )RU D[OH L ZH GHILQH UL
UL
' '
L
L
6WHHUFRPSOLDQFHJLYHVULVHWRDQDGGLWLRQDOVWHHUDQJOHGXHWRWKHH[WHUQDOWRUTXH WKDW DFWV DERXW WKH NLQJSLQ VWHHULQJ D[LV )RU WKH SDLU RI IURQW ZKHHOV WKLV WRUTXHUHVXOWVIURPWKHVLGHIRUFHRIFRXUVHDOVRIURPWKHKHUHQRWFRQVLGHUHG GULYLQJRUEUDNLQJIRUFHV WKDWH[HUWVDPRPHQWDERXWWKHNLQJSLQWKURXJKWKH PRPHQWDUPZKLFKLVFRPSRVHGRIWKHFDVWHUOHQJWKHDQGWKHSQHXPDWLFWUDLOW :LWK WKH WRWDO VWHHULQJ VWLIIQHVV F
IHOW DERXW WKH NLQJSLQV ZLWK WKH VWHHULQJ
ZKHHO KHOG IL[HG WKH DGGLWLRQDO VWHHU DQJOH EHFRPHV ZKHQ IRU VLPSOLFLW\ WKH LQIOXHQFHRIFDPEHURQWKHSQHXPDWLFWUDLOLVGLVUHJDUGHG
F
'&
) H % W F \
,QDGGLWLRQWKHVLGHIRUFHEXWDOVRWKHIRUHDQGDIWIRUFH PD\LQGXFHDVWHHU DQJOHGXHWRVXVSHQVLRQFRPSOLDQFH7KHVRFDOOHGVLGHIRUFHVWHHUUHDGV
VIL
'F ) VIL
\L
)RUWKHIURQWD[OHZHVKRXOGVHSDUDWHWKHLQIOXHQFHVRIPRPHQWVWHHUDQGVLGH IRUFHVWHHU)RUWKLVUHDVRQVLGHIRUFHVWHHUDWWKHIURQWLVGHILQHGWRRFFXUDVD UHVXOWRIWKHVLGHIRUFHDFWLQJLQDSRLQWRQWKHNLQJSLQD[LV %HVLGHWKHZKHHODQJOHVLQGLFDWHGDERYHWKHZKHHOVPD\KDYHEHHQJLYHQ LQLWLDODQJOHVWKDWDOUHDG\H[LVWDWVWUDLJKWDKHDGUXQQLQJ7KHVHDUHWKHWRHDQJOH
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
R
SRVLWLYHSRLQWLQJRXWZDUGV DQGWKHLQLWLDOFDPEHUDQJOH
R
SRVLWLYHOHDQLQJ
RXWZDUGV )RUWKHOHIWDQGULJKWZKHHOVZHKDYHWKHLQLWLDODQJOHV
'& ' &
L/R
L/R
LR
L5R
LR
L5R
' '
LR
LR
$GGLQJDOOUHOHYDQWFRQWULEXWLRQV WR WRJHWKHU\LHOGVWKHWRWDOVWHHU DQJOHIRUHDFKRIWKHZKHHOV 7KHHIIHFWLYHFRUQHULQJVWLIIQHVVRIDQD[OH&HIILLVQRZGHILQHGDVWKHUDWLR RIWKHD[OHVLGHIRUFHDQGWKHYLUWXDOVOLSDQJOH7KLVDQJOHLVGHILQHGDVWKHDQJOH EHWZHHQWKHGLUHFWLRQRIPRWLRQRIWKHFHQWUHRIWKHD[OHLDFWXDOO\DWURDGOHYHO ZKHQWKHYHKLFOHYHORFLW\ZRXOGEHYHU\ORZDQGDSSURDFKHV]HURWKHQDOVR)\L
ß
DQGWKHGLUHFWLRQRIPRWLRQDWWKHDFWXDOVSHHGFRQVLGHUHG7KHYLUWXDOVOLS
DQJOHRIWKHIURQWD[OHKDVEHHQLQGLFDWHGLQ)LJDQGLVGHVLJQDWHGDV
D
:H
KDYHLQJHQHUDO
)
& '
\L
HIIL
DL
7KHD[OHVLGHIRUFHVLQWKHVWHDG\VWDWHWXUQFDQEHGHULYHGE\FRQVLGHULQJWKH ODWHUDOIRUFHDQGPRPHQWHTXLOLEULXPRIWKHYHKLFOH
O&D PD O
) '
L
\L
\
7KHD[OHVLGHIRUFHLVWKHVXPRIWKHOHIWDQGULJKWLQGLYLGXDOW\UHVLGHIRUFHV:H KDYH
) ' ) '
& % ò& &
ò
\L/
\L5
)
L
L
)
L
L
) )
]L
]L
L
L
& %
ZKHUHWKHDYHUDJHZKHHOVOLSDQJOH
'
L
DL
%
% ò& % % ò& &
LR
LR
L
) L
L
) L
L
) )
]L
]L
L
L
& %
LR
LR
LQGLFDWHGLQWKHILJXUHLV
L
DQGWKHDYHUDJHDGGLWLRQDOVWHHUDQJOHDQGWKHDYHUDJHFDPEHUDQJOHDUH
' '
L
L
UL
%
FL
%
VIL
UL
7KHXQNQRZQTXDQWLW\LVWKHYLUWXDOVOLSDQJOH
DL
ZKLFKFDQEHGHWHUPLQHGIRU
DJLYHQODWHUDODFFHOHUDWLRQD\1H[WZHXVHWKHHTXDWLRQV VXEVWLWXWHWKHUHVXOWLQJH[SUHVVLRQV DQG LQ DQG DGGXSWKHVHWZRHTXDWLRQV7KHUHVXOWLV D UHODWLRQVKLSEHWZHHQWKHD[OHVOLS DQJOH
DL
DQGWKHD[OHVLGHIRUFH)\L:HREWDLQIRUWKHVOLSDQJOHRID[OHL
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
)\L ' &HII L ) O L&) L% L&) L KN &) L H L%WL O % &&) L F VIL% L ' \L % FL &) L O&DL F %F &PJKN O&DL DL
'
L LR% L LR
7KHFRHIILFLHQWRI)\LFRQVWLWXWHVWKHHIIHFWLYHD[OHFRUQHULQJFRPSOLDQFHZKLFK LVWKHLQYHUVHRIWKHHIIHFWLYHD[OHFRUQHULQJVWLIIQHVV 7KHTXDQWLWDWLYH HIIHFWRIHDFKRIWKHVXVSHQVLRQVWHHULQJDQGW\UHIDFWRUVLQFOXGHGFDQEHHDVLO\ DVVHVVHG7KHVXEVFULSWLUHIHUVWRWKHFRPSOHWHD[OH&RQVHTXHQWO\WKHFRUQHULQJ DQGFDPEHUVWLIIQHVVHVDSSHDULQJLQWKLVH[SUHVVLRQDUHWKHVXPRIWKHVWLIIQHVVHV RIWKHOHIWDQGULJKWW\UH
& '& & '& )
L
)
) L
L/
)
L/
%& %&
)
L5
) L5
'& '&
)
)
%& %&
L/R
L/R
)
L5R
) L5R
LQ ZKLFK DQG KDYH EHHQ WDNHQ LQWR DFFRXQW 7KH ORDG WUDQVIHU FRHIILFLHQW
L
IROORZVIURP(T ([SUHVVLRQ VKRZVWKDWWKHLQIOXHQFH
RIODWHUDOORDGWUDQVIHURQO\RFFXUVLILQLWLDOO\ DW VWUDLJKWDKHDGUXQQLQJVLGH IRUFHV DUH DOUHDG\ SUHVHQW WKURXJK WKH LQWURGXFWLRQ RI HJ RSSRVLWH VWHHU DQG FDPEHUDQJOHV,IWKHVHDQJOHVDUHDEVHQWWKHLQIOXHQFHRIORDGWUDQVIHULVSXUHO\ QRQOLQHDUDQGLVRQO\IHOWDWKLJKHUOHYHOVRIODWHUDODFFHOHUDWLRQV,QWKHQH[W VXEVHFWLRQ WKLV QRQOLQHDU HIIHFW ZLOO EH LQFRUSRUDWHG LQ WKH HIIHFWLYH D[OH FKDUDFWHULVWLF
(IIHFWLYH1RQ/LQHDU$[OH&KDUDFWHULVWLFV
7RLOOXVWUDWHWKHPHWKRGRIHIIHFWLYHD[OHFKDUDFWHULVWLFVZHZLOOILUVWGLVFXVVWKH GHWHUPLQDWLRQ RI WKH HIIHFWLYH FKDUDFWHULVWLF RI D IURQW D[OH VKRZLQJ VWHHULQJ FRPSOLDQFH7KHVWHHULQJZKHHOLVKHOGIL[HG'XHWRW\UHVLGHIRUFHVDQGVHOI DOLJQLQJWRUTXHVOHIWDQGULJKW GLVWRUWLRQVZLOODULVHUHVXOWLQJLQDQLQFUHPHQWDO VWHHUDQJOH
F
RIWKHIURQWZKHHOV
ZLOOEHQHJDWLYHLQ)LJIRUWKHFDVHRI
F
MXVWVWHHUFRPSOLDQFH 6LQFHORDGWUDQVIHULVQRWFRQVLGHUHGLQWKLVH[DPSOHWKH VLWXDWLRQDWWKHOHIWDQGULJKWZKHHOVDUHLGHQWLFDOLQLWLDOWRHDQGFDPEHUDQJOHV EHLQJGLVUHJDUGHG 7KHIURQWW\UHVOLSDQJOHLVGHQRWHGZLWK DQJOHRIWKHD[OHLVGHQRWHGZLWK
D
'
ZKHUHERWK
&
7KHµYLUWXDO¶VOLS
DQGHTXDOVFI)LJ
F
DQG
D
F
DUHUHODWHGZLWK)\DQG0]7KHVXEVFLSWUHIHUVWRWKH
IURQWD[OHDQGWKXVWRWKHSDLURIW\UHV&RQVHTXHQWO\)\DQG0]GHQRWHWKHVXP RIWKHOHIWDQGULJKWW\UHVLGHIRUFHVDQGPRPHQWV7KHREMHFWLYHLVWRILQGWKH
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
9 D1
0]1
1
1
)\1
)LJ:KHHOVXVSHQVLRQDQGVWHHULQJFRPSOLDQFHUHVXOWLQJLQDGGLWLRQDOVWHHUDQJOH
)
IXQFWLRQ
\
D
ZKLFKLVWKHHIIHFWLYHIURQWD[OHFKDUDFWHULVWLF)LJXUHVKRZV
D JUDSKLFDO DSSURDFK $FFRUGLQJ WR (T WKH SRLQWV RQ WKH
) )
WRREWDLQWKHVRXJKW F
PXVWEHVKLIWHGKRUL]RQWDOO\RYHUDOHQJWK
\
\
FXUYH D
7KH
VORSH RI WKH FXUYH DW WKH RULJLQ FRUUHVSRQGV WR WKH HIIHFWLYH D[OH FRUQHULQJ VWLIIQHVVIRXQGLQWKHSUHFHGLQJVXEVHFWLRQ$OWKRXJKWKHFKDQJHVZLWKUHVSHFW WR WKH RULJLQDO FKDUDFWHULVWLF PD\ EH VPDOO WKH\ FDQ VWLOO EH RI FRQVLGHUDEOH LPSRUWDQFHVLQFHLWLVWKHGLIIHUHQFHRIVOLSDQJOHVIURQWDQGUHDUZKLFKODUJHO\ GHWHUPLQHVWKHYHKLFOH¶VKDQGOLQJEHKDYLRXU 7KH HIIHFWLYH D[OH FKDUDFWHULVWLF IRU WKH FDVH RI UROO VWHHU FDQ EH HDVLO\ HVWDEOLVKHGE\VXEWUDFWLQJ
UL
IURP
L
,QVWHDGRIXVLQJWKHOLQHDUUHODWLRQVKLSV
DQG QRQOLQHDU FXUYHV PD\ EH DGRSWHG SRVVLEO\ REWDLQHG IURP PHDVXUHPHQWV)RUWKHFDVHRIUROOFDPEHUWKHVLWXDWLRQEHFRPHVPRUHFRPSOH[
)\
1
( 1)
) \1D1
)
1
(
\
0
)
1
]
1
D
1
F
1
−
1
F
1
1
1
D
F
( 1)
0
1
]
)
)LJ(IIHFWLYHIURQWD[OHFKDUDFWHULVWLF
\
LQIOXHQFHGE\VWHHULQJFRPSOLDQFH
D
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
) \
) \
tyre
tyre
PD \ L
R
0 0
) ]R
)]
) ]R
)]
) \
axle
)] )
]
)
\axle
0
axle
)LJ7KHLQIOXHQFHRIORDGWUDQVIHURQWKHUHVXOWLQJD[OHFKDUDFWHULVWLF
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
)
\
YV
)
]
FXUYH
KRZHYHUWKHDYHUDJHVLGHIRUFHZLOOEHVPDOOHUWKDQWKHRULJLQDOYDOXHLWKDGLQ WKHDEVHQFHRIORDGWUDQVIHU7KHJUDSKLQGLFDWHVWKDWDQLQFUHDVH
RIWKHVOLS
DQJOHZRXOGEHQHHGHGWRFRPSHQVDWHIRUWKHDGYHUVHHIIHFWRIORDGWUDQVIHU7KH ORZHUGLDJUDPJLYHVDW\SLFDOH[DPSOHRIWKHFKDQJHLQFKDUDFWHULVWLFDVDUHVXOW RIORDGWUDQVIHU$WWKHRULJLQWKHVORSHLVQRWDIIHFWHGEXWDWODUJHUVOLSDQJOHV DQLQFUHDVLQJO\ORZHUGHULYDWLYHDSSHDUVWRRFFXU7KHSHDNGLPLQLVKHVDQGPD\ HYHQGLVDSSHDUFRPSOHWHO\7KHZD\WRGHWHUPLQHWKHUHVXOWLQJFKDUDFWHULVWLFLV WKHVXEMHFWRIWKHQH[WH[HUFLVH
([HUFLVH&RQVWUXFWLRQRIHIIHFWLYHD[OHFKDUDFWHULVWLFDWORDGWUDQVIHU )RUDVHULHVRIW\UHYHUWLFDOORDGV)]WKHFKDUDFWHULVWLFVRIWKHWZRW\UHVPRXQWHGRQ VD\WKHIURQWD[OHRIDQDXWRPRELOHDUHJLYHQ,QDGGLWLRQLWLVNQRZQKRZWKHORDG WUDQVIHU
)]DWWKHIURQWD[OHGHSHQGVRQWKHFHQWULIXJDOIRUFH.
PJ)\)]
PJ
)\)] DFWLQJDWWKHFHQWUHRIJUDYLW\)URPWKLVGDWDWKHUHVXOWLQJFRUQHULQJFKDUDF
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
WHULVWLFRIWKHD[OHFRQVLGHUHGDWVWHDG\VWDWHFRUQHULQJ FDQEHGHWHUPLQHG )LQG WKH UHVXOWLQJ FKDUDFWHULVWLF RI RQH D[OH IURP WKH VHW RI LQGLYLGXDOW\UH FKDUDFWHULVWLFVDWGLIIHUHQWW\UHORDGV)]DQGWKHORDGWUDQVIHUFKDUDFWHULVWLFERWK VKRZQLQ)LJ +LQW)LUVWGUDZLQWKHORZHUGLDJUDPWKHD[OHFKDUDFWHULVWLFVIRUYDOXHVRI
) ]
DQG1DQGWKHQGHWHUPLQHZKLFKSRLQWRQHDFKRIWKHVH FXUYHV LV YDOLG FRQVLGHULQJ WKH ORDG WUDQVIHU FKDUDFWHULVWLF OHIWKDQG GLDJUDP 'UDZWKHUHVXOWLQJD[OHFKDUDFWHULVWLF ,WPD\EHKHOSIXOWRHPSOR\WKH0DJLF)RUPXOD DQGWKHSDUDPHWHUVVKRZQ EHORZ VLGHIRUFH
) \
ZLWKIDFWRUV
%
'VLQ>&DUFWDQ^% &) &' &
FRUQHULQJVWLIIQHVV &) ZLWKSDUDPHWHUV
F
(%
'
DUFWDQ%
)](
`@
ZLWK
FVLQ>DUFWDQ^)]F`@
>1UDG@F
>1@
,QDGGLWLRQZHKDYHJLYHQIRUWKHORDGWUDQVIHU
) ]
)\D[OHXSWROLIWRIIRI
WKH LQQHU W\UH DIWHU ZKLFK WKH RWKHU D[OH PD\ WDNH RYHU WR DFFRPPRGDWH WKH LQFUHDVHGWRWDOORDGWUDQVIHU 'UDZWKHLQGLYLGXDOFXUYHVRI)\/DQG)\5IRUWKHOHIWDQGULJKWW\UH DVDIXQFWLRQ RI
ZKLFKDSSHDUWRDULVHXQGHUWKHORDGWUDQVIHUFRQGLWLRQFRQVLGHUHGKHUH
)LQDOO\ SORW WKHVH IRUFHV DV D IXQFWLRQ RI WKH YHUWLFDO ORDG )] UDQJLQJ IURP 1 1RWHWKHYDULDWLRQRIWKHODWHUDOIRUFHRIDQLQGLYLGXDOOHIWRUULJKW W\UH LQWKLVVDPHUDQJHRIYHUWLFDOORDGZKLFKPD\EHFRYHUHGLQDOHIWDQGLQDULJKW KDQGWXUQDWLQFUHDVLQJVSHHGRIWUDYHOXQWLODQGSRVVLEO\EH\RQG WKHPRPHQWWKDW RQHRIWKHZKHHOVWKHLQQHUZKHHO OLIWVIURPWKHJURXQG
)
)
\W\UH
5000 [N]
0
)
[N] | ) |
]W\UH ]
10 o
8000 7000 6000 5000 4000 3000 2000 1000 0
4000 3000 2000 1000 0 1000 2000 3000 4000
\D[OH
)
]
4000 3000 2000 1000
5000
? ? 0
10 o
)LJ 7KH FRQVWUXFWLRQ RI WKH UHVXOWLQJ D[OH FRUQHULQJ FKDUDFWHULVWLFV DW ORDGWUDQVIHU ([HUFLVH
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
9HKLFOH+DQGOLQJDQG6WDELOLW\
,Q WKLV VHFWLRQ DWWHQWLRQ LV SDLG WR WKH PRUH IXQGDPHQWDO DVSHFWV RI YHKLFOH KRUL]RQWDO PRWLRQV ,QVWHDG RI GLVFXVVLQJ UHVXOWV RI FRPSXWHU VLPXODWLRQV RI FRPSOLFDWHG YHKLFOH PRGHOV ZH UDWKHU WDNH WKH VLPSOHVW SRVVLEOH PRGHO RI DQ DXWRPRELOHWKDWUXQVDWFRQVWDQWVSHHGRYHUDQHYHQKRUL]RQWDOURDGDQGWKHUHE\ JDLQFRQVLGHUDEOHLQVLJKWLQWRWKHEDVLFDVSHFWVRIYHKLFOHKDQGOLQJDQGVWDELOLW\ ,PSRUWDQWHDUO\ZRUNRQWKHOLQHDUWKHRU\RIYHKLFOHKDQGOLQJDQGVWDELOLW\KDV EHHQSXEOLVKHGE\5LHNHUWDQG6FKXQFN :KLWFRPEDQG0LOOLNHQ DQG6HJHO 3HYVQHU VWXGLHGWKHQRQOLQHDUVWHDG\VWDWHFRUQHULQJ EHKDYLRXUDWODUJHUODWHUDODFFHOHUDWLRQVDQGLQWURGXFHGWKHKDQGOLQJGLDJUDP 2QHRIWKHILUVWPRUHFRPSOHWHYHKLFOHPRGHOVWXGLHVKDVEHHQFRQGXFWHG E\ 3DFHMND DQGE\5DGWDQG3DFHMND )RUPRUH LQWURGXFWRU\RUVSHFLDOLVHGVWXG\WKHUHDGHUPD\EHUHIHUUHGWR ERRNV RQ WKH VXEMHFW SXEOLVKHG HDUOLHU FI HJ *LOOHVSLH 0LWVFKNH 0LOOLNHQ DQG.RUWPDQG/XJQHU 7KHGHULYDWLRQRIWKHHTXDWLRQVRIPRWLRQIRUWKHWKUHHGHJUHHRIIUHHGRP PRGHO RI )LJ ZLOO EH WUHDWHG ILUVW DIWHU ZKLFK WKH VLPSOH PRGHO ZLWK WZR GHJUHHV RI IUHHGRP LV FRQVLGHUHG DQG DQDO\VHG 7KLV DQDO\VLV FRPSULVHV WKH VWHDG\VWDWHUHVSRQVHWRVWHHULQJLQSXWDQGWKHVWDELOLW\RIWKHUHVXOWLQJPRWLRQ $OVRWKHIUHTXHQF\UHVSRQVHWRVWHHULQJIOXFWXDWLRQVDQGH[WHUQDOGLVWXUEDQFHV ZLOOEHGLVFXVVHGILUVWIRUWKHOLQHDUYHKLFOHPRGHODQGVXEVHTXHQWO\IRUWKHQRQ OLQHDUPRGHOZKHUHODUJHODWHUDODFFHOHUDWLRQVDQGGLVWXUEDQFHVDUHLQWURGXFHG 7KHVLPSOHPRGHOWREHHPSOR\HGLQWKHDQDO\VLVLVSUHVHQWHGLQ)LJ7KH WUDFNZLGWKKDVEHHQQHJOHFWHGZLWKUHVSHFWWRWKHUDGLXVRIWKHFRUQHULQJPRWLRQ
; <
) \1
O
E
D
0]
2
Y
)\2
]
1
9
2 )[2
0 1
1 X
U
\
0
) 2
[
[
) \2
]1
)\1
)
[2
2
)LJ6LPSOHFDUPRGHOZLWKVLGHIRUFHFKDUDFWHULVWLFVIRUIURQWDQGUHDUGULYHQ D[OH
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
ZKLFKDOORZVWKHXVHRIDWZRZKHHOYHKLFOHPRGHO7KHVWHHUDQGVOLSDQJOHVZLOO EHUHVWULFWHGWRUHODWLYHO\VPDOOYDOXHV7KHQWKHYDULDWLRQRIWKHJHRPHWU\PD\ EHUHJDUGHGWRUHPDLQOLQHDUWKDWLVFRV
.DQGVLQ .
DQGVLPLODUO\IRUWKH
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
PD\ WKHQ EH UHJDUGHG FI
)LJ 7KHIRUFHV)\DQG)[DQGWKHPRPHQW0]DUHGHILQHGWRDFWXSRQWKH VLQJOHIURQWZKHHODQGVLPLODUO\ZHGHILQH)\HWFIRUWKHUHDUZKHHO
'LIIHUHQWLDO(TXDWLRQVIRU3ODQH9HKLFOH0RWLRQV
,QWKLVVHFWLRQWKHGLIIHUHQWLDOHTXDWLRQVIRUWKHWKUHHGHJUHHRIIUHHGRPYHKLFOH PRGHORI)LJZLOOEHGHULYHG,QILUVWLQVWDQFHWKHIRUHDQGDIWPRWLRQZLOO DOVR EH OHIW IUHH WR YDU\ 7KH UHVXOWLQJ VHW RI HTXDWLRQV RI PRWLRQ PD\ EH RI LQWHUHVW IRU WKH UHDGHU WR IXUWKHU VWXG\ WKH YHKLFOH¶V G\QDPLF UHVSRQVH DW VRPHZKDWKLJKHUIUHTXHQFLHVZKHUHWKHUROOG\QDPLFVRIWKHYHKLFOHERG\PD\ EHFRPHRILPSRUWDQFH)URPWKHVHHTXDWLRQVWKHHTXDWLRQVIRUWKHVLPSOHWZR GHJUHHRIIUHHGRPPRGHORI)LJXVHGLQWKHVXEVHTXHQWVHFWLRQFDQEHHDVLO\ DVVHVVHG ,Q 6XEVHFWLRQ WKH HTXDWLRQV IRU WKH FDU ZLWK WUDLOHU ZLOO EH HVWDEOLVKHG7KHSRVVLEOHLQVWDELOLW\RIWKHPRWLRQZLOOEHVWXGLHG :HZLOOHPSOR\/DJUDQJH¶VHTXDWLRQVWRGHULYHWKHHTXDWLRQVRIPRWLRQ)RU DV\VWHPZLWKQGHJUHHVRIIUHHGRPQJHQHUDOLVHG FRRUGLQDWHVTLDUHVHOHFWHG ZKLFKDUHVXIILFLHQWWRFRPSOHWHO\GHVFULEHWKHPRWLRQZKLOHSRVVLEOHNLQHPDWLF
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
FRQVWUDLQWVUHPDLQVDWLVILHG7KHPRYLQJV\VWHPSRVVHVVHVNLQHWLFHQHUJ\7DQG SRWHQWLDOHQHUJ\8([WHUQDOJHQHUDOLVHGIRUFHV4LDVVRFLDWHGZLWKWKHJHQHUDOLVHG FRRUGLQDWHVTLPD\DFWRQWKHV\VWHPDQGGRZRUN:,QWHUQDOIRUFHVDFWLQJIURP GDPSHUVWRWKHV\VWHPVWUXFWXUHPD\EHUHJDUGHGDVH[WHUQDOIRUFHVWDNLQJSDUW LQWKHWRWDOZRUN:7KHHTXDWLRQRI/DJUDQJHIRUFRRUGLQDWHTLUHDGV
M7 & M7 % M8 ' GW MT 0 MT MT G
L
L
4
L
L
7KH V\VWHP GHSLFWHG LQ )LJ DQG GHVFULEHG LQ WKH SUHFHGLQJ VXEVHFWLRQ SHUIRUPVDPRWLRQRYHUDIODWOHYHOURDG3URSHUFRRUGLQDWHVDUHWKH&DUWHVLDQ FRRUGLQDWHV;DQG<RIUHIHUHQFHSRLQW$WKH\DZDQJOH
RIWKHPRYLQJ[D[LV
ZLWKUHVSHFWWRWKHLQHUWLDO;D[LVDQGILQDOO\WKHUROODQJOH
DERXWWKHUROOD[LV
)RUPRWLRQVQHDUWKH;D[LVDQGWKXVVPDOO\DZDQJOHV(T LVDGHTXDWHWR GHULYHWKHHTXDWLRQVRIPRWLRQ)RUFDVHVZKHUH
PD\DWWDLQODUJHYDOXHVHJ
ZKHQ PRYLQJ DORQJ D FLUFXODU SDWK LW LV SUHIHUUHG WR XVH PRGLILHG HTXDWLRQV ZKHUHWKHYHORFLWLHVXYDQGURIWKHPRYLQJD[HVV\VWHPDUHXVHGDVJHQHUDOLVHG PRWLRQYDULDEOHVLQDGGLWLRQWRWKHFRRUGLQDWH
7KHUHODWLRQVEHWZHHQWKHWZR
VHWVRIYDULDEOHVDUHWKHGRWVUHIHUULQJWRGLIIHUHQWLDWLRQZLWKUHVSHFWWRWLPH
' ;0 FRV % <0 VLQ 0 VLQ % <0 FRV Y ' &; U' 0
X
7KHNLQHWLFHQHUJ\FDQEHH[SUHVVHGLQWHUPVRIXYDQGU3UHSDUDWLRQRIWKH ILUVWWHUPVRI(T IRUWKHFRRUGLQDWHV;<DQG
M7 ' M;J M7 ' M<J M7 ' M0 M7 ' M
M7 MX % M7 MY ' MX M;J MY M;J M7 MX % M7 MY ' MX M<J MY M<J M7 MU M7 Y & M7 X MX MY
7KH\DZDQJOH
M7 FRV & M7 VLQ MX MY M7 VLQ % M7 FRV MX MY
\LHOGV
PD\QRZEHHOLPLQDWHGE\PXOWLSO\LQJWKHILQDOHTXDWLRQVIRU
; DQG < VXFFHVVLYHO\ ZLWK FRV
DQG VLQ
DQG VXEVHTXHQWO\ DGGLQJ DQG
VXEWUDFWLQJWKHP7KHUHVXOWLQJHTXDWLRQVUHSUHVHQWWKHHTXLOLEULXPLQWKH[DQG \RUXDQGY GLUHFWLRQVUHVSHFWLYHO\ :HREWDLQWKHIROORZLQJVHWRIPRGLILHG/DJUDQJHDQHTXDWLRQVIRUWKHILUVW WKUHHYDULDEOHVXYDQGUDQGVXEVHTXHQWO\IRUWKHUHPDLQLQJUHDOFRRUGLQDWHV
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
IRURXUV\VWHPRQO\
M7 & U M7 ' 4 GW MX MY G M7 % U M7 ' 4 GW MY MX G M7 & Y M7 % X M7 ' 4 GW MU MX MY G M7 & M7 % M8 ' 4 GW M0 M M G
X
Y
U
7KHJHQHUDOLVHGIRUFHVDUHIRXQGIURPWKHYLUWXDOZRUN
' 34
:
M
'
T
M
M
ZLWKTMUHIHUULQJWRWKHTXDVLFRRUGLQDWHV[DQG\DQGWKHFRRUGLQDWHV
DQG
1RWHWKDW[DQG\FDQQRWEHIRXQGIURPLQWHJUDWLQJXDQGY)RUWKDWUHDVRQWKH WHUP µTXDVL¶ FRRUGLQDWH LV XVHG )RU WKH YHKLFOH PRGHO ZH ILQG IRU WKH YLUWXDO ZRUNDVDUHVXOWRIWKHYLUWXDOGLVSODFHPHQWV [ \
:
' 3)
[
[
% 3)
\
\
%30
DQG
% 30
]
ZKHUHDSSDUHQWO\
4 4 4
X
Y
U
4
' ' ' '
3 ) ' ) FRV & ) VLQ % ) 3 ) ' ) VLQ % ) FRV % ) 3 0 ' D ) VLQ % D ) FRV % 0 & E ) % 0 3 0 ' & N % N 0 [
[
\
[
]
\
[
\
[
\
\
]
\
]
7KHORQJLWXGLQDOIRUFHVDUHDVVXPHGWREHWKHVDPHDWWKHOHIWDQGULJKWZKHHOV DQGWKHHIIHFWRIDGGLWLRQDOVWHHUDQJOHV
L
DUHQHJOHFWHGKHUH6KRFNDEVRUEHUV
LQWKHZKHHOVXVSHQVLRQVDUHUHSUHVHQWHGE\WKHUHVXOWLQJOLQHDUPRPHQWVDERXW WKHUROOD[HVZLWKGDPSLQJFRHIILFLHQWVN
L
DWWKHIURQWDQGUHDUD[OHV
:LWKWKHUROODQJOH DQGWKHUROOD[LVLQFOLQDWLRQDQJOH
U
.K K ODVVXPHG
VPDOOWKHNLQHWLFHQHUJ\EHFRPHV
7
' ò P^X & KN U % Y % KN0 ` % %ò , 0 % ò , U % ò , U &
[
\
]
U
%
U
U0
&,
U0
[]
7KHSRWHQWLDOHQHUJ\8LVEXLOWXSLQWKHVXVSHQVLRQVSULQJVLQFOXGLQJWKHUDGLDO W\UHFRPSOLDQFHV DQGWKURXJKWKHKHLJKWRIWKHFHQWUHRIJUDYLW\:HKDYHDJDLQ IRUVPDOODQJOHV
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
8
'
ò F
%F
& ò PJKN
7KHHTXDWLRQVRIPRWLRQDUHILQDOO\HVWDEOLVKHGE\XVLQJWKHH[SUHVVLRQV DQG LQWKHHTXDWLRQV 7KHHTXDWLRQVZLOOEHOLQHDULVHGLQWKH DQG
DVVXPHGO\ VPDOO DQJOHV
)RU WKH YDULDEOHV X Y U DQG
ZH REWDLQ
VXFFHVVLYHO\
P X 0
& UY & KN U0& KNU0 '
P Y0
% UX % KN & KNU
, U0 ]
,[
% , & , ]
U
[]
'
)
)
[
[
&)
%) %) \
& PKNX0 & UY '
D)
% PKN % PKNY0 % UX % , & , % N % N
]
%)
\
U
[
[
D
E
\
%D) %0 &E) %0 \
]
\
& PKN % , & , U % 0 % F % F & PJKN '
U0 []
F
\
]
]
G
1RWH WKDW WKH VPDOO DGGLWLRQDO UROO DQG FRPSOLDQFH VWHHU DQJOHV
KDYH EHHQ L
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
VXEVFULSW DQG UHVSHFWLYHO\DQGWKHD[OHVLGHIRUFHVDSSHDULQJLQ(TE DUHWREHUHJDUGHG)RUDOLQHDUPRGHOWKHORDGWUDQVIHUFDQEHQHJOHFWHGLILQLWLDO OHIWULJKWRSSRVLWH ZKHHODQJOHVDUHGLVUHJDUGHG:HKDYHDWVWHDG\VWDWHHIIHFW RIGDPSLQJYDQLVKHV
)
]L
'
&F
L
%)
\L
K
L
VL
7KHIURQWDQGUHDUVOLSDQJOHVIROORZIURPWKHODWHUDOYHORFLWLHVRIWKHZKHHOD[OHV DQGWKHZKHHOVWHHUDQJOHVZLWKUHVSHFWWRWKHPRYLQJORQJLWXGLQDO[D[LV7KH ORQJLWXGLQDOYHORFLWLHVRIWKHZKHHOD[OHVPD\EHUHJDUGHGWKHVDPHOHIWDQGULJKW
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
DQGHTXDOWRWKHYHKLFOHORQJLWXGLQDOVSHHGX7KLVLVDOORZHGZKHQVL_U_©X7KHQ WKHH[SUHVVLRQVIRUWKHDVVXPHGO\VPDOOVOLSDQJOHVUHDG
0 ' % & Y % DU & H
'
X
& & EU
Y
X
7KHDGGLWLRQDOUROODQGFRPSOLDQFHVWHHUDQJOHV
L
DQGWKHZKHHOFDPEHUDQJOHV
DUH REWDLQHG IURP (T ZLWK RU FRUUHVSRQGLQJ QRQOLQHDU L H[SUHVVLRQV,QLWLDOZKHHODQJOHVDUHDVVXPHGWREHHTXDOWR]HUR7KHLQIOXHQFH RIWKHVWHHUDQJOHYHORFLW\DSSHDULQJLQWKHH[SUHVVLRQIRUWKHIURQWVOLSDQJOHLV UHODWLYHO\VPDOODQGPD\EHGLVUHJDUGHG7KHVPDOOSURGXFWVRIWKHFDVWHUOHQJWK HDQGWKHWLPHUDWHRIFKDQJHRI
L
KDYHEHHQQHJOHFWHGLQWKHDERYHH[SUHVVLRQV
(TXDWLRQV PD\ EH IXUWKHU OLQHDULVHG E\ DVVXPLQJ WKDW DOO WKH GHYLDWLRQVIURPWKHUHFWLOLQHDUPRWLRQDUHVPDOO7KLVDOORZVWKHQHJOHFWLRQRIDOO SURGXFWV RI YDULDEOH TXDQWLWLHV ZKLFK YDQLVK ZKHQ WKH YHKLFOH PRYHV VWUDLJKW DKHDG 7KH VLGH IRUFHV DQG PRPHQWV DUH WKHQ ZULWWHQ DV LQ (T ZLWK WKH RUSURYLGHG,IWKHPRPHQWGXHWRFDPEHULVQHJOHFWHGDQGWKH
VXEVFULSWVL
SQHXPDWLFWUDLOLVLQWURGXFHGLQWKHDOLJQLQJWRUTXHZHKDYH
)
\L
0
]L
' ) %) ' ' 0 ' && \
L
]
\
L
&
L
0
)
L
L
%& ' &W ) ' &W & L
L
) L
L
\
L
L
L
)
L
L
7KHWKUHHOLQHDUHTXDWLRQVRIPRWLRQIRUWKHV\VWHPRI)LJZLWKWKHIRUZDUG VSHHGXNHSWFRQVWDQWILQDOO\WXUQ RXWWRUHDGH[SUHVVHGVROHO\LQWHUPVRIWKH WKUHHPRWLRQYDULDEOHVYUDQG
P Y 0
% X U % KN ' & % & ^ % F )
)
% F X % H0 & Y & DU X % F ` % &Y % EU X % F ` % & %&
^
VF
VU
VF
VU
)
)
D
, U0 ]
% , & , ' D&W & ^%F X % H0 & Y & DU X % F & E%W & ^%F &Y%EU X % F ` % D & &E & ]
U
[]
)
)
VF
VU
VF
VU
)
`
)
%
E
,[
% PKN % PKNY0 % X U % , & , U0% N % N 0 % % F % F & PJKN '
]
U
[]
,QWKHVHHTXDWLRQVWKHDGGLWLRQDOVWHHUDQJOHV
F
KDYHEHHQHOLPLQDWHGE\XVLQJ L
H[SUHVVLRQV ZLWK )XUWKHUPRUHWKHUHVXOWLQJFRPSOLDQFHVWHHU
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
DQGUROOVWHHUFRHIILFLHQWVIRUL
F F
VFL
VIL
H
&
&
F
VIL
&
L
&
)
F
'
%W
L
RUKDYHEHHQLQWURGXFHG
L
L
L
H
L
%W
F
L
&
)
F
VUL
%
F
L
VIL
'
L
&
F
VIL
&
L
ZKHUH WKH VWHHU VWLIIQHVV DW WKH UHDU F
H
&
L
%W
H
F
L
%W
F
L
&
) L
L
L
&
)
L
L
PD\ EH WDNHQ HTXDO WR LQILQLW\
)XUWKHUPRUHZHKDYHWKHUROOD[LVLQFOLQDWLRQDQJOH
U
'
K
&K
O
,Q&KDSWHUVDQGWKHWUDQVLHQWSURSHUWLHVRIWKHW\UHZLOOEHDGGUHVVHG7KH UHOD[DWLRQOHQJWKGHQRWHGE\
L
LVDQLPSRUWDQWSDUDPHWHUWKDWFRQWUROVWKHODJRI
WKHUHVSRQVHRIWKHVLGHIRUFHWRWKHLQSXWVOLSDQJOH)RUWKH/DSODFHWUDQVIRUPHG YHUVLRQ RI WKH HTXDWLRQV ZLWK WKH /DSODFH YDULDEOH V UHSUHVHQWLQJ GLIIHUHQWLDWLRQZLWKUHVSHFWWRWLPHZHPD\LQWURGXFHW\UHODJE\UHSODFLQJWKH VOLSDQJOH
L
E\WKHILOWHUHGWUDQVLHQWVOLSDQJOH7KLVPD\EHDFFRPSOLVKHGE\
UHSODFLQJ WKH FRUQHULQJ VWLIIQHVVHV &) L DSSHDULQJ LQ DQG E\ WKH µWUDQVLHQWVWLIIQHVVHV¶
&
)
L
ß
&
)
%V
L
L
X
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
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
/LQHDU$QDO\VLVRIWKH7ZR'HJUHHRI)UHHGRP0RGHO
)URPWKHHTXDWLRQVEDQGF WKHUHGXFHGVHWRIHTXDWLRQVIRUWKHWZRGHJUHH RIIUHHGRPPRGHOFDQEHGHULYHGLPPHGLDWHO\7KHUROODQJOH DQGLWVGHULYDWLYH DUHVHWHTXDOWR]HURDQGIXUWKHUPRUHZHZLOODVVXPHWKHIRUZDUGVSHHGX
.9
WR UHPDLQ FRQVWDQW DQG QHJOHFW WKH LQIOXHQFH RI WKH ODWHUDO FRPSRQHQW RI WKH ORQJLWXGLQDOIRUFHV)[L7KHHTXDWLRQVRIPRWLRQRIWKHVLPSOHPRGHORI)LJ IRUYDQGUQRZUHDG
%
P Y 0
'
, U 0
') % & E)
XU
D)
\
\
)
D
\
E
\
ZLWK Y GHQRWLQJ WKH ODWHUDO YHORFLW\ RI WKH FHQWUH RI JUDYLW\ DQG U WKH \DZ YHORFLW\ 7KH V\PERO P VWDQGV IRU WKH YHKLFOH PDVV DQG ,
,] GHQRWHV WKH
PRPHQWRILQHUWLDDERXWWKHYHUWLFDOD[LVWKURXJKWKHFHQWUHRIJUDYLW\)RUWKH PDWWHURIVLPSOLFLW\WKHUHDUZDUGVKLIWVRIWKHSRLQWVRIDSSOLFDWLRQRIWKHIRUFHV )\DQG)\RYHUDOHQJWKHTXDOWRWKHSQHXPDWLFWUDLOWDQGWUHVSHFWLYHO\WKDW LVWKHDOLJQLQJWRUTXHV KDYHEHHQGLVUHJDUGHG/DWHUZHFRPHEDFNWRWKLV7KH VLGHIRUFHVDUHIXQFWLRQVRIWKHUHVSHFWLYHVOLSDQJOHV
)
\
'
)
\
DQG
)
\
'
)
\
DQGWKHVOLSDQJOHVDUHH[SUHVVHGE\
' &
X
Y
% DU
DQG
' & Y & EU X
QHJOHFWLQJWKHHIIHFWRIWKHWLPHUDWHRIFKDQJHRIWKHVWHHUDQJOHDSSHDULQJLQ (T )RUUHODWLYHO\ORZIUHTXHQF\PRWLRQVWKHHIIHFWLYHD[OHFKDUDFWHULVWLFV RUHIIHFWLYHFRUQHULQJVWLIIQHVVHVDFFRUGLQJWR(TV PD\EHHPSOR\HG :KHQRQO\VPDOOGHYLDWLRQVZLWKUHVSHFWWRWKHXQGLVWXUEHGVWUDLJKWDKHDG PRWLRQDUHFRQVLGHUHGWKHVOLSDQJOHVPD\EHDVVXPHGWRUHPDLQVPDOOHQRXJK WR DOORZ OLQHDULVDWLRQ RI WKH FRUQHULQJ FKDUDFWHULVWLFV )RU WKH VLGH IRUFH WKH UHODWLRQVKLSZLWKWKHVOLSDQJOHUHGXFHVWRWKHOLQHDUHTXDWLRQ
)
\L
'
&
L
L
'
&
)
L
L
ZKHUH&LGHQRWHVWKHFRUQHULQJVWLIIQHVV7KLVFDQEHUHSODFHGE\WKHV\PERO&)
L
ZKLFKPD\EHSUHIHUUHGLQPRUHJHQHUDOFDVHVZKHUHDOVRFDPEHUDQGDOLJQLQJ VWLIIQHVVHVSOD\DUROH 7KHWZROLQHDUILUVWRUGHUGLIIHUHQWLDOHTXDWLRQVQRZUHDG
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
PY 0
% & % & Y %
X
, U0
% D
&
X
%E
&
% D& & E& U '
PX
X
U
% D& & E& Y '
X
&
D&
$IWHUHOLPLQDWLRQRIWKHODWHUDOYHORFLW\YZHREWDLQWKHVHFRQGRUGHUGLIIHUHQWLDO HTXDWLRQIRUWKH\DZUDWHU
, P X U
% ^, & % & % P D
% ^& & O & P X
X
&
%E
D&
&
` U0
%
& E& ` U '
%& & O
0
PXD&
+HUHDVEHIRUHWKHGRWVUHIHUWRGLIIHUHQWLDWLRQZLWKUHVSHFWWRWLPH LVWKHVWHHU DQJOHRIWKHIURQWZKHHODQGO
DE UHSUHVHQWVWKHZKHHOEDVH7KHHTXDWLRQV
PD\EHVLPSOLILHGE\LQWURGXFLQJWKHIROORZLQJTXDQWLWLHV
' & %& &V ' & D&& E &T ' & D %& E PN ' , &
+HUH&GHQRWHVWKHWRWDOFRUQHULQJVWLIIQHVVRIWKHYHKLFOHVLVWKHGLVWDQFHIURP WKHFHQWUHRIJUDYLW\WRWKHVRFDOOHGQHXWUDOVWHHUSRLQW6)LJ TLVDOHQJWK FRUUHVSRQGLQJ WR DQ DYHUDJH PRPHQW DUP DQG N LV WKH UDGLXV RI J\UDWLRQ (TXDWLRQV DQG QRZUHGXFHWR
PY0
% X U % & Y % & V U ' X
PN
U0
%
FT X
&
X
U
%
&V
Y
X
'
& D
DQGZLWKYHOLPLQDWHG
% P & T % N
P N X U
%
X U0 &&O
&PX
& V U
'PX
0
D&
% X& & O
7KHQHXWUDOVWHHUSRLQW6LVGHILQHGDVWKHSRLQWRQWKHORQJLWXGLQDOD[LVRIWKH YHKLFOH ZKHUH DQ H[WHUQDO VLGH IRUFH FDQ EH DSSOLHG ZLWKRXW FKDQJLQJ WKH YHKLFOH¶V \DZ DQJOH ,I WKH IRUFH DFWV LQ IURQW RI WKH QHXWUDO VWHHU SRLQW WKH YHKLFOHLVH[SHFWHGWR\DZLQWKHGLUHFWLRQRIWKHIRUFHLIEHKLQGWKHQDJDLQVWWKH IRUFH7KHSRLQWLVRILQWHUHVWZKHQGLVFXVVLQJWKHVWHHULQJFKDUDFWHULVWLFVDQG VWDELOLW\
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
/LQHDU6WHDG\6WDWH&RUQHULQJ6ROXWLRQV
:HDUHLQWHUHVWHGLQWKHSDWKFXUYDWXUH5 WKDWUHVXOWVIURPDFRQVWDQWVWHHU DQJOH DWDJLYHQFRQVWDQWVSHHGRIWUDYHO96LQFHZHKDYHDWVWHDG\VWDWH
'
5
.
U 9
U
X
WKHH[SUHVVLRQIRUWKHSDWKFXUYDWXUHEHFRPHVXVLQJ ZLWKXUHSODFHGE\ 9DQGWKHWLPHGHULYDWLYHVRPLWWHG
& & O
'
5
& &
&P9
D&
& E&
%\WDNLQJWKHLQYHUVHWKHH[SUHVVLRQIRUWKHVWHHUDQJOHUHTXLUHGWRQHJRWLDWHD FXUYHZLWKDJLYHQUDGLXV5LVREWDLQHG
'
O
5
&P9
D&
& E&
O& &
,WLVFRQYHQLHQWWRLQWURGXFHWKHVRFDOOHGXQGHUVWHHUFRHIILFLHQWRUJUDGLHQW
)RURXUPRGHOWKLVTXDQWLW\LVGHILQHGDV
' & PJ
D&
O
& E&
PJ&
O & &
& &
' &V
ZLWKJGHQRWLQJWKHDFFHOHUDWLRQGXHWRJUDYLW\$IWHUKDYLQJGHILQHGWKHODWHUDO DFFHOHUDWLRQ ZKLFK LQ WKH SUHVHQW OLQHDU DQDO\VLV HTXDOV WKH FHQWULSHWDO DFFHOHUDWLRQ
D
\
'
9U
'
9
5
(T FDQEHZULWWHQLQWKHPRUHFRQYHQLHQWIRUP
'
O 5
%
9
JO
'
O 5
%
D
\
J
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
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
α 1− α 2
δ [rad]
[rad]
understeer
η
η
1
O
5
neutral steer
0
D\ J
0
δ
D\ J
O
5
oversteer )LJ
7KHVWHHUDQJOHYHUVXVODWHUDODFFHOHUDWLRQDWFRQVWDQWSDWKFXUYDWXUHOHIWJUDSK 7KH GLIIHUHQFHLQVOLSDQJOHYHUVXVODWHUDODFFHOHUDWLRQDQGWKHUHTXLUHGVWHHUDQJOHDWD JLYHQSDWKFXUYDWXUHULJKWJUDSK 7KHXQGHUVWHHUJUDGLHQW
$FFRUGLQJWR(T WKHVWHHUDQJOHFKDQJHVVLJQZKHQIRUDQRYHUVWHHUHG FDUWKHVSHHGLQFUHDVHVEH\RQGWKHFULWLFDOVSHHGWKDWLVH[SUHVVHGE\
9
FULW
'
JO
&
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
9
FKDU
'
JO
!
([SUHVVLRQ IRUWKHXQGHUVWHHUJUDGLHQW LVVLPSOLILHGZKHQWKHIROORZLQJ H[SUHVVLRQVIRUWKHIURQWDQGUHDUD[OHORDGVDUHXVHG
)
]
'
E O
P J DQG )
]
'
D O
PJ
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
O
V
9
α2
6
)\2
D\
β δ
α1
)\1 5
)LJ7ZRZKHHOYHKLFOHPRGHOLQDFRUQHULQJPDQRHXYUH
:HREWDLQ
'
)
]
&
&
)
]
&
ZKLFK VD\V WKDW D YHKLFOH H[KLELWV DQ XQGHUVWHHU QDWXUH ZKHQ WKH UHODWLYH FRUQHULQJ FRPSOLDQFH RI WKH W\UHV DW WKH IURQW LV ODUJHU WKDQ DW WKH UHDU ,W LV LPSRUWDQWWRQRWHWKDWLQ DQG WKHTXDQWLWLHV)]GHQRWHWKHYHUWLFDO D[OHORDGVWKDWRFFXUDWVWDQGVWLOODQGWKXVUHSUHVHQWWKHPDVVGLVWULEXWLRQRIWKH YHKLFOH&KDQJHVRIWKHVHORDGVGXHWRDHURG\QDPLFGRZQIRUFHVDQGIRUHDQGDIW ORDGWUDQVIHUDWEUDNLQJRUGULYLQJVKRXOGQRWEHLQWURGXFHGLQH[SUHVVLRQ ,Q WKH VDPH GLDJUDP WKH GLIIHUHQFH LQ VOLS DQJOH IURQW DQG UHDU PD\ EH LQGLFDWHG:HILQGIRUWKHVLGHIRUFHV
)
\
'
E O
PD
\
'
D )
]
\
J
)
\
'
D O
PD
\
'
D )
]
\
J
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
DQGKHQFHIRUWKHVOLSDQJOHV
'
)
]
&
D
\
J
'
)
]
&
D
\
J
7KHGLIIHUHQFHQRZUHDGVZKHQFRQVLGHULQJWKHUHODWLRQ
& '
D
\
J
$SSDUHQWO\WKHVLJQRIWKLVGLIIHUHQFHLVGLFWDWHGE\WKHXQGHUVWHHUFRHIILFLHQW &RQVHTXHQWO\LWPD\EHVWDWHGWKDWDFFRUGLQJWRWKHOLQHDUPRGHODQXQGHUVWHHUHG YHKLFOH UHDU
!
! PRYHVLQDFXUYHZLWKVOLSDQJOHVODUJHUDWWKHIURQWWKDQDWWKH
)RUDQHXWUDOO\VWHHUHGYHKLFOHWKHDQJOHVUHPDLQWKHVDPH
DQGZLWKDQRYHUVWHHUHGFDUWKHUHDUVOLSDQJOHVDUHELJJHU
!
$VLVVKRZQ
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
ZLOOEHXVHG
,QIOXHQFHRIWKH3QHXPDWLF7UDLO
7KHGLUHFWLQIOXHQFHRIWKHSQHXPDWLFWUDLOVWLPD\QRWEHQHJOLJLEOH,QUHDOLW\ WKH W\UH VLGH IRUFHV DFW D VPDOO GLVWDQFH EHKLQG WKH FRQWDFW FHQWUHV $V D FRQVHTXHQFHWKHQHXWUDOVWHHUSRLQWVKRXOGDOVREHFRQVLGHUHGWREHORFDWHGDWD GLVWDQFHDSSUR[LPDWHO\HTXDOWRWKHDYHUDJHYDOXHRIWKHSQHXPDWLFWUDLOVPRUH
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
WR WKH UHDU ZKLFK PHDQV DFWXDOO\ PRUH XQGHUVWHHU 7KH FRUUHFW YDOXHV RI WKH SRVLWLRQVRIWKHQHXWUDOVWHHUSRLQWDQG RI WKHXQGHUVWHHUFRHIILFLHQW IRXQGE\XVLQJWKHHIIHFWLYHD[OHGLVWDQFHVDN
DWEN
EWDQGON
FDQEH
DNENLQ
WKH(TV DQG LQVWHDGRIWKHRULJLQDOTXDQWLWLHVDEDQGO
6WDELOLW\RIWKH0RWLRQ
6WDELOLW\RIWKHVWHDG\VWDWHFLUFXODUPRWLRQFDQEHH[DPLQHGE\FRQVLGHULQJWKH GLIIHUHQWLDOHTXDWLRQ RU 7KHVWHHUDQJOHLVNHSWFRQVWDQWVRWKDWWKH HTXDWLRQJHWVWKHIRUP
D U
% D U0% D U '
E
)RU WKLV VHFRQGRUGHU GLIIHUHQWLDO HTXDWLRQ VWDELOLW\ LV DVVXUHG ZKHQ DOO FRHIILFLHQWV DL DUH SRVLWLYH 2QO\ WKH ODVW FRHIILFLHQW D PD\ EHFRPH QHJDWLYH ZKLFK FRUUHVSRQGV WR GLYHUJHQW LQVWDELOLW\ VSLQRXW ZLWKRXW RVFLOODWLRQV $V DOUHDG\ LQGLFDWHG WKLV ZLOO LQGHHG RFFXU ZKHQ IRU DQ RYHUVWHHUHG YHKLFOH WKH FULWLFDOVSHHG LVH[FHHGHG7KHFRQGLWLRQIRUVWDELOLW\UHDGV
D
'
& & O
%
9
'
JO
& & O
! O5
VV
ZLWKWKHVXEVFULSWVVUHIHUULQJWRVWHDG\VWDWHFRQGLWLRQVRU
9
9 FULW
'
JO
&
7KH QH[W VHFWLRQ ZLOO IXUWKHU DQDO\VH WKH G\QDPLF QDWXUH RI WKH VWDEOH DQG XQVWDEOHPRWLRQV ,WLVRILPSRUWDQFHWRQRWHWKDWZKHQWKHFRQGLWLRQRIDQDXWRPRELOHVXEMHFWHG WRGULYLQJRUEUDNLQJIRUFHVLVFRQVLGHUHGWKHFRUQHULQJVWLIIQHVVHVIURQWDQGUHDU ZLOOFKDQJHGXHWRWKHDVVRFLDWHGIRUHDQGDIWD[OHORDGWUDQVIHUDQGWKHUHVXOWLQJ VWDWHRIFRPELQHGVOLS,QH[SUHVVLRQ IRUWKHXQGHUVWHHUFRHIILFLHQW WKH TXDQWLWLHV)]LUHSUHVHQWWKHVWDWLFYHUWLFDOD[OHORDGVREWDLQHGWKURXJK(TV DQG DUH WR UHPDLQ XQFKDQJHG ,Q 6XEVHFWLRQ WKH HIIHFW RI ORQJLWXGLQDO IRUFHVRQYHKLFOHVWDELOLW\ZLOOEHIXUWKHUDQDO\VHG
)UHH/LQHDU0RWLRQV
7R VWXG\ WKH QDWXUH RI WKH IUHH PRWLRQ DIWHU D VPDOO GLVWXUEDQFH LQ WHUPV RI QDWXUDO IUHTXHQF\ DQG GDPSLQJ WKH HLJHQYDOXHV WKDW LV WKH URRWV RI WKH FKDUDFWHULVWLFHTXDWLRQRIWKHOLQHDUVHFRQGRUGHUV\VWHPDUHWREHDVVHVVHG7KH
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
Im
Im
ζω R ωR
ωQ λ1
λ2 9!9FULW
Re
Re
oversteer
understeer
Re
9 )LJ
Im
9
3RVVLEOHHLJHQYDOXHVIRUWKHRYHUDQGXQGHUVWHHUHGFDUDWORZHUDQGKLJKHUVSHHGV
FKDUDFWHULVWLFHTXDWLRQRIWKHV\VWHPGHVFULEHGE\WKHHTXDWLRQV RU UHDGVDIWHUXVLQJWKHUHODWLRQ EHWZHHQVDQG
P
N 9
% P & T % N
9
%& & O
%
9 JO
'
)RU D VLQJOH PDVVGDPSHUVSULQJ V\VWHP VKRZQ LQ )LJ ZLWK U WKH PDVV GLVSODFHPHQW
WKHIRUFHGGLVSODFHPHQWRIWKHVXSSRUW0WKHPDVV'WKHVXP
RI WKH WZR GDPSLQJ FRHIILFLHQWV ' DQG ' DQG . WKH VXP RI WKH WZR VSULQJ
VWLIIQHVVHV . DQG . D GLIIHUHQWLDO HTXDWLRQ VLPLODU LQ VWUXFWXUH WR (T
DULVHV
0 U
% ' U0% . U '
'
0
%.
DQGWKHFRUUHVSRQGLQJFKDUDFWHULVWLFHTXDWLRQ
0
%' %.'
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
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
DSDLURIFRPSOH[URRWVLQWKHXQGHUVWHHUFDVHH[FHSWDWORZVSHHGVZKHUHWKH XQGHUVWHHUHGYHKLFOHFDQVKRZDSDLURIUHDOQHJDWLYHURRWV $VLQGLFDWHGLQWKHILJXUHWKHFRPSOH[URRWLVFKDUDFWHULVHGE\WKHQDWXUDO IUHTXHQF\
R
RI WKH XQGDPSHG V\VWHP '
UHVXOWLQJDFWXDOQDWXUDOIUHTXHQF\
Q
WKH GDPSLQJ UDWLR
DQG WKH
([SUHVVLRQVIRUWKHVHTXDQWLWLHVLQWHUPV
RIWKHPRGHOSDUDPHWHUVDUHUDWKHUFRPSOH[+RZHYHULIZHWDNHLQWRDFFRXQW WKDWLQQRUPDOFDVHV_V_©ODQGT
. N .ò OZHPD\VLPSOLI\WKHVHH[SUHVVLRQVDQG
ILQGWKHIROORZLQJXVHIXOIRUPXODH 7KHQDWXUDOIUHTXHQF\RIWKHXQGDPSHGV\VWHP
R
'
. 0
.
& P9
@
%
9
JO
7KHGDPSLQJUDWLR
'
0
.
'
R
%
9
JO
7KHQDWXUDOIUHTXHQF\
Q
'
R
&
.
& P
JO
7KHLQIOXHQFHRISDUDPHWHUVKDVEHHQLQGLFDWHGLQ)LJ$QDUURZSRLQWLQJ XSZDUGVUHSUHVHQWVDQLQFUHDVHRIWKHTXDQWLW\LQWKHVDPHFROXPQRIWKHPDWUL[ 7KH\DZUDWHUHVSRQVHWRDVWHSFKDQJHLQVWHHUDQJOHLVW\SLILHGE\WKHULVH
ωR
.
2
'
neutral oversteer
0
U
.
1
'
1
δ analogous system
0
ζ 1 0
ωR
under
ωQ
2
over (η < 0) neutral (η = 0)
9 & P
ζ
ωQ _
_ _
η
under (η > 0)
9FULW
9
parameter influence
)LJ7KHLQIOXHQFHRISDUDPHWHUVRQQDWXUDOIUHTXHQF\DQGGDPSLQJ
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
9
U
= 60 m/s W
40
U
VV
U
99
char
20
9!9
char
0
η W
1
U
2
[s]
W
&
δ
P N
0
W
)LJ 6WHS UHVSRQVH RI \DZ UDWH WR VWHHU DQJOH 3DUDPHWHUV DFFRUGLQJ WR7DEOH 3DUDPHWHULQIOXHQFHRQWKHULVHWLPHWU
WLPHWULQGLFDWHGLQ)LJDQGH[SUHVVHGLQWHUPVRIWKHSDUDPHWHUVDVIROORZV
W
U'
PN D& O
%
'
9 9
JO
PN D O
& O
%
E
9
&D
&
P9
&
7KHSDUDPHWHULQIOXHQFHKDVEHHQLQGLFDWHGLQWKHILJXUH7KHUHVXOWVFRUUHVSRQG TXDOLWDWLYHO\ZHOOZLWKWKHUHVSRQVHWLPHVIRXQGLQYHKLFOHPRGHOVLPXODWLRQ VWXGLHV$UHPDUNDEOHUHVXOWLVWKDWIRUDQXQGHUVWHHUHGDXWRPRELOHWKHUHVSRQVH WLPHLVVPDOOHUWKDQIRUDQRYHUVWHHUHGFDU
)RUFHG/LQHDU9LEUDWLRQV
7KHFRQYHUVLRQRIWKHHTXDWLRQVRIPRWLRQ LQWRWKHVWDQGDUGVWDWHVSDFH UHSUHVHQWDWLRQ LV XVHIXO ZKHQ WKH OLQHDU V\VWHP SURSHUWLHV DUH WKH VXEMHFW RI LQYHVWLJDWLRQ7KHV\VWHPDWKDQGLVRIWKHVHFRQGRUGHUDQGKHQFHSRVVHVVHVWZR VWDWHYDULDEOHVIRUZKLFKZHFKRRVHYDQGU7KHV\VWHPLVVXEMHFWHGWRDVLQJOH LQSXWVLJQDOWKHVWHHUDQJOH 9DULRXVYDULDEOHVPD\EHRILQWHUHVWWRDQDO\VHWKH YHKLFOH¶V UHVSRQVH WR VWHHULQJ LQSXW RVFLOODWLRQV 7KH IROORZLQJ TXDQWLWLHV DUH VHOHFWHG WR LOOXVWUDWH WKH PHWKRG DQG WR VWXG\ WKH G\QDPLF EHKDYLRXU RI WKH YHKLFOHWKHODWHUDODFFHOHUDWLRQD\RIWKHFHQWUHRIJUDYLW\RIWKHYHKLFOHWKH\DZ UDWH U DQG WKH YHKLFOH VOLS DQJOH QRWDWLRQWKHHTXDWLRQEHFRPHV
GHILQHG DW WKH FHQWUH RI JUDYLW\ ,Q PDWUL[
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
[0 ' $[ % %X \ ' &[ % 'X
ZLWK
D
[0 '
Y 0 U0
X'
\'
Y0
\
U
'
%9U U
&Y9
DQG
&
$' &
9
P9 &V PN
%
P9
&T 9
&
&V
PN
%'
P & D
9
PN
&' &
&
&V
&
P9
P9
P
&
9
''
7KHIUHTXHQF\UHVSRQVHIXQFWLRQVKDYHEHHQFRPSXWHGXVLQJ0DWODEVRIWZDUH )LJXUHSUHVHQWVWKHDPSOLWXGHDQGSKDVHUHVSRQVHIXQFWLRQVIRUHDFKRIWKH WKUHHRXWSXWTXDQWLWLHVDQGDWWKUHHGLIIHUHQWYDOXHVRIVSHHGRIWUDYHO7KHYDOXHV RIWKHFKRVHQPRGHOSDUDPHWHUVDQGDQXPEHURIFKDUDFWHULVWLFTXDQWLWLHVKDYH EHHQOLVWHGLQ7DEOH ([SOLFLWH[SUHVVLRQVRIWKHIUHTXHQF\UHVSRQVHIXQFWLRQVLQWHUPVRIPRGHO SDUDPHWHUVDUHKHOSIXOWRXQGHUVWDQGDQGSUHGLFW WKHFKDUDFWHULVWLFDVSHFWVRI WKHVHIXQFWLRQVZKLFKPD\EHHVWDEOLVKHGE\PHDQVRIFRPSXWDWLRQVRUSRVVLEO\ WKURXJKIXOOVFDOHH[SHULPHQWV )URP WKH GLIIHUHQWLDO HTXDWLRQ WKH IUHTXHQF\ UHVSRQVH IXQFWLRQ LV HDVLO\GHULYHG&RQVLGHULQJWKHTXDQWLWLHVIRUPXODWHGE\ DQG DQG WKHVWHDG\VWDWHUHVSRQVHU VV
95 REWDLQHGIURP ZHILQG
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
7DEOH3DUDPHWHUYDOXHVDQGW\SLI\LQJTXDQWLWLHV SDUDPHWHUV
GHULYHGW\SLI\LQJTXDQWLWLHV
D
P
O
P
E
P
) ]
1
&
1UDG
)]
1
&
1UDG
T
P
P
NJ
V
P
N
P
M
'
R
>UDGV@
>@
Q
>UDGV@
WU>V@
E
UDGa H[WUDVWHHUJODWHUDODFFHO
U
9>PV@
U
@
%
P9D
M
& O
VV
&
%
VV
M
R
'
U
9O
%
R
9 JO
6LPLODUO\WKHIRUPXODIRUWKHUHVSRQVHRIODWHUDODFFHOHUDWLRQD\FDQEHGHULYHG
D
\
M
'
D
\
& O
@
VV
& PN
%E
&
M
9
%
D
VV
M
R
'
\
9 O
%
R
9
JO
DQGIRUWKHVOLSDQJOH
M
'
PN DP9
@
VV
&
& R
9
& E&
M
%
O
M
R
&D
O
9
E & O
'& E
VV
P
%
9
JO
%\FRQVLGHULQJ(T LWFDQQRZEHH[SODLQHGIRULQVWDQFH WKDW DWKLJKHU IUHTXHQFLHVWKHV\VWHPVKRZVIHDWXUHVRIDILUVWRUGHUV\VWHPEHFDXVHRIWKHM WHUPLQWKHQXPHUDWRUWKH\DZUDWHDPSOLWXGHUHVSRQVHWHQGVWRDGHFD\DWD G%
SHURFWDYHUDWHLISORWWHGLQORJORJVFDOH DQGWKHSKDVHODJDSSURDFKHV
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
yaw velocity response to steer angle U
δ
1/s
15 10
V=60m/s 40
5 20
0 45 phase
0
deg.
20
-45 60
-90 0
D\J
δ
0.5
1 frequency
1.5
2
Hz
lateral acceleration response to steer angle
40 30
V=60m/s 40 20
20 1 rad
10 0 45
phase deg.
0 V=20m/s
-45 -90 -145
40 60
0
0.5
1 frequency
1.5
2
Hz
vehicle slip angle response to steer angle
β δ _
6 4
V=60m/s 40 20
2 0 0 phase deg.
-90
20
-180
60
-270
0
0.5
1 frequency
1.5
2
Hz
)LJ)UHTXHQF\UHVSRQVHIXQFWLRQVYHKLFOHSDUDPHWHUVDFFRUGLQJWR7DEOH
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
GHJUHHV7KHSKDVHLQFUHDVHDWORZIUHTXHQFLHVDQGKLJKHUVSHHGVLVGXHWRWKH SUHVHQFHRIWKHVSHHG9LQWKDWVDPHWHUP$WVSHHGVEH\RQGDSSUR[LPDWHO\WKH FKDUDFWHULVWLFVSHHGWKHFRUUHVSRQGLQJODVW WHUPLQWKHGHQRPLQDWRUKDVOHVV LQIOXHQFHRQWKHLQLWLDOVORSHRIWKHSKDVHFKDUDFWHULVWLF7KHODWHUDODFFHOHUDWLRQ UHVSRQVH VKRZQLQWKHFHQWUHJUDSKRI)LJJLYHVDILQLWHDPSOLWXGHDW IUHTXHQFLHVWHQGLQJWRLQILQLW\EHFDXVHRIWKHSUHVHQFHRI
LQWKHQXPHUDWRU
)RUWKHVDPHUHDVRQWKHSKDVHODJJRHVEDFNWR]HURDWODUJHIUHTXHQFLHV7KH VLGHVOLSSKDVHUHVSRQVHWHQGVWRGHJUHHVDWODUJHUVSHHGV ZKLFKLVGXHWR LQWKHQXPHUDWRURI 7KLVLQFRQWUDVWWRWKDW
WKHQHJDWLYHFRHIILFLHQWRIM
FRHIILFLHQWRIWKH\DZUDWHUHVSRQVH ,WLVRILQWHUHVWWRVHHWKDWWKHVWHDG\VWDWHVOLSDQJOHUHVSRQVHLQGLFDWHGLQ FKDQJHV VLJQ DW D FHUWDLQ VSHHG 9 $W ORZ VSHHGV ZKHUH WKH W\UH VOLS DQJOHVDUHVWLOOYHU\VPDOOWKHYHKLFOHVOLSDQJOHREYLRXVO\LVQHJDWLYHIRUSRVLWLYH VWHHU DQJOH FRQVLGHULQJ SRVLWLYH GLUHFWLRQV DV DGRSWHG LQ )LJ $W ODUJHU YHORFLWLHVWKHW\UHVOLSDQJOHVLQFUHDVHDQGDVDUHVXOW FKDQJHVLQWRWKHSRVLWLYH GLUHFWLRQ
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
DQG
DQG
'R WKLV LQ WHUPV RI WKH
DQGWKHDVVRFLDWHGGLIIHUHQWLDOHTXDWLRQ)LQGWKH
VVDQGSORWWKLVDVDIXQFWLRQRIWKHVSHHG
IUHTXHQF\UHVSRQVHIXQFWLRQ
M
96KRZDOVRWKH 9
IRUWKHDPSOLWXGHDQGSKDVHDWDVSHHG
PV8VHWKHYHKLFOHSDUDPHWHUVVXSSOLHGLQ7DEOH
O
E α
) \
δ
β Y
U
D 9
α
X
δ )\
)LJµ)RXUZKHHO¶VWHHULQJWRPDNHVOLSDQJOH
([HUFLVH
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
1RQ/LQHDU6WHDG\6WDWH&RUQHULQJ6ROXWLRQV
)URP(TV DQG ZLWKWKHVDPHUHVWULFWLRQVDVVWDWHGEHORZ(T WKH IROORZLQJ IRUFH EDODQFH HTXDWLRQV FDQ EH GHULYHG IROORZV DOVR IURP (TV 7KHHIIHFWRIWKHSQHXPDWLFWUDLOVZLOOEHGHDOWZLWKODWHURQ
) \ )\ D \ ' ' ) ] )] J :KHUH.
' . PJ
PD\UHSUHVHQWVWKHFHQWULIXJDOIRUFH7KHNLQHPDWLFUHODWLRQVKLS
& &
' O 5
IROORZV IURP (TV DQG ,Q )LJ WKH YHKLFOH PRGHO KDV EHHQ GHSLFWHGLQDVWHDG\VWDWHFRUQHULQJPDQRHXYUH,WFDQHDVLO\EHREVHUYHGIURP WKLVGLDJUDPWKDWUHODWLRQ KROGVDSSUR[LPDWHO\ZKHQWKHDQJOHVDUHVPDOO 7KHUDWLRRIWKHVLGHIRUFHDQGYHUWLFDOORDGDVVKRZQLQ SORWWHGDVD IXQFWLRQ RI WKH VOLS DQJOH PD\ EH WHUPHG DV WKH QRUPDOLVHG W\UH RU D[OH FKDUDFWHULVWLF 7KHVH FKDUDFWHULVWLFV VXEWUDFWHG KRUL]RQWDOO\ IURP HDFK RWKHU SURGXFHWKHµKDQGOLQJFXUYH¶&RQVLGHULQJWKHHTXDOLWLHV WKHRUGLQDWHPD\ EHUHSODFHGE\D\J7KHUHVXOWLQJGLDJUDPZLWKDEVFLVVD
E
LVWKHQRQOLQHDU
YHUVLRQRIWKHULJKWKDQGGLDJUDPRI)LJURWDWHG DQWLFORFNZLVH 7KH GLDJUDP PD\ EH FRPSOHWHG E\ DWWDFKLQJ WKH JUDSK WKDW VKRZV IRU D VHULHV RI VSHHGV9WKHUHODWLRQVKLSEHWZHHQODWHUDODFFHOHUDWLRQLQJXQLWV D \ JDQGWKH UHODWLYHSDWKFXUYDWXUHO5DFFRUGLQJWR(T
)LJXUH VKRZV WKH QRUPDOLVHG D[OH FKDUDFWHULVWLFV DQG WKHFRPSOHWHG
KDQGOLQJGLDJUDP7KHKDQGOLQJFXUYHFRQVLVWVRIDPDLQEUDQFKDQGWZRVLGH OREHV 7KH GLIIHUHQW SRUWLRQV RI WKH FXUYHV KDYH EHHQ FRGHG WR LQGLFDWH WKH FRUUHVSRQGLQJSDUWVRIWKHRULJLQDOQRUPDOLVHGD[OHFKDUDFWHULVWLFVWKH\RULJLQDWH IURP 1HDU WKH RULJLQ WKH V\VWHP PD\ EH DSSUR[LPDWHG E\ D OLQHDU PRGHO &RQVHTXHQWO\WKHVORSHRIWKHKDQGOLQJFXUYHLQWKHRULJLQZLWKUHVSHFWWRWKH YHUWLFDOD[LVLVHTXDOWRWKHXQGHUVWHHUFRHIILFLHQW
,QFRQWUDVWWRWKHVWUDLJKW
KDQGOLQJ OLQH RI WKH OLQHDU V\VWHP )LJ WKH QRQOLQHDU V\VWHP VKRZV D FXUYHGOLQH7KHVORSHFKDQJHVDORQJWKHFXUYHZKLFKPHDQVWKDWWKHGHJUHHRI XQGHUVWHHUFKDQJHVZLWKLQFUHDVLQJODWHUDODFFHOHUDWLRQ7KHGLDJUDPRI)LJ VKRZV WKDW WKH YHKLFOH FRQVLGHUHG FKDQJHV IURP XQGHUVWHHU WR RYHUVWHHU :H GHILQH
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
XQGHUVWHHU RYHUVWHHU
LI
LI
M M9 M M9
!
5
5
7KHIDPLO\RIVWUDLJKWOLQHVUHSUHVHQWVWKHUHODWLRQVKLSEHWZHHQDFFHOHUDWLRQDQG FXUYDWXUHDWGLIIHUHQWOHYHOVRIVSHHG7KHVSHHGOLQHEHORQJLQJWR9 EHHQLQGLFDWHGZKHHOEDVHO HTXDO WR WKH VWHHU DQJOH
NPKKDV
P 7KLVOLQHLVVKLIWHGWRWKHOHIWRYHUDGLVWDQFH UDG DQG WKUHH SRLQWV RI LQWHUVHFWLRQ ZLWK WKH
KDQGOLQJFXUYHDULVH7KHVHSRLQWV,,,DQG,,,LQGLFDWHWKHSRVVLEOHHTXLOLEULXP FRQGLWLRQV DW WKH FKRVHQ VSHHG DQG VWHHU DQJOH 7KH FRQQHFWHG YDOXHV RI WKH UHODWLYHSDWKFXUYDWXUHO5FDQEHIRXQGE\JRLQJEDFNWRWKHVSHHGOLQH$VZLOO
1
L
=1 (front)
L
=2 (rear)
) \L
) ]L
α L [rad]
0.1
D\ 9 J J5 1 oversteer
understeer
α1− α 2= δ − δ0
δ =0.04 rad =50 km/h
9
II
( O=3 m) 9
I
O
0.1
O 5
5
III )LJ
+DQGOLQJ GLDJUDP UHVXOWLQJ IURPQRUPDOL]HGW\UHFKDUDFWHULVWLFV(TXLOLEULXP SRLQWV,,,DQG,,,VWHDG\WXUQV RIZKLFKRQO\,LVVWDEOHDULVHIRUVSHHG9 NPKDQGVWHHUDQJOH
UDG)URPWKHGLIIHUHQWOLQHW\SHV WKH PDQQHULQ
ZKLFKWKHFXUYHVDUHREWDLQHGIURPWKHXSSHUGLDJUDPPD\EHUHWULHYHG
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
EHVKRZQIXUWKHURQRQO\SRLQW,UHIHUVWRDVWDEOHFRUQHULQJPRWLRQ,QSRLQWV ,,DQG,,,5 WKHPRWLRQLVXQVWDEOH $WDJLYHQVSHHG9DFHUWDLQVWHHUDQJOH LVQHHGHGWRQHJRWLDWHDFLUFXODU SDWKZLWKJLYHQUDGLXV57KHVWHHUDQJOHUHTXLUHGFDQEHUHDGGLUHFWO\IURPWKH KDQGOLQJGLDJUDP7KHVWHHUDQJOHQHHGHGWRQHJRWLDWHWKHVDPHFXUYHDWYHU\ORZ
ß
VSHHG9
WHQGVWRO57KLVVWHHUDQJOHLVGHQRWHGZLWK
DEVFLVVDRIWKHKDQGOLQJFXUYH
&RQVHTXHQWO\WKH
PD\DVZHOOEHUHSODFHGE\
7KLVRSHQV
WKH SRVVLELOLW\ WR GHWHUPLQH WKH KDQGOLQJ FXUYH ZLWK WKH DLG RI VLPSOH H[SHULPHQWDO PHDQV LH PHDVXULQJ WKH VWHHULQJ ZKHHO LQSXW UHGXFHG WR HTXLYDOHQWURDGZKHHOVWHHUDQJOHE\PHDQVRIWKHVWHHULQJUDWLRZKLFKPHWKRG DXWRPDWLFDOO\LQFOXGHVVWHHULQJFRPSOLDQFHHIIHFWV DWYDULRXVVSHHGVUXQQLQJ RYHUWKHVDPHFLUFXODUSDWK 6XEWUDFWLQJ QRUPDOL]HG FKDUDFWHULVWLFV PD\ JLYH ULVH WR YHU\ GLIIHUHQWO\
1 ) \L
oversteer
rear slide
2
) ]L
understeer
2
front slide
1 understeer
rear slide
1 2
oversteer
2
front slide
understeer
1 oversteer
α1 , α 2 )LJ
α1 − α2
$QXPEHURIKDQGOLQJFXUYHVDULVLQJIURPWKHSDLUVRIQRUPDOL]HGW\UHFKDUDFWHULVWLFV
VKRZQOHIW2QO\WKHPDLQEUDQFKRIKDQGOLQJFXUYHLVGUDZQIURQWUHDU
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
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
D\
J
D\
9
J
O
5
δ
δ
( )
9 ~ X ~
O
O
O
5
5
D\ J
D\ J
δ
δ
5
δ
9
9
O
O
O
5
5
O
5
)LJ7\SHVRITXDVLVWHDG\VWDWHPDQRHXYUHV
5
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
6WDELOLW\RIWKH0RWLRQDW/DUJH/DWHUDO$FFHOHUDWLRQV
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
M)\L
L'
)]L M
L '
L
7KH FRQGLWLRQV IRU VWDELOLW\ WKDW LV VHFRQG DQG ODVW FRHIILFLHQW RI HTXDWLRQ FRPSDUDEOHZLWK(T PXVWEHSRVLWLYHUHDGDIWHUKDYLQJLQWURGXFHGWKH
UDGLXVRIJ\UDWLRQNN
N %D
,P
% N %E
M
M 5
!
!
9
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
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
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
LVVPDOODQG
LV
ODUJH$SSDUHQWO\ZHILQGWKDWWKHYHKLFOHWKDWLQFUHDVHVLWVVSHHGZKLOHUXQQLQJ DWDFRQVWDQWWXUQLQJUDGLXVZLOOILUVWFURVVWKHVWDELOLW\ERXQGDU\DQGPD\WKHQ
stable
D\ J
δPD[
9
unstable stable 9 O
α1− α2
5
δ
9
δ α1− α2
O
5
δ
α1− α2 )LJ
O
5
&RQVWUXFWLRQRIVWDELOLW\ERXQGDU\XSSHUGLDJUDPIURP)LJ 2QWKHLVRODWHG EUDQFKDVWDEOHUDQJHPD\RFFXUODUJHVWHHUDQJOHDVLQGLFDWHGLQPLGGOHGLDJUDP 7KHORZHUGLDJUDPVKRZVWKHFDVHZLWKFRPSOHWHXQGHUVWHHUIHDWXULQJDVWDEOHPDLQ EUDQFK
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
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
RIWKHQRUPDOLVHGD[OHFKDUDFWHULVWLFVVWDUWVWRVKRZ
D QHJDWLYH VLJQ VR WKDW FRQGLWLRQ LV VWLOO VDWLVILHG 6LPLODUO\ WKH OLPLW RYHUVWHHUYHKLFOHRIWKHXSSHUJUDSKUHPDLQVXQVWDEOHEH\RQGWKHSHDN2QWKH LVRODWHG SDUW RI WKH KDQGOLQJ FXUYH RI WKH ORZHU GLDJUDP WKH PRWLRQ UHPDLQV XQVWDEOH,WZLOOEHFOHDUWKDWWKHLVRODWHGEUDQFKHVYDQLVKZKHQZHGHDOZLWKD[OH FKDUDFWHULVWLFVWKDWGRQRWVKRZDSHDNDQGGHFD\LQJSDUWRIWKHFXUYH ,WPD\VHHPWKDWWKHHVWDEOLVKPHQWRIXQVWDEOHVROXWLRQVKDVQRSDUWLFXODU YDOXH,WZLOOEHFRPHFOHDUKRZHYHUWKDWWKHH[LVWHQFHDQGWKHORFDWLRQRIERWK VWDEOH DQG XQVWDEOH VLQJXODU SRLQWV SOD\ DQ LPSRUWDQW UROH LQ VKDSLQJ WKH WUDMHFWRULHVLQWKHSKDVHSODQH$OVRWKHQDWXUHRIVWDELOLW\RULQVWDELOLW\LQWKH VLQJXODUSRLQWVDUHRILPSRUWDQFH
([HUFLVH &RQVWUXFWLRQRIWKHFRPSOHWHKDQGOLQJGLDJUDPIURPSDLUVRID[OH FKDUDFWHULVWLFV
front
a, b
(dry)
0.8 ) \1
c (wet, new tyre)
) ]1
0.4
0
[rad]
0.1
0
0.2
α1
0.3
a
rear
b (different type)
0.8 ) \2
c (wet, worn tyre)
) ]2
0.4
0
0
0.1
[rad]
0.2
α2
0.3
)LJ7KUHHVHWVRIK\SRWKHWLFDOD[OHFRUQHULQJFKDUDFWHULVWLFV([HUFLVH
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
:HFRQVLGHUWKUHHVHWVRIK\SRWKHWLFDOD[OHFKDUDFWHULVWLFVDEDQGF VKRZQLQWKH JUDSKRI)LJ7KHGLPHQVLRQVRIWKHYHKLFOHPRGHODUHD
E
òO
P)RU
WKHW\UHVZHPD\HPSOR\D[OHFKDUDFWHULVWLFVGHVFULEHGE\WKH0DJLF)RUPXOD
)\
'VLQ>&DUFWDQ^%
(%
:HGHILQHWKHSHDNVLGHIRUFH' VRWKDW%
DUFWDQ%
`@
:) DQGWKHFRUQHULQJVWLIIQHVV& ]
)
%&'
F) )]
F) & )RUWKHVL[W\UHD[OHFRQILJXUDWLRQVWKHSDUDPHWHUYDOXHVKDYH
EHHQJLYHQLQWKHWDEOHEHORZ
D[OH
FDVH
F)
&
(
DE
F
D
E
F
IURQW
UHDU
'HWHUPLQHIRUHDFKRIWKHWKUHHFRPELQDWLRQVWZRGU\RQHZHW
7KHKDQGOLQJFXUYHFI)LJ
7KHFRPSOHWHKDQGOLQJGLDJUDPFI)LJ
7KHSRUWLRQRIWKHFXUYHVZKHUHWKHYHKLFOHVKRZVDQRYHUVWHHUQDWXUH
7KHVWDELOLW\ERXQGDU\DVVRFLDWHGZLWKWKHVHRYHUVWHHUUDQJHV LQWKHD\JYHUVXV
,QGLFDWHLQWKHGLDJUDPRULQDVHSDUDWHJUDSK
O5 GLDJUDP
ULJKWKDQGVLGHRIWKHKDQGOLQJGLDJUDP FI)LJ
D WKHFRXUVHRIWKHVWHHUDQJOH UHTXLUHGWRQHJRWLDWHDFXUYHZLWKUDGLXV5 P DVDIXQFWLRQRIWKHVSHHG9,IDSSOLFDEOHLQGLFDWHWKHVWDELOLW\ERXQGDU\WKDW LVWKHFULWLFDOVSHHG9FULWEHORQJLQJWRWKLVUDGLXV E WKHFRXUVHRIVWHHUDQJOH DVDIXQFWLRQRIUHODWLYHSDWKFXUYDWXUHO5DWDIL[HG VSHHG9
NPKDQGLIDSSOLFDEOHDVVHVVWKHFULWLFDOUDGLXV5FULW
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
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
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
$VVHVVPHQWRIWKH,QIOXHQFHRIWKH3QHXPDWLF7UDLORQWKH+DQGOLQJ&XUYH
6R IDU WKH GLUHFW LQIOXHQFH RI WKH SQHXPDWLF WUDLOV KDYH QRW EHHQ WDNHQ LQWR DFFRXQW$VZLWKWKHOLQHDUDQDO\VLVZHPD\GRWKLVE\FRQVLGHULQJWKHHIIHFWLYH D[OHSRVLWLRQV
' &
D
D
W
' %
E
E
W
' %
DQG O
D
E
7KHGLIILFXOW\ZHKDYHWRIDFHQRZLVWKHIDFWWKDWWKHVHSQHXPDWLFWUDLOVWLZLOO YDU\ ZLWK WKH UHVSHFWLYH VOLS DQJOHV :H KDYH LI WKH UHVLGXDO WRUTXHV DUH QHJOHFWHG
W
L
'&
L
0 )
]L
L
\L
L
,QWURGXFLQJWKHHIIHFWLYHD[OHORDGV
)
]
'
E
PJ
)
]
O
'
D
PJ
O
\LHOGVIRUWKHODWHUDOIRUFHEDODQFHLQVWHDGRI )
\
)
)
'
\
)
]
'
D
\
J
]
RUDIWHUVRPHUHDUUDQJHPHQWV
D
D
ZLWK 4
)
\
)
]
'
'
E
E
O
D E
O
D E
)
\
)
.
]
'
D 4
\
J
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
7KHFRUUHFWHGQRUPDOLVHGVLGHIRUFHFKDUDFWHULVWLFVDVLQGLFDWHGLQ FDQEH FRPSXWHGEHIRUHKDQGDQGGUDZQDVIXQFWLRQVRIWKHVOLSDQJOHVDQGWKHQRUPDO SURFHGXUHWRDVVHVVWKHKDQGOLQJ FXUYHFDQEHIROORZHG7KLVFDQEHGRQHE\ RUZHPLJKWVHOHFWDOHYHORI4D\ J
WDNLQJWKHYHU\JRRGDSSUR[LPDWLRQ4
WKHQDVVHVVWKHYDOXHVRIWKHVOLSDQJOHVWKDWEHORQJWRWKDWOHYHORIWKHFRUUHFWHG QRUPDOLVHG VLGH IRUFHV DQG FRPSXWH 4 DFFRUGLQJ WR DQG IURP WKDW WKH FRUUHFWYDOXHRID\J
/DUJH'HYLDWLRQVZLWK5HVSHFWWRWKH6WHDG\VWDWH0RWLRQ
7KHYDULDEOHVUDQGYPD\EHFRQVLGHUHGDVWKHWZRVWDWHYDULDEOHVRIWKHVHFRQG RUGHUQRQOLQHDUV\VWHPUHSUHVHQWHGE\WKHHTXDWLRQV 7KURXJKFRPSXWHU QXPHULFDO LQWHJUDWLRQ WKH UHVSRQVH WR D JLYHQ DUELWUDU\ YDULDWLRQ RI WKH VWHHU DQJOHFDQEHHDVLO\REWDLQHG)RUPRWLRQVZLWKFRQVWDQWVWHHUDQJOH SRVVLEO\ DIWHUDVWHSFKDQJH WKHV\VWHPLVDXWRQRPRXVDQGWKHSKDVHSODQHUHSUHVHQW DWLRQPD\EHXVHGWRILQGWKHVROXWLRQ)RUWKDWZHSURFHHGE\HOLPLQDWLQJWKH WLPHIURP(TV 7KHUHVXOWLVDILUVWRUGHUQRQOLQHDUHTXDWLRQXVLQJN
,P GY GU
'
N
)
\
%
)
\
D)
\
&
&
P9U
E)
\
6LQFH)\DQG)\DUHIXQFWLRQVRI
DQG
LWPD\EHHDVLHUWRWDNH
DQG
DV
WKHVWDWHYDULDEOHV:LWK ZHREWDLQ G
G
'
GYGU GYGU
& %
E
D
ZKLFKEHFRPHVZLWK G
G
'
)
\
\
)
)]
)]
& & % & & %
9 JO
9 JO
)RUWKHVDNHRIVLPSOLFLW\ZHKDYHDVVXPHG,P
N
DE
%\XVLQJ(T WKHWUDMHFWRULHVVROXWLRQFXUYHV FDQEHFRQVWUXFWHGLQWKH
SODQH7KHLVRFOLQHPHWKRGWXUQVRXWWREHVWUDLJKWIRUZDUGDQGVLPSOHWR
HPSOR\7KHSDWWHUQRIWKHWUDMHFWRULHV LV VWURQJO\LQIOXHQFHGE\WKHVRFDOOHG VLQJXODUSRLQWV,QWKHVHSRLQWVWKHPRWLRQILQGVDQHTXLOLEULXP,QWKHVLQJXODU SRLQWVWKHPRWLRQLVVWDWLRQDU\ DQG FRQVHTXHQWO\WKHGLIIHUHQWLDOVRIWKHVWDWH YDULDEOHVYDQLVK )URP WKH KDQGOLQJ GLDJUDP .PJ DQG O5 DUH UHDGLO\ REWDLQHG IRU JLYHQ FRPELQDWLRQV RI 9 DQG
8VHG LQ FRPELQDWLRQ ZLWK WKH QRUPDOL]HG W\UH
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
FKDUDFWHULVWLFV)\)]DQG) \)]WKHYDOXHVRI
DQG
DUHIRXQGZKLFKIRUP
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
G
SODQH
NHSWFRQVWDQW7KH
IROORZLQJ WKUHH LVRFOLQHV PD\ DOUHDG\ SURYLGH VXIILFLHQW LQIRUPDWLRQ WR GUDZ
HVWLPDWHGFRXUVHVRIWKHWUDMHFWRULHV:HKDYHIRUN YHUWLFDOLQWHUFHSWVG
'
JO 9
)
\
G
'&
JO
)
\
)
G
ß
DE
% %
]
)
% &
LQWHUFHSWVXQGHU G
\
]
9
)
ß4
)
KRUL]RQWDOLQWHUFHSWVG
'
)
]
\
G
)
]
)LJXUHLOOXVWUDWHVWKHZD\WKHVHLVRFOLQHVDUHFRQVWUXFWHG7KHV\VWHPRI )LJZLWKN
D
E
UDGDQG9
NPKKDVEHHQFRQVLGHUHG1RWH
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
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
α
.<0
trajectories
U
2
DU % 9
α
. U >0 L
=1
II
0.1
0.1
α
JO) \LL
2
δ
2 9)]L
I
0.1
0
II area of attraction
I
α
α
0.1
isoclines
2
Y% 9
III
III
separatrix area of repulsion
)LJ
,VRFOLQHVIRUWKHFRQVWUXFWLRQRIWUDMHFWRULHVLQWKHSKDVHSODQH$OVRVKRZQWKHWKUHH VLQJXODU SRLQWV , ,, DQG ,,, FI )LJ DQG WKH VHSDUDWULFHV FRQVWLWXWLQJ WKH ERXQGDU\RIWKHGRPDLQRIDWWUDFWLRQ3RLQW,UHSUHVHQWVWKHVWDEOHFRUQHULQJPRWLRQ DWVWHHUDQJOH
)URP WKH RULJLQDO HTXDWLRQV LW FDQ EH IRXQG WKDW WKH LVRFOLQH IRUPVWKHERXQGDU\EHWZHHQDUHDVZLWKU s!DQGU sLQGLFDWHGLQ)LJ 1RZLWLVHDV\WRDVFHUWDLQWKHGLUHFWLRQDORQJWKHWUDMHFWRULHV:HQRWHWKDWWKH V\VWHP H[KLELWV D ERXQGHG GRPDLQ RI DWWUDFWLRQ 7KH ERXQGDULHV DUH FDOOHG VHSDUDWULFHV 2QFH RXWVLGH WKH GRPDLQ WKH PRWLRQ ILQGV LWVHOI LQ DQ XQVWDEOH VLWXDWLRQ5HPDLQVWKHGLVWXUEDQFHOLPLWHGVRWKDWUHVXOWLQJLQLWLDOFRQGLWLRQVRI WKHVWDWHYDULDEOHVVWD\ZLWKLQWKHERXQGDULHVWKHQXOWLPDWHO\WKHVWHDG\VWDWH FRQGLWLRQLVUHDFKHGDJDLQ )RUV\VWHPVZLWKQRUPDOL]HGFKDUDFWHULVWLFVVKRZLQJHYHU\ZKHUHDSRVLWLYH VORSHDKDQGOLQJFXUYHDULVHVWKDWFRQVLVWVRIRQO\WKHPDLQEUDQFKWKURXJKWKH RULJLQ,IWKHUHDUD[OHFKDUDFWHULVWLFDWOHDVWLQWKHHQG LVKLJKHUWKDQWKHIURQW D[OH FKDUDFWHULVWLF WKH YHKLFOH ZLOO VKRZ DW OHDVW LQ WKH OLPLW DQ XQGHUVWHHU QDWXUHDQGXQVWDEOHVLQJXODUSRLQWVFDQQRWRFFXU7KLVDWOHDVWLIIRUWKHFDVHRI LQLWLDORYHUVWHHU WKHVSHHGUHPDLQVXQGHUWKHFULWLFDOVSHHG,QVXFKFDVHVWKH GRPDLQRIDWWUDFWLRQLVWKHRUHWLFDOO\XQERXQGHGVRWKDWIRUDOOLQLWLDOFRQGLWLRQV XOWLPDWHO\WKHVWDEOHHTXLOLEULXPLVDWWDLQHG7KHGRPDLQRI)LJDSSHDUVWR EHRSHQRQWZRVLGHVZKLFKPHDQVWKDWLQLWLDOFRQGLWLRQVLQDFHUWDLQUDQJHRI UY YDOXHV GR QRW UHTXLUH WR EH OLPLWHG LQ RUGHU WR UHDFK WKH VWDEOH SRLQW 2EYLRXVO\GLVWXUEDQFHLPSXOVHVDFWLQJLQIURQWRIWKHFHQWUHRIJUDYLW\PD\JLYH ULVHWRVXFKFRPELQDWLRQVRILQLWLDOFRQGLWLRQV ,Q )LJV DQG WKH LQIOXHQFH RI DQ LQFUHDVH LQ VWHHU DQJOH
RQ WKH
VWDELOLW\PDUJLQGLVWDQFHEHWZHHQVWDEOHSRLQWDQGVHSDUDWUL[ KDVEHHQVKRZQ
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
δ
=0
δ
= 0.04
δ
= 0.06
δ
= 0.15
δ
= 0.2
α2 U
0.1
0.1
α1
Y
δ
= 0.1
)LJ,QIOXHQFHRIVWHHULQJRQWKHVWDELOLW\PDUJLQV\VWHPRI)LJWRS
δ
δ
=0
= 0.04
δ
= 0.15
0.1
0.1
)LJ,QIOXHQFHRIVWHHULQJRQWKHVWDELOLW\PDUJLQV\VWHPRI)LJERWWRP
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
IRU WKHWZRYHKLFOHVFRQVLGHUHGLQ)LJ7KHV\VWHPRI)LJLVFOHDUO\ PXFKPRUHVHQVLWLYH$QLQFUHDVHLQ EXWDOVRDQLQFUHDVHLQVSHHG9 UHGXFHV
WKHVWDELOLW\PDUJLQXQWLOLWLVWRWDOO\YDQLVKHGDVVRRQDVWKHWZRVLQJXODUSRLQWV PHUJHDOVRWKHFRUUHVSRQGLQJSRLQWV,DQG,,RQWKHKDQGOLQJFXUYHRI)LJ DQGWKHGRPDLQEUHDNVRSHQ$VDUHVXOWDOOWUDMHFWRULHVVWDUWLQJDERYHWKHORZHU VHSDUDWUL[ WHQG WR OHDYH WKH DUHD 7KLV FDQ RQO\ EH VWRSSHG E\ HLWKHU TXLFNO\ UHGXFLQJ WKH VWHHU DQJOH RU HQODUJLQJ
WR DURXQG UDG RU PRUH 7KH ODWWHU
VLWXDWLRQDSSHDUVWREHVWDEOHDJDLQIRFXV DVKDVEHHQVWDWHGEHIRUH)RUWKH XQGHUVWHHUHGYHKLFOHRI)LJVWDELOLW\LVSUDFWLFDOO\DOZD\VHQVXUHG )RUDIXUWKHUDSSUHFLDWLRQRIWKHSKDVHGLDJUDPLWLVRILQWHUHVWWRGHWHUPLQH WKHQHZLQLWLDOVWDWHURYR DIWHUWKHDFWLRQRIDODWHUDOLPSXOVHWRWKHYHKLFOHFI )LJ )RUDQLPSXOVH6DFWLQJDWDGLVWDQFH[LQIURQWRIWKHFHQWUHRIJUDYLW\ WKHLQFUHDVHLQUDQGYEHFRPHV
U
'
6[
Y
,
'
6
P
ZKLFKUHVXOWVLQWKHGLUHFWLRQ D
U Y
'
[
DE
E N
[
DU
U D
6
E
∆U
D
δ
Y
[E
+1
+
∆Y
Y
-1 6
)LJ
<0
6
>0
boundary of domain of attraction
/DUJHGLVWXUEDQFHLQDFXUYH1HZLQLWLDOVWDWHYHFWRU
Y
U DIWHUWKHDFWLRQRID
ODWHUDO LPSXOVH 6 2QFH RXWVLGH WKH GRPDLQ RI DWWUDFWLRQ WKH PRWLRQ EHFRPHV XQVWDEOHDQGPD\JHWRXWRIFRQWURO
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
7KHILJXUHVKRZVWKHFKDQJHLQVWDWHYHFWRUIRUGLIIHUHQWSRLQWVRIDSSOLFDWLRQ DQGGLUHFWLRQRIWKHLPSXOVH6N
,P
DE (YLGHQWO\DQLPSXOVHDFWLQJDWWKH
UHDULQRXWZDUGGLUHFWLRQ FRQVWLWXWHVWKHPRVWGDQJHURXVGLVWXUEDQFH2QWKH RWKHUKDQGDQLPSXOVHDFWLQJLQIURQWRIWKHFHQWUHRIJUDYLW\DERXWKDOIZD\WKH IURQWD[OHGRHVQRWDSSHDUWREHDEOHWRJHWWKHQHZVWDUWLQJSRLQWRXWVLGHRIWKH GRPDLQRIDWWUDFWLRQQRPDWWHUWKHLQWHQVLW\RIWKHLPSXOVH :KHQWKHVOLSDQJOHVEHFRPHODUJHUWKHIRUZDUGVSHHGXPD\QRORQJHUEH FRQVLGHUHGDVDFRQVWDQWTXDQWLW\7KHQWKHV\VWHPLVGHVFULEHGE\DWKLUGRUGHU VHW RI HTXDWLRQV ,Q WKH SDSHU 3DFHMND WKH VROXWLRQV IRU WKH VLPSOH
E
DXWRPRELOHPRGHOKDYHEHHQSUHVHQWHGDOVRIRU\DZDQJOHV!
7KH9HKLFOHDW%UDNLQJRU'ULYLQJ
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
)/
)]1
)[1
K
)]2 )[2 O
)LJ7KHDXWRPRELOHVXEMHFWHGWRORQJLWXGLQDOIRUFHVDQGWKHUHVXOWLQJORDGWUDQVIHU
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
YDULDWLRQVRIWKHFRUQHULQJVWLIIQHVVZLWKYHUWLFDOORDG 7KH ORDG WUDQVIHU IURP WKH UHDU D[OH WR WKH IURQW D[OH WKDW UHVXOWV IURP D IRUZDUGORQJLWXGLQDOIRUFH)/DFWLQJDWWKHFHQWUHRIJUDYLW\DWDKHLJKWKDERYH WKH URDG VXUIDFH )/ SRVVLEO\ FRUUHVSRQGLQJ WR WKH LQHUWLDO IRUFH DW EUDNLQJ EHFRPHV
)] ' K )/ O
7KHXQGHUVWHHUJUDGLHQWUHDGVDFFRUGLQJWR(T
'
)] R ) & ]R & )] & )]
7KHVWDWLFD[OHORDGV)]LRL
RU DUHFDOFXODWHGDFFRUGLQJWR(T ZKLOH
WKHDFWXDOORDGV)]LIURQWDQGUHDUEHFRPH
)] ' )] R % ) ]
)] ' )] R & )]
$WPRGHUDWHEUDNLQJZLWKGHFHOHUDWLRQD[
)/PWKHORDGWUDQVIHUUHPDLQVVPDOO
DQG ZH PD\ XVH WKH OLQHDULVHG DSSUR[LPDWLRQ RI WKH YDULDWLRQ RI FRUQHULQJ VWLIIQHVVZLWKYHUWLFDOORDG
& L ' &LR %
) ]L
L
L'
ZLWK
M& L
M) ]L
)
]LR
7KH XQGHUVWHHU JUDGLHQW FDQ QRZ EH H[SUHVVHG LQ WHUPV RI WKH ORQJLWXGLQDODFFHOHUDWLRQD[ZKLFKPLJKWEHPLQXVWKHIRUZDUGFRPSRQHQWRI WKHDFFHOHUDWLRQGXHWRJUDYLW\SDUDOOHOWRWKHURDG :HREWDLQ
'
D[ J
R%
ZLWKWKHGHWHUPLQLQJIDFWRU DSSUR[LPDWHO\H[SUHVVHGDV
'
K )] R % E &R
DQG
R
K )] R D &R
GHQRWLQJWKHRULJLQDOYDOXHQRWLQFOXGLQJWKHHIIHFWRIORQJLWXGLQDOIRUFHV
2EYLRXVO\VLQFH
LVXVXDOO\SRVLWLYHQHJDWLYHORQJLWXGLQDODFFHOHUDWLRQVD[
FRUUHVSRQGLQJWREUDNLQJZLOOUHVXOWLQDGHFUHDVHRIWKHGHJUHHRIXQGHUVWHHU 7RLOOXVWUDWHWKHPDJQLWXGHRIWKHHIIHFWZHXVHWKHSDUDPHWHUYDOXHVJLYHQ LQ7DEOHDERYH(T DQGDGGWKHFJKHLJKWK VWLIIQHVVYHUVXVORDGJUDGLHQWV WDNHWKHYDOXH
L
PDQGWKHFRUQHULQJ
&LR)]LR7KHUHVXOWLQJIDFWRUDSSHDUVWR
7KLVFRQVWLWXWHVDQLQFUHDVH RI
HTXDOWRD[J
$SSDUHQWO\ WKH HIIHFW RI D[ RQ WKH XQGHUVWHHU JUDGLHQW LV FRQVLGHUDEOH ZKHQ
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
UHJDUGLQJWKHRULJLQDOYDOXH
R
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
7KH0RPHQW0HWKRG
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¶VERRN 7KHJUDSKVKRZVFXUYHVRIWKHUHVXOWLQJW\UHPRPHQW1YHUVXVWKHUHVXOWLQJ W\UHVLGHIRUFH<LQQRQGLPHQVLRQDOIRUP7KHUHVXOWLQJIRUFHDQGPRPHQWUHVXOW
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
0.4 -4
δ
9
)\1 β
1
-6
β = -8o
1 PJ
O
0
10
20
30
45
-1
60
-0.5 0 0.5 1 2
0
<
-2
δstw = 90o 120
-120
4 6 8
)\2 -0.4 -1
0
000
)LJ7KH
< = D\ PJ J
1
GLDJUDPSRUWUD\LQJWKHFDU¶VSRWHQWLDOPDQRHXYULQJFDSDFLW\
IURP WKH LQGLYLGXDO VLGH IRUFHV DQG DFW IURP JURXQG WR YHKLFOH )RU JUHDWHU DFFXUDF\RQHPD\WDNHWKHHIIHFWRIWKHSQHXPDWLFWUDLOVLQWRFRQVLGHUDWLRQ7ZR VHWVRIFXUYHVKDYHEHHQSORWWHGRQHVHWIRUFRQVWDQWYDOXHVRIWKHYHKLFOHVLGH VOLSDQJOH ZLWKWKHVWHHULQJZKHHODQJOH
VWZ
DVSDUDPHWHUDQGWKHRWKHUVHWIRU
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
DQGIURQWVWHHU
DQJOH FDQUHPDLQXQFKDQJHGZKLOHWKHUHDUVWHHUDQJOHSURGXFHWKHGHVLUHGUHDU W\UHVOLSDQJOH2IFRXUVHWKHGLDJUDPQHHGVWREHDGDSWHGLQFDVHRIUHDUZKHHO
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
700LOOLNHQ
)LJ7KH076)ODW7UDF5RDGZD\6LPXODWRU
VWHHULQJ$QRWKHUPRUH SUDFWLFOHVROXWLRQZRXOGEHWREULQJWKHYHKLFOHLQWKH GHVLUHGDWWLWXGH
6E E\EULHIO\LQGXFLQJODUJHEUDNHRUGULYHVOLSDWWKHUHDU
WKDWORZHUVWKHFRUQHULQJIRUFHDQGOHWVWKHFDUVZLQJWRWKHGHVLUHGVOLSDQJOH ZKLOHDWWKHVDPHWLPHWKHVWHHULQJZKHHOLVWXUQHGEDFNZDUGVWRHYHQQHJDWLYH YDOXHV 7KH 000 GLDJUDP ZKLFK LV DFWXDOO\ D *RXJK SORW IRU D VLQJOH W\UH FI )LJVDQG HVWDEOLVKHGIRUWKHZKROHFDUDWGLIIHUHQWVWHHUDQJOHVPD\EH DVVHVVHGH[SHULPHQWDOO\HLWKHUWKURXJKRXWGRRURULQGRRUH[SHULPHQWV2QWKH SURYLQJJURXQGDYHKLFOHPD\EHDWWDFKHGDWWKHVLGHRIDKHDY\WUXFNRUUDLOZD\ YHKLFOH DQG VHW DW GLIIHUHQW VOLS DQJOHV ZKLOH WKH IRUFH DQG PRPHQW DUH EHLQJ PHDVXUHG WHWKHUHG WHVWLQJ FI 0LOOLNHQ )LJXUH GHSLFWV WKH UHPDUNDEOH ODERUDWRU\ 000 WHVW PDFKLQH 7KLV 076 )ODW7UDF 5RDGZD\ 6LPXODWRU
70XVHVIRXUIODWEHOWVZKLFKFDQEHVWHHUHGDQGGULYHQLQGHSHQGHQWO\
7KHFDULVFRQVWUDLQWLQLWVFHQWUHRIJUDYLW\EXWLVIUHHWRUROODQGSLWFK
7KH&DU7UDLOHU&RPELQDWLRQ
,QWKLVVHFWLRQZHZLOOGLVFXVVWKHUROHRIWKHW\UHLQFRQQHFWLRQZLWKWKHG\QDPLF EHKDYLRXU RI D FDU WKDW WRZV D WUDLOHU 0RUH VSHFLILFDOO\ ZH ZLOO VWXG\ WKH SRVVLEOHXQVWDEOHPRWLRQVWKDWPD\VKRZXSZLWKVXFKDFRPELQDWLRQ/LQHDU GLIIHUHQWLDOHTXDWLRQVDUHVXIILFLHQWWRDQDO\VHWKHVWDELOLW\RIWKHVWUDLJKWDKHDG PRWLRQ:HZLOODJDLQHPSOR\/DJUDQJH¶VHTXDWLRQVWRVHWXSWKHHTXDWLRQVRI PRWLRQ7KHRULJLQDOHTXDWLRQV PD\EHHPSOR\HGEHFDXVHWKH\DZDQJOH
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
LVDVVXPHGWRUHPDLQVPDOO7KHJHQHUDOLVHGFRRUGLQDWHV<
DQG DUHXVHGWR
GHVFULEH WKH FDU¶V ODWHUDO SRVLWLRQ DQG WKH \DZ DQJOHV RI FDU DQG WUDLOHU UHVSHFWLYHO\ 7KH IRUZDUG VSHHG G;GW
.9 . X LV FRQVLGHUHG WR EH FRQVWDQW
)LJXUHJLYHVDWRSYLHZRIWKHV\VWHPZLWKWKUHHGHJUHHVRIIUHHGRP7KH
nDQGWKHYHKLFOH
DOWHUQDWLYHVHWRIWKUHHYDULDEOHVYUDQGWKHDUWLFXODWLRQDQJOH
YHORFLW\9DSDUDPHWHU ZKLFKDUHQRWFRQQHFWHGWRWKHLQHUWLDOD[HVV\VWHP ; < KDV EHHQ LQGLFDWHG DV ZHOO DQG ZLOO EH HPSOR\HG ODWHU RQ 7KH NLQHWLF HQHUJ\IRUWKLVV\VWHPEHFRPHVLIZHQHJOHFWDOOWKHWHUPVRIWKHVHFRQGRUGHU RIPDJQLWXGHSURGXFWVRIYDULDEOHV
7 ' ò P;0 % <0 % ò , 0 % ò P F^;0 % <0&K 0 &I 0 `% ò ,F 0
7KHSRWHQWLDOHQHUJ\UHPDLQV]HUR
8'
DQGWKHYLUWXDOZRUNGRQHE\WKHH[WHUQDOURDGFRQWDFWIRUFHVDFWLQJRQWKHWKUHH D[OHVUHDGV
: ' )\
< % D % )\
< & E % )\
<&K &J
:LWKWKHXVHRIWKHHTXDWLRQV DQG WKHIROORZLQJHTXDWLRQVRIPRWLRQ DUHHVWDEOLVKHGIRUWKHJHQHUDOLVHGFRRUGLQDWHV<
P % P F < & PF K % I ' )\ % )\ % )\
, % PF I
F
DQG
, % PF K
& PF I < & K ' & J)\
& PF K < & I ' D)\ & E)\ & K)\
7KLVFRQVWLWXWHVDV\VWHPRIWKHVL[WKRUGHU%\LQWURGXFLQJWKHYHORFLWLHVYDQG
0
α3
ψ
PF
θ
,F
)\3
K ϕ
J
< E α2
I
)\2
U
; O
;
D
Y
P,
9
α1
X
< )LJ6LQJOHWUDFNPRGHORIFDUWUDLOHUFRPELQDWLRQ
)\1
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
UWKHRUGHUFDQEHUHGXFHGWRIRXU,QDGGLWLRQWKHDQJOHRIDUWLFXODWLRQ
n ZLOO
EHXVHG:HKDYHWKHUHODWLRQV
<0 ' 9 % Y
'U
' &n n P%P F Y%9U 0 &PF ^ K%I U&I 0 n ` ' )\ % )\ % )\ 0
DQGZLWKWKHVHWKHHTXDWLRQVIRUYUDQG
,%P F K K% I ` U0 &PF K Y%9U 0 %I n ' D)\ & E)\ & K)\
^
, % PF I
F
n& U0 % PF I Y0 % 9U & K U0 ' J)\
7KH ULJKWKDQG PHPEHUV DUH VWLOO WR EH H[SUHVVHG LQ WHUPV RI WKH PRWLRQ YDULDEOHV:LWKWKHD[OHFRUQHULQJVWLIIQHVVHV&&DQG&ZHKDYH
)\ ' &
' & & Y % DU
)\ ' &
)\ ' &
9 Y & EU ' && 9 0 ' & & Y & KU & J U & n % n 9
)URPWKHUHVXOWLQJVHWRIOLQHDUGLIIHUHQWLDOHTXDWLRQVWKHFKDUDFWHULVWLFHTXDWLRQ PD\EHGHULYHGZKLFKLVRIWKHIRXUWKGHJUHH,WVJHQHUDOVWUXFWXUHLV
D V %D V %D V %D V%D '
7KHVWDELOLW\RIWKHV\VWHPFDQEHLQYHVWLJDWHGE\FRQVLGHULQJWKHUHDOSDUWVRI WKH URRWV RI WKLV HTXDWLRQ RU ZH PLJKW HPSOR\ WKH FULWHULXP IRU VWDELOLW\ DFFRUGLQJWR5RXWK+XUZLW]$FFRUGLQJWRWKLVFULWHULXPWKHV\VWHPRIRUGHUQ LVVWDEOHZKHQDOOWKHFRHIILFLHQWVDLDUHSRVLWLYHDQGWKH+XUZLW]GHWHUPLQDQWV +Q+QHWFDUHSRVLWLYH)RURXUIRXUWKRUGHUV\VWHPWKHFRPSOHWHFULWHULXPIRU VWDELOLW\UHDGV
D + ' D
DL !
D D D
IRU
D ' D D D &D D &D D D
!
L'
,Q)LJWKHERXQGDULHVRIVWDELOLW\KDYHEHHQSUHVHQWHGLQWKHFDUDYDQD[OH FRUQHULQJ VWLIIQHVV YV VSHHG SDUDPHWHU SODQH 7KH WKUHH FXUYHV EHORQJ WR WKH
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
WKUHHGLIIHUHQWVHWVRISDUDPHWHUVIRUWKHSRVLWLRQ I RI WKHFDUDYDQ¶VFHQWUHRI JUDYLW\DQGWKHFDUDYDQ¶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
D
7KHRVFLOODWRU\LQVWDELOLW\FRQQHFWHGZLWKWKHµVQDNLQJ¶SKHQRPHQRQDULVHV DV VRRQ DV IURP D VWDEOH FRQGLWLRQ WKH VHFRQG KLJKHVW +XUZLW] GHWHUPLQDQW EHFRPHVQHJDWLYH+Q
+WKHQDOVR+Q FI.ORWWHU RU/HLSKRO]
:KHQWKHFULWLFDOVSHHGLVVXUSDVVHGVHOIH[FLWHGRVFLOODWLRQVDUHFUHDWHG ZKLFKVKRZVDQDPSOLWXGHWKDWLQWKHDFWXDOQRQOLQHDUFDVHGRHVQRWDSSHDUWR OLPLWLWVHOI7KLVLVLQFRQWUDVWWRWKHFDVHRIWKHZKHHOVKLPP\SKHQRPHQRQWR
= 2.0 F = 1200
1.7 1600
I
&3
P
1.7 1200
divergent instability
stable
100000
[m] [ kg ]
[ N/rad ]
40000 oscillatory instability
9FULW
0 0
10
20
30
40
50
[ m/s ]
)LJ 6WDELOLW\ ERXQGDULHV IRU WKH FDU FDUDYDQ FRPELQDWLRQ LQ WKH FDUDYDQFRUQHULQJ VWLIIQHVVYHUVXVFULWLFDOVSHHGGLDJUDP9HKLFOHSDUDPHWHUVDFFRUGLQJWR7DEOH LQDGGLWLRQK
PJ
PNF
P,F
PFNF FI)LJ
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
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
([HUFLVH6WDELOLW\RIDWUDLOHU &RQVLGHUWKHWUDLOHURI)LJWKDWLVWRZHGE\DKHDY\VWHDGLO\PRYLQJYHKLFOHDWD IRUZDUGVSHHG9DORQJDVWUDLJKWOLQH7KHWUDLOHULVFRQQHFWHGWRWKHYHKLFOHE\PHDQV RIDKLQJH7KHDWWDFKPHQWSRLQWVKRZVDODWHUDOIOH[LELOLW\WKDWLVUHSUHVHQWHGE\WKH ODWHUDOVSULQJZLWKVWLIIQHVVF\)XUWKHUPRUHD\DZWRUVLRQDOVSULQJDQGGDPSHUDUH
n
n
SURYLGHGZLWKFRHIILFLHQWVF DQGN 'HULYHWKHHTXDWLRQVRIPRWLRQRIWKLVV\VWHPZLWKJHQHUDOLVHGFRRUGLQDWHV\DQG
n$VVXPHVPDOOGLVSODFHPHQWVVRWKDWWKHHTXDWLRQVFDQEHNHSWOLQHDU7KHGDPSLQJ FRXSOHNn n s PD\EHFRQVLGHUHGDVDQH[WHUQDOPRPHQWDFWLQJRQWKHWUDLOHURUZHPD\ XVHWKHGLVVLSDWLRQIXQFWLRQ ' òNn n s DQGDGGM'MT J WRWKHOHIWKDQGVLGHRI
L
/DJUDQJH¶V HTXDWLRQ 2EYLRXVO\ WKH LQWURGXFWLRQ RI WKLV H[WUD WHUP ZLOO EH EHQHILFLDOSDUWLFXODUO\ZKHQWKHV\VWHPWREHPRGHOOHGLVPRUHFRPSOH[ $VVHVVWKHFRQGLWLRQIRUVWDELOLW\IRUWKLVIRXUWKRUGHUV\VWHP6LPSOLI\WKHV\VWHP E\SXWWLQJJ
IDQGF
n
N
n
1RZILQGWKHH[SOLFLWFRQGLWLRQDOVWDWHPHQWIRUWKH
FRUQHULQJVWLIIQHVV&
α
F
\
ϕ
& I ,P
9
\
FN
ϕ
ϕ
J
)\ )LJ2QWKHVWDELOLW\RIDWUDLOHU([HUFLVH
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
9HKLFOH'\QDPLFVDW0RUH&RPSOH[7\UH6OLS&RQGLWLRQV
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
9HKLFOHVWDELOLW\DWH[FHVVLYHEUDNLQJDQGZKHHOORFN&KDSWHU
0RWRUF\FOHVWHDG\VWDWHFRUQHULQJ&KDSWHU
:KHHOVKLPP\&KDSWHU
6WHHULQJYLEUDWLRQV&KDSWHU
0RWRUF\FOHZHDYHDQGZREEOH&KDSWHU
7\UHRXWRIURXQGQHVV&KDSWHU
&RUQHULQJRQXQHYHQURDGV&KDSWHUV
$%6RQXQHYHQURDGV&KDSWHU
7UDYHUVLQJVKRUWREVWDFOHV&KDSWHU
3DUNLQJ&KDSWHU
&KDSWHU %$6,&7<5(02'(//,1* &216,'(5$7,216
,QWURGXFWLRQ
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
SULPDU\WDVNIXQFWLRQVDQGVHFRQGDU\HIIHFWV
TXDVL VWHDG\VWDWH
]
WUDQVLHQWYLEUDWRU\VWDWH
V\PPHWULF
ORDGFDUU\LQJ
UDGLDOGHIOHFWLRQ
FXVKLRQLQJ
LQSODQH
EUDNLQJGULYLQJ
WDQJHQWLDOVOLS
G\QDPLFFRXSOLQJ
UROOLQJUHVLVWDQFH
DQGGLVWRUVLRQ
QDWXUDOYLEUDWLRQV
DQWL
FRUQHULQJ
ODWHUDODQG
SKDVHODJ
V\PPHWULF
SQHXPDWLFWUDLO
VSLQWXUQVOLS
GHVWDELOLVDWLRQ
RXWRISODQH
RYHUWXUQLQJFRXSOH
DQGGLVWRUVLRQ
QDWXUDOYLEUDWLRQV
%$6,&7<5(02'(//,1*&216,'(5$7,216
LQSXWYHFWRU
ρ κ Ω
RXWSXWYHFWRU
UDGLDOGHIOHFWLRQ
)
]
ORQJLWXGLQDOVOLS
)
[
VSHHGRIUHYROXWLRQ
0
α ODWHUDOVOLSDQJOH ϕ VSLQWXUQVOLS γ FDPEHUDQJOH
)
\
\
QRUPDOORDG ORQJLWXGLQDOIRUFH UROOLQJUHVLVWPRPHQW FRUQHULQJVLGH IRUFH
0
]
VHOI DOLJQLQJWRUTXH
0
[
RYHUWXUQLQJFRXSOH
)LJ,QSXWRXWSXWTXDQWLWLHVURDGVXUIDFHFRQVLGHUHGIODW
EHWUHDWHGDVLQSXWYDULDEOHVLQWRWKHV\VWHP)LJXUHSUHVHQWVWKHµYHFWRUV¶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µEXLOWLQ¶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
%$6,&7<5(02'(//,1*&216,'(5$7,216
[ 0D
)
Ω
]D
)
[
[D
0D Z
0\
[
D
]
)
]
V\PPHWULFEHKDYLRXU
D
Ω Z
0\
wheel
)[
,
tangential distortion and slip radial deflection rolling radius variation rolling resistance
~ U
)
]
9
F
α
~
)LJ
-
-
\
)\
ψ
0]
M M [
Z
~
&]
non-uniformity
[
)\
[
Z
\
)\ 0[ 0]
\
γ ψ
~
0]
DQWLV\PPEHKDYLRXU
lateral distortion and slip torsional deflections turn and camber slip tyre inertia
M \ M
Z
:KHHOD[OHPRWLRQDQGURDGVXUIDFHFRRUGLQDWHVDQGSRVVLEO\XQFRXSOHGW\UHV\VWHP EORFNVYDOLGIRUVPDOOGHYLDWLRQVIURPWKHVWHDG\VWDWHVWUDLJKWDKHDGPRWLRQ
FRPELQHGVOLS 7KHVHLQWHUDFWLRQVEHFRPHLPSRUWDQWLIDWOHDVWRQHRIWKHLQSXW PRWLRQV RU PRUH SUHFLVHO\ RQH RI WKH DVVRFLDWHG VOLS FRPSRQHQWV EHFRPHV UHODWLYHO\ODUJH%HFDXVHRILWVUHODWLYHVLPSOLFLW\WKHVWHDG\VWDWHEHKDYLRXUZLOO EHWUHDWHGILUVW&KDSWHU 7KHGLVFXVVLRQRQW\UHG\QDPLFEHKDYLRXUVWDUWVLQ &KDSWHU
'HILQLWLRQRI7\UH,QSXW4XDQWLWLHV
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
%$6,&7<5(02'(//,1*&216,'(5$7,216
VXUIDFHFRQGLWLRQVJURRYHV 7KHWLPHUDWHRIFKDQJHRIWKLVUHODWLYHSRVLWLRQLV QHHGHGQRWRQO\IRUSRVVLEOHK\VWHUHVLVHIIHFWVEXWPDLQO\IRUWKHGHWHUPLQDWLRQ RIWKHVRFDOOHGµVOLS¶RIWKHZKHHOZLWKUHVSHFWWRWKHJURXQG ,IWKHURDGVXUIDFHQHDUWKHFRQWDFWSDWFKFDQEHDSSUR[LPDWHGE\DIODWSODQH WKDW LV ZKHQ WKH VPDOOHVW FRQVLGHUHG ZDYHOHQJWK RI WKH GHFRPSRVHG VXUIDFH YHUWLFDOSURILOHLVODUJHZLWKUHVSHFWWRWKHFRQWDFWOHQJWKDQGLWVDPSOLWXGHVPDOO WKHGLVWDQFHRIWKHZKHHOFHQWUHWRWKHURDGSODQHDQGWKHDQJOHEHWZHHQZKHHO SODQHDQGWKHQRUPDOWRWKHURDGVXUIDFHZLOOVXIILFHLQDGGLWLRQWRWKHVHYHUDO VOLSTXDQWLWLHVDQGWKHUXQQLQJVSHHGRIWKHZKHHO )RUWKHGHILQLWLRQRIWKHYDULRXVPRWLRQDQGSRVLWLRQLQSXWTXDQWLWLHVOLVWHGLQ )LJLWLVKHOSIXOWRFRQVLGHU)LJ$QXPEHURISODQHVKDYHEHHQGUDZQ 7KHURDGSODQHDQGWKHZKHHOFHQWUHSODQHZLWKOLQHRILQWHUVHFWLRQDORQJWKH XQLWYHFWRUO DQGWZRSODQHVQRUPDOWRWKHURDGSODQHRQHRIZKLFKFRQWDLQVWKH YHFWRUODQGWKHRWKHUWKHXQLWYHFWRUVZKLFKLVGHILQHGDORQJWKHZKHHOVSLQD[LV )URPWKHILJXUHIROORZVWKHGHILQLWLRQRIWKHFRQWDFWFHQWUH&DOVRGHVLJQDWHGDV WKHSRLQWRILQWHUVHFWLRQRIWKHWKUHHSODQHV 7KHXQLWYHFWRUWOLHVLQWKHURDG SODQHDQGLVGLUHFWHGSHUSHQGLFXODUWRO9HFWRUUIRUPVWKHFRQQHFWLRQEHWZHHQ ZKHHOFHQWUH$DQGFRQWDFWFHQWUH&,WVOHQJWKULVGHILQHGDVWKHORDGHGUDGLXV RIWKHW\UH7KHSRVLWLRQDQGDWWLWXGHRIWKHZKHHOZLWKUHVSHFWWRWKHLQHUWLDOWULDG LVFRPSOHWHO\GHVFULEHGE\WKHYHFWRUVEDDQGV7KHURDGSODQHLVGHILQHGDW WKHFRQWDFWFHQWUHE\WKHSRVLWLRQYHFWRURIWKDWSRLQWFDQGWKHQRUPDOWR WKH
QRUPDOWR
n
JURXQGSODQH
γ
9
%
D
E
Ω
)LJ
.
ζ
O LQHRI
l
l
s
9F [ \
ωη
L QHUWLDO WULDG
α
W
&
ωζ
ωξ
$
U
F
2
L QWHUVHFWLRQ
ZKHHOFHQWUHSODQH
ψ
]
ξ
0[ [
)[
&
)\
)] ]
0]
\ 0\
η SODQHWKURXJKZKHHOVSL QD[LV
s
JURXQGSODQH
'HILQLWLRQRISRVLWLRQDWWLWXGHDQGPRWLRQRIWKHZKHHODQGWKHIRUFHVDQGPRPHQWV DFWLQJIURPWKHURDGRQWKHZKHHO'LUHFWLRQVVKRZQDUHGHILQHGDVSRVLWLYH
%$6,&7<5(02'(//,1*&216,'(5$7,216
URDGLQWKDWSRLQWUHSUHVHQWHGE\WKHXQLWYHFWRUQSRVLWLYHXSZDUGV )LJXUH DOVR VKRZV WZR V\VWHPV RI D[HV EHVLGHV WKH LQHUWLDO WULDG )LUVWO\ ZH KDYH LQWURGXFHGWKHURDGFRQWDFWD[HVV\VWHP&[\] RIZKLFKWKH[D[LVSRLQWV IRUZDUGVDORQJWKHOLQHRILQWHUVHFWLRQO WKH]D[LVSRLQWVGRZQZDUGVQRUPDO WRWKHURDGSODQHQ DQGWKH\D[LVSRLQWVWRWKHULJKWDORQJWKHWUDQVYHUVHXQLW YHFWRUW6HFRQGO\WKHZKHHOD[OHV\VWHPRID[HV$
KDVEHHQGHILQHG
ZLWKWKH D[LVSDUDOOHOWRWKH[D[LVWKH D[LVDORQJWKHZKHHOVSLQGOHD[LVV DQGWKH D[LVDORQJWKHUDGLXVU 6LJQ FRQYHQWLRQV LQ WKH OLWHUDWXUH DUH QRW XQLIRUP )RU WKH VDNH RI FRQYHQLHQFH DQG WR UHGXFH VRXUFHV RI PDNLQJ HUURUV ZH KDYH FKRVHQ D VLJQ FRQYHQWLRQWKDWDYRLGVZRUNLQJZLWKQHJDWLYHTXDQWLWLHVDVPXFKDVSRVVLEOH 7KHUDGLDOGHIOHFWLRQRIWKHW\UH LVGHILQHGDVWKHUHGXFWLRQRIWKHW\UHUDGLXV IURPWKHXQORDGHGVLWXDWLRQUIWRWKHORDGHGFDVHU
U U
I
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
DW]HURORQJLWXGLQDOVOLS
%HFDXVHRIWKHRFFXUUHQFHRIUROOLQJUHVLVWDQFHPHDVXUHPHQWVRIW\UHFKDUDFW HULVWLFV ZRXOG WKHQ UHTXLUH WKH DSSOLFDWLRQ RI D GULYLQJ WRUTXH WR UHDFK WKH FRQGLWLRQRIVOLSHTXDOWR]HUR7KLVPD\EHFRPHRILPSRUWDQFHHVSHFLDOO\ZKHQ H[SHULPHQWVDUHFRQGXFWHGDWODUJHFDPEHUDQJOHVZKHUHWKHGUDJPD\EHFRPH FRQVLGHUDEOHPRWRUF\FOHW\UHV $QDOWHUQDWLYHRIWHQXVHGGHILQLWLRQWDNHVWKH HIIHFWLYH UROOLQJ UDGLXV UH GHILQHG DW IUHH UROOLQJ 0D
DV WKH VOLS UDGLXV
8QGHUQRUPDOFRQGLWLRQVWKHUHVXOWLQJ)[YV GLDJUDPVDFFRUGLQJWRWKHODWWHU WZR GHILQLWLRQV DUH YHU\ FORVH $ VPDOO KRUL]RQWDO VKLIW RI WKH FXUYHV LV VXIILFLHQWO\DFFXUDWHWRFKDQJHIURPRQHGHILQLWLRQWRWKHRWKHU7KHGUDZEDFN RIWKHODVWGHILQLWLRQLVWKDWZKHQWHVWLQJRQYHU\ORZIULFWLRQLF\ VXUIDFHVWKH UROOLQJUHVLVWDQFHPD\EHWRRODUJHWROHWWKHZKHHOURWDWHZLWKRXWWKHDSSOLFDWLRQ RI D GULYLQJ WRUTXH &RQVHTXHQWO\ WKH VWDWH RI IUHH UROOLQJ FDQQRW EH UHDOLVHG
%$6,&7<5(02'(//,1*&216,'(5$7,216
UH UI
UI UH U
Ω
6
ρ
9[
$
)]
0\
9V[
)[
)LJ(IIHFWLYHUROOLQJUDGLXVDQGORQJLWXGLQDOVOLSYHORFLW\
XQGHUWKHVHFRQGLWLRQV1HYHUWKHOHVVZHZLOODGRSWWKHODVWGHILQLWLRQZKHUHUV UHDQGFRQVHTXHQWO\SRLQW6LVORFDWHGDWDGLVWDQFHUHIURPWKHZKHHOFHQWUH )LJXUHGHSLFWVWKLVFRQILJXUDWLRQ $FFRUGLQJWRWKLVGHILQLWLRQZHZLOOKDYHWKHVLWXDWLRQWKDWZKHQDZKHHOUROOV IUHHO\WKDWLV DW 0 D ORQJLWXGLQDOVOLS
DWFRQVWDQWVSHHGRYHUDIODWHYHQURDGVXUIDFHWKH
LVHTXDOWR]HUR7KLVQRWZLWKVWDQGLQJWKHIDFWWKDWDWIUHH
UROOLQJ VRPH IRUH DQG DIW GHIRUPDWLRQV ZLOO RFFXU EHFDXVH RI WKH SUHVHQFH RI K\VWHUHVLVLQWKHW\UHWKDWJHQHUDWHVDUROOLQJUHVLVWDQFHPRPHQW0\7KURXJKWKLV DUROOLQJUHVLVWDQFHIRUFH)U
0\UDULVHVZKLFKQHFHVVDULO\LVDFFRPSDQLHGE\
WDQJHQWLDOGHIRUPDWLRQV:HPD\DJUHHWKDWDWWKHLQVWDQWRIREVHUYDWLRQSRLQW 6 WKDWOLHVRQWKHVOLSFLUFOHDQGLVDWWDFKHGWRWKHZKHHOULPKDVUHDFKHG LWV ORZHVWSRVLWLRQWKDWLVRQWKHOLQHDORQJWKHUDGLXVYHFWRUU$WIUHHUROOLQJLWV YHORFLW\ KDV WKHQ EHFRPH HTXDO WR ]HUR DQG SRLQW 6 KDV EHFRPH WKH FHQWUH RI URWDWLRQRIWKHPRWLRQRIWKHZKHHOULP:HKDYHDWIUHHUROOLQJRQDIODWURDGIRU DZKHHOLQXSULJKWSRVLWLRQ
DQGRUZLWKRXWZKHHO\DZUDWH
)LJDYHORFLW\RIWKHZKHHOFHQWUHLQIRUZDUG[RU
9[ ' UH ZLWK
0
FI
GLUHFWLRQ
GHQRWLQJ WKH VSHHG RI UHYROXWLRQ RI WKH ZKHHO ERG\ WR EH GHILQHG
KHUHDIWHU%\XVLQJWKLVUHODWLRQVKLSWKHYDOXHRIWKHHIIHFWLYHUROOLQJUDGLXVFDQ EH DVVHVVHG IURP DQ H[SHULPHQW 7KH IRUZDUG VSHHG DQG WKH ZKHHO VSHHG RI UHYROXWLRQDUHERWKPHDVXUHGZKLOHWKHZKHHOD[OHLVPRYHGDORQJDVWUDLJKWOLQH RYHUDIODWURDG'LYLVLRQRIERWKTXDQWLWLHVOHDGVWRWKHYDOXHRIUH7KHHIIHFWLYH UROOLQJUDGLXVZLOOEHDIXQFWLRQRIWKHQRUPDOORDGDQGWKHVSHHGRIWUDYHO:H PD\SRVVLEO\KDYHWRWDNHLQWRDFFRXQWWKHGHSHQGHQF\RQWKHFDPEHUDQJOHDQG WKHVOLSDQJOH
%$6,&7<5(02'(//,1*&216,'(5$7,216
,IDWEUDNLQJRUGULYLQJWKHORQJLWXGLQDOVOLSLVQRORQJHU]HURSRLQW6ZLOO PRYHZLWKDORQJLWXGLQDOVOLSVSHHG9V[ZKLFKGLIIHUVIURP]HUR:HREYLRXVO\ REWDLQLIDJDLQ
0
9V[ ' 9[ & U H
7KHORQJLWXGLQDOVOLSVRPHWLPHVFDOOHGWKHVOLSUDWLR LVGHQRWHGE\ DQGPD\ EHWHQWDWLYHO\GHILQHGDVWKHUDWLRRIORQJLWXGLQDOVOLSYHORFLW\9V[RISRLQW6DQG WKHIRUZDUGVSHHGRIWKHZKHHOFHQWUH9[
'&
9V[ 9[
RUZLWK
'&
9[& UH 9[
7KLVDJDLQKROGVIRUDZKHHORQDIODWURDGDQGZLWK
0
$PRUHJHQHUDODQG
SUHFLVHGHILQLWLRQRI ZLOOEHJLYHQODWHURQ7KHVLJQRIWKHORQJLWXGLQDOVOLS KDVEHHQFKRVHQVXFKWKDWDWGULYLQJZKHQ)[! LVSRVLWLYHDQGDWEUDNLQJ ZKHQ)[ LVQHJDWLYH:KHQWKHZKHHOLVORFNHG
ZHREYLRXVO\KDYH
,QWKHOLWHUDWXUHWKHV\PEROVRU6 LVPRUHFRPPRQO\XVHGWRGHQRWHWKH VOLSUDWLR 7KHDQJXODUVSHHGRIUROOLQJ
U
PRUHSUHFLVHO\GHILQHGIRUWKHFDVHRIPRYLQJ
RYHUXQGXODWHGURDGVXUIDFHVLVWKHWLPHUDWHRIFKDQJHRIWKHDQJOHEHWZHHQWKH UDGLXVFRQQHFWLQJ6DQG$WKLVUDGLXVLVWKRXJKWWREHDWWDFKHGWRWKHZKHHO DQG WKH UDGLXV U GHILQHG LQ )LJ DOZD\V O\LQJ LQ WKH SODQH QRUPDO WR WKH URDG WKURXJKWKHZKHHOVSLQD[LV )LJXUHLOOXVWUDWHVWKHVLWXDWLRQ 7KH OLQHDU VSHHG RI UROOLQJ 9U LV GHILQHG DV WKH YHORFLW\ ZLWK ZKLFK DQ
LPDJLQDU\SRLQW& WKDWLVSRVLWLRQHGRQWKHOLQHDORQJWKHUDGLXVYHFWRUUDQG FRLQFLGHV ZLWK SRLQW 6 DW WKH LQVWDQW RI REVHUYDWLRQ PRYHV IRUZDUG LQ [ GLUHFWLRQ ZLWKUHVSHFWWRSRLQW6WKDWLVIL[HGWRWKHZKHHOULP
9U ' UH
U
)RUDW\UH IUHHO\UROOLQJRYHUDIODWURDGZHKDYH DGGLWLRQ9U
9[1RWHWKDWDWZKHHOORFN
U
DQGZLWK
0
LQ
WKHDQJXODUVSHHGRIUROOLQJ
LVQRWHTXDOWR]HURZKHQWKHZKHHOPRYHVRYHUDURDGZLWKDFXUYHGYHUWLFDO
U
SURILOHWKHQQRWDOZD\VWKHVDPHSRLQWRIWKHZKHHOLVLQFRQWDFWZLWKWKHURDG )RUDFDPEHUHGZKHHOVKRZLQJD\DZUDWH
0 SXUHUROOLQJFDQRFFXURQDIODW
URDGHYHQZKHQWKHVSHHGRIWKHZKHHOFHQWUH9[ RIUROOLQJDULVHVWKDWLVHTXDOWR9U
U
H
0 VLQ
,QWKDWFDVHDOLQHDUVSHHG
DQGFRQVHTXHQWO\DQDQJXODUVSHHG
%$6,&7<5(02'(//,1*&216,'(5$7,216
$
Ω
U
UH
9[ 9F[
ΩU
&
&
6
9F[
9V[
)LJ5ROOLQJDQGVOLSSLQJRIDW\UHRYHUDQXQGXODWHGURDGVXUIDFH
RIUROOLQJ
U
0 VLQ
,QWKHQRUPDOFDVHRIDQDSSUR[LPDWHO\KRUL]RQWDOURDGVXUIDFHWKHZKHHO VSHHGRIUHYROXWLRQ
PD\EHGHILQHGDVWKHDQJXODUVSHHGRIWKHZKHHOERG\
ULP VHHQZLWKUHVSHFWWRDYHUWLFDOSODQHWKDWSDVVHVWKURXJKWKHZKHHOVSLQGOH D[LV 2Q D IODW OHYHO URDG WKH DQJXODU VSHHG RI UROOLQJ UHYROXWLRQRIWKHZKHHO
U
DQG WKH VSHHG RI
DUHHTXDOWRHDFKRWKHU7KHDEVROXWHVSHHGRIURWDWLRQ
RIWKHZKHHODERXWWKHVSLQGOHD[LV
ZLOOEHGLIIHUHQWIURP
ZKHQWKHZKHHO
LV FDPEHUHG DQG D \DZ UDWH RFFXUV RI WKH SODQH WKURXJK WKH VSLQGOH D[LV DQG QRUPDOWRWKHURDGDERXWWKHQRUPDOWRWKHURDG7KHQFI)LJ
' & %0
VLQ
7KLVHTXDWLRQIRUPVDFRUUHFWEDVLVIRUDJHQHUDOGHILQLWLRQRI OHYHOURDGVXUIDFHV,WVFRPSXWDWLRQLVVWUDLJKWIRUZDUGLI
DOVRRQQRQ
LVDYDLODEOHIURP
ZKHHOG\QDPLFVFDOFXODWLRQV
7KHORQJLWXGLQDOUXQQLQJVSHHG9F[ LVGHILQHGDVWKHORQJLWXGLQDOFRPSRQHQW
RIWKHYHORFLW\RISURSDJDWLRQRIWKHLPDJLQDU\SRLQW& RQUDGLXVYHFWRUU LQ WKHGLUHFWLRQRIWKH[D[LVYHFWRUO ,QFDVHWKHZKHHOLVPRYHGLQVXFKDZD\
WKDWWKHVDPHSRLQWUHPDLQVLQFRQWDFWZLWKWKHURDGZHZRXOGKDYH9F[
9 V[
7KLVFRUUHVSRQGVWRZKHHOORFNZKHQWKHURDGLVIODWDQGWKHYHKLFOHSLWFKUDWH LV]HUR)RUDIUHHO\UROOLQJW\UHWKHORQJLWXGLQDOUXQQLQJVSHHGHTXDOVWKHOLQHDU
VSHHGRIUROOLQJ9F[ 9U2QDIODWURDGDQGDW]HURFDPEHURU]HUR\DZUDWH
ZHREWDLQ9F[
'&
9V[ 9F[(
0
9[7KHJHQHUDOGHILQLWLRQIRUORQJLWXGLQDOVOLSQRZUHDGV
%$6,&7<5(02'(//,1*&216,'(5$7,216
7KHODWHUDOVOLSLVGHILQHGDVWKHUDWLRRIWKHODWHUDOYHORFLW\9F\RIWKHFRQWDFW
FHQWUH&DQGWKHORQJLWXGLQDOUXQQLQJVSHHG9F[ :HKDYHLQWHUPVRIWKHVOLS DQJOH
'&
WDQ
9F\
9F[(
ZKLFKIRUDZKHHOQRWVKRZLQJFDPEHUUDWH 0 QRUUDGLDOGHIOHFWLRQUDWH \DZUDWH
0 DWQRQ]HURFDPEHUDQJOH
0 DQG
ZKHQUXQQLQJRQDIODWURDGUHGXFHVWR
WKHUDWLRRIODWHUDODQGIRUZDUGVSHHGRIWKHZKHHOFHQWUH
'&
WDQ
9\ 9[
,QSUDFWLFHSRLQWV&DQG& OLHFORVHO\WRJHWKHUDQGPDNLQJGLVWLQFWLRQEHWZHHQ WKH ORQJLWXGLQDO RU WKH ODWHUDO YHORFLWLHV RI WKHVH SRLQWV LV RQO\ RI DFDGHPLF
LQWHUHVWDQGPD\EHQHJOHFWHG,QVWHDGRI9F[ LQWKHGHQRPLQDWRUZHPD\ZULWH9F[ DQG LI ZH ZLVK LQVWHDG RI 9F\ LQ WKH QXPHUDWRU WKH ODWHUDO VSHHG RI SRLQW 6 SDUDOOHOWRURDGSODQH ZKLFKLV9V\7KLVPD\HYHQEHDEHWWHUFKRLFHLIODUJH YDOXHVRIFDPEHUDUHFRQVLGHUHGZKLOHDYHUWLFDOW\UHGHIOHFWLRQUDWHRFFXUV7KH GHILQLWLRQVRIWKHVOLSFRPSRQHQWVWKHQUHGXFHWR
9V[ 9F[
'&
'&
WDQ
9V\ 9F[
7KHVOLSYHORFLWLHV9 V[DQG9V\IRUPWKHFRPSRQHQWVRIWKHVOLSVSHHGYHFWRU9V DQG
DQGWDQ
9V '
9V[
WKHFRPSRQHQWVRIWKHVOLSYHFWRUVV:HKDYH
9V\
DQG
VV '
WDQ
nLVGHILQHGDVWKHFRPSRQHQW
7KH VSLQ VOLS YHFWRU
]
RIWKHDEVROXWHVSHHGRIURWDWLRQ
RIWKHZKHHOERG\DORQJWKHQRUPDOWRWKHURDGSODQHQGLYLGHGE\WKH
IRUZDUGUXQQLQJVSHHG:HREWDLQWKHH[SUHVVLRQLQWHUPVRI\DZUDWH FDPEHUDQJOH FI)LJ
0 DQG
%$6,&7<5(02'(//,1*&216,'(5$7,216
n' &
]
(
9F[
0 '& &
VLQ
(
9F[
7KHPLQXVVLJQLVLQWURGXFHGDJDLQWRUHPDLQFRQVLVWHQWZLWKWKHGHILQLWLRQVRI ORQJLWXGLQDO DQG ODWHUDO VOLS 7KHQ ZH ZLOO KDYH DV D UHVXOW RI D SRVLWLYH
nDSRVLWLYHPRPHQW0 ,WWXUQVRXWWKDWWKHQDOVRWKHUHVXOWLQJVLGH ]
IRUFH)\LVSRVLWLYH7KH\DZUDWH
0 LVGHILQHGDVWKHVSHHGRIURWDWLRQRIWKHOLQH
RILQWHUVHFWLRQXQLWYHFWRUO DERXWWKH]D[LVQRUPDOWRWKHURDGFI)LJ ,I VLGHVOLSGRHVQRWRFFXU
/ DQGWKHZKHHOPRYHVRYHUDIODWURDGHTXDWLRQ
PD\EHZULWWHQDV
9 n ' & % (U VLQ ' & % (U VLQ 5 9F[ 5 UH 9F[ :KHQWKHW\UHUROOVIUHHO\WKHQ9V[
9F[
9U ZHREYLRXVO\REWDLQ
n ' & % VLQ 5 UH
ZLWK5GHQRWLQJWKHPRPHQWDU\FXUYDWXUHRIWKHSDWKRI& RUDSSUR[LPDWHO\
γ Ω ωζ
ω]
Q
ωξ ωη [
O
ψ
9[
5
\
] )LJ5RWDWLRQDOVOLSUHVXOWLQJIURPSDWKFXUYDWXUHDQGZKHHOFDPEHUVOLSDQJOH
%$6,&7<5(02'(//,1*&216,'(5$7,216
RIWKHFRQWDFWFHQWUH& )RUDW\UHZHVKDOOGLVWLQJXLVKEHWZHHQVSLQGXHWRSDWKFXUYDWXUHDQGVSLQ GXHWRZKHHOFDPEHU)RUDKRPRJHQHRXVEDOOWKHHIIHFWRIERWKLQSXWTXDQWLWLHV LVWKHVDPH)RUIXUWKHUXVHZHGHILQHWXUQVOLSDV
nW ' &
0
(
9F[
'&
LI
LV FRQVWDQW
5
:KHHOFDPEHURUZKHHOLQFOLQDWLRQDQJOH LVGHILQHGDVWKHDQJOHEHWZHHQWKH ZKHHOFHQWUHSODQHDQGWKHQRUPDOWRWKHURDG:LWK)LJZHILQG
VLQ
' & Q @V
RURQOHYHOURDGV
VLQ
' V]
ZKHUHV]UHSUHVHQWVWKHYHUWLFDOFRPSRQHQWRIWKHXQLWYHFWRUVDORQJWKHZKHHO VSLQD[LV
$VVHVVPHQWRI7\UH,QSXW0RWLRQ&RPSRQHQWV
7KHORFDWLRQRIWKHFRQWDFWFHQWUH&DQGWKH PDJQLWXGH RIWKHZKHHOUDGLXVU UHVXOWIURPWKHURDGJHRPHWU\DQGWKHSRVLWLRQRIWKHZKHHOD[OH:HFRQVLGHUWKH DSSUR[LPDWHDVVXPSWLRQWKDWWKHURDGSODQHLVGHILQHGE\WKHSODQHWRXFKLQJWKH VXUIDFHDWSRLQW4ORFDWHGYHUWLFDOO\EHORZWKHZKHHOFHQWUH$7KHSRVLWLRQRI R
R
R
R
SRLQW4ZLWKUHVSHFWWRWKHLQHUWLDOIUDPH2 [ \ ] LVJLYHQE\YHFWRUT7KH QRUPDOWRWKHURDGSODQHLVGHILQHGE\XQLWYHFWRUQ7KHORFDWLRQRIDUHIHUHQFH SRLQW%RIWKHYHKLFOHLVGHILQHGE\YHFWRUEDQGWKHORFDWLRQRIWKHZKHHOFHQWUH $E\EDFI)LJ 7KHRULHQWDWLRQRIWKHZKHHOVSLQD[LVLVJLYHQE\XQLW YHFWRUVDQGWKHORFDWLRQRIWKHFRQWDFWFHQWUH&E\
F ' E%D%U
ZKHUH U LV VWLOO WR EH GHWHUPLQHG 7KH H[SUHVVLRQ IRU U LV GHULYHG IURP WKH HTXDWLRQV
U ' UO î V
ZLWK
O' QîV
ZLWK UHVXOWLQJ IURP WKH FRQGLWLRQ WKDW *O *
DQG ZLWK LQ RUGHU WR
%$6,&7<5(02'(//,1*&216,'(5$7,216
REWDLQWKHPDJQLWXGHRIWKHORDGHGUDGLXVU
F Q' T Q @
@
ZKLFK LQGLFDWHV WKDW FRQWDFW SRLQW & DQG URDG SRLQW 4 OLH RQ WKH VDPH SODQH SHUSHQGLFXODUWRQ2QIODWOHYHOURDGVWKHDERYHHTXDWLRQVEHFRPHDORWVLPSOHU 7
VLQFHLQWKDWFDVHQ
DQGWKH]FRPSRQHQWVRIFDQGTEHFRPH]HUR
7KHUDGLDOW\UHGHIOHFWLRQ LVQRZUHDGLO\REWDLQHGIURP
' UI & U
ZLWKUIWKHIUHHXQORDGHGUDGLXV)RUDJLYHQW\UHWKHHIIHFWLYHUROOLQJUDGLXVUH LVDIXQFWLRQRIDPRQJVWRWKHUWKLQJVWKHXQORDGHGUDGLXVWKHUDGLDOGHIOHFWLRQ WKHFDPEHUDQJOHDQGWKHVSHHGRIWUDYHO 7KHYHFWRUIRUWKHVSHHGRISURSDJDWLRQRIWKHFRQWDFWFHQWUH9FUHSUHVHQWLQJ WKHPDJQLWXGHDQGGLUHFWLRQRIWKHYHORFLW\ZLWKZKLFKSRLQW&PRYHVRYHUWKH URDGVXUIDFHLVREWDLQHGE\GLIIHUHQWLDWLRQZLWKUHVSHFWWRWLPHRISRVLWLRQYHFWRU F
9F ' F0 ' E0 % D0 % U0 ' 9 % U0
:LWK 9 WKH YHORFLW\ YHFWRU RI WKH ZKHHO FHQWUH $ )LJ 7KH VSHHG RI
SURSDJDWLRQRISRLQW& UHSUHVHQWHGE\WKHYHFWRU9F EHFRPHVFI)LJDQG DVVXPHUHUFRQVWDQW
9F( ' 9 %
UH U0 U
7KHYHORFLW\YHFWRURISRLQW6WKDWLVIL[HGWRWKHZKHHOERG\UHVXOWVIURP
9V ' 9 % ZLWK
UH U
î
U
EHLQJ WKH WKH DQJXODU YHORFLW\ RI WKH ZKHHO ERG\ ZLWK UHVSHFW WR WKH
LQHUWLDOIUDPH2QWKHRWKHUKDQGWKLVYHORFLW\LVHTXDOWRWKHVSHHGRISRLQW&
PLQXVWKHOLQHDUVSHHGRIUROOLQJ
9V ' 9 F( & 9U O
IURPZKLFK9UIROORZV
9U ' O @ 9 F( & 9V
9U ' 9F[( & 9V[
RU
7KHOLQHDUVSHHGRIUROOLQJLVDFFRUGLQJWR UHODWHGWRWKHDQJXODUVSHHGRI
%$6,&7<5(02'(//,1*&216,'(5$7,216
UROOLQJ
U
'
UH 9U
2IFRXUVHRQIODWURDGV
U
ZKLFKLVWKHZKHHOVSHHGRIUHYROXWLRQDQGPD\
EHGLUHFWO\FDOFXODWHGE\XVLQJWKHUHODWLRQVKLS 7KHODWHUDOVOLSVSHHG9F\LVREWDLQHGE\WDNLQJWKHODWHUDOFRPSRQHQWRI9F
9F\ ' 9F @ W ZLWK
W' OîQ
7KHODWHUDOVOLSWDQ
WDQ
'&
9F\ (
9 F[
UHDGV
9F W 9 (F O @
'&
@
7KHORQJLWXGLQDOVOLSVSHHG9V[LVREWDLQHGLQDVLPLODUZD\
9V[ ' 9V @O
7KHORQJLWXGLQDOVOLS QRZEHFRPHV
'&
9V[
'& (
9F[
9V O 9F( O @
@
7KHWXUQVOLSDFFRUGLQJWRGHILQLWLRQ LVGHULYHGDVIROORZV
nW ' &
0
9F[(
0 ' & O (@ W
9F O
@
ZLWKLQWKHQXPHUDWRUWKHWLPHGHULYDWLYHRIWKHXQLWYHFWRUO7KHZKHHOFDPEHU DQJOHLVREWDLQHGDVLQGLFDWHGEHIRUH
VLQ
' &Q@V
([HUFLVH6OLSDQGUROOLQJVSHHGRIDZKHHOVWHHUHGDERXWDYHUWLFDOD[LV 7KHYHKLFOHGHSLFWHGLQ)LJUXQVRYHUDIODWOHYHOURDG7KHUHDUIUDPHPRYHVZLWK
XY
2 [ \ ] ZKLFKDWWKH LQVWDQWFRQVLGHUHGLVSRVLWLRQHGSDUDOOHOWRWKHPRYLQJWULDG%[\] DWWDFKHGWRWKH 0 UHDUIUDPH 7KHIURQWIUDPHFDQEHWXUQHGZLWKDUDWH G G W ZLWKUHVSHFWWRWKH YHORFLWLHV DQGUZLWKUHVSHFWWRDQLQHUWLDOWULDGFKRRVH
R
R
R
R
UHDUIUDPH$WWKHLQVWDQWFRQVLGHUHGWKHIURQWIUDPHLVVWHHUHGRYHUDQDQJOH ,WLVDVVXPHGWKDWWKHHIIHFWLYHUROOLQJUDGLXVLVHTXDOWRWKHORDGHGUDGLXV
U
H
U
%$6,&7<5(02'(//,1*&216,'(5$7,216
2R \R
[R E
U %
\
O
Y
H
X D
I
W
Q WV
δ
[
O
)LJ7RSYLHZRIYHKLFOH([HUFLVH
&
& 7KHORQJLWXGLQDOVOLSDWWKHIURQWZKHHOVLVDVVXPHGWREHHTXDOWR]HUR9V[
'HULYHH[SUHVVLRQVIRUWKHODWHUDOVOLSVSHHG9F\WKHOLQHDUVSHHGRIUROOLQJ9UDQG WKHODWHUDOVOLSWDQ
IRUWKHULJKWIURQWZKHHO
([HUFLVH 6OLS DQG UROOLQJ VSHHG RI D ZKHHO VWHHUHG DERXW DQ LQFOLQHGD[LV PRWRUF\FOH 7KHZKHHOVKRZQLQ)LJ UXQV RYHUDIODWOHYHOURDGVXUIDFH,WVFHQWUH $PRYHV
g
DORQJDKRUL]RQWDOVWUDLJKWOLQHDWDKHLJKW+ZLWKDVSHHGX7KHUDNHDQJOH LV
D
7KHVWHHUD[LV%$YHFWRU WUDQVODWHVZLWKWKHVDPHVSHHGX7KHUHLVQRZKHHOVOLS LQORQJLWXGLQDOGLUHFWLRQ9V[
$JDLQZHDVVXPHUH U)RUWKHVDNHRIVLPSOLI\LQJ
WKHFRPSOH[SUREOHPLWLVDVVXPHGWKDWWKHZKHHOFHQWUHKHLJKW+LVDJLYHQFRQVWDQW
δ
%
X ε
+ IL[HG
a
$ s
X r
& )LJ 6LGHYLHZRIIURQWSDUWRIYHKLFOHPRWRUF\FOH ZLWKZKHHOWXUQHGRYHUDQJOH DERXW
D
VWHHUD[LV ([HUFLVH
%$6,&7<5(02'(//,1*&216,'(5$7,216
'HULYHH[SUHVVLRQVIRUWKHODWHUDOVOLSVSHHG9F\WKHOLQHDUVSHHGRIUROOLQJ9UDQG WKHWXUQVOLSVSHHG
0 LQWHUPVRI+XDQG 0 IRU
H[SUHVVLRQV IRU WKH VOLS DQJOH
DQG$OVRVKRZWKH
WKH FDPEHU DQJOH
DQG WKH VSLQ VOLS
n ZLWK
FRQWULEXWLRQVERWKIURPWXUQLQJDQGFDPEHU1RWHWKDWLQUHDOLW\KHLJKW+GHSHQGVRQ DQGFKDQJHVGXHWR
0
)XQGDPHQWDO'LIIHUHQWLDO(TXDWLRQVIRU D5ROOLQJDQG6OLSSLQJ%RG\
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
E E E E
LQ)LJDV\VWHPRID[HV2 [ \ ] LVDVVXPHGWREHIL[HGWRWKHURDG
E
E
E
7KH[ DQG\ D[HVOLHLQWKHURDGVXUIDFHDQGWKH] D[LVSRLQWVGRZQZDUGV $QRWKHUFRRUGLQDWHV\VWHP&[\] LVLQWURGXFHGRIZKLFKWKHD[HV[DQG\OLH
E
E E
LQWKH[ 2 \ SODQHDQG]SRLQWVGRZQZDUGV7KH[D[LVLVGHILQHGWROLHLQ WKHZKHHOFHQWUHSODQHDQGWKH\D[LVIRUPVWKHYHUWLFDOSURMHFWLRQRIWKHZKHHO VSLQGOH D[LV 7KH RULJLQ & ZKLFK LV WKH VRFDOOHG FRQWDFW FHQWUH RU SHUKDSV EHWWHUWKHSRLQWRILQWHUVHFWLRQWUDYHOVZLWKDQDVVXPHGO\FRQVWDQWVSHHG9FRYHU
E
E E
WKH[ 2 \ SODQH7KHWUDYHOOHGGLVWDQFHVLV
V ' 9F W
ZKHUHWGHQRWHVWKHWLPH7KHWDQJHQWWRWKHRUELWRI&PDNHVDQDQJOH ZLWKWKH
E
IL[HG[ D[LV:LWKUHVSHFWWRWKLVWDQJHQWWKH[D[LVLVURWDWHGRYHUDQDQJOH
GHILQHGDVWKHVOLSDQJOH7KHDQJXODUGHYLDWLRQRIWKH[D[LVZLWKUHVSHFWWRWKH
E
[ D[LVWKDWLVWKH\DZDQJOH EHFRPHV
%$6,&7<5(02'(//,1*&216,'(5$7,216
projection of wheel axis
wheel centre plane
R
2
ψ
\
&
9W
contact area deformed grid line
\ \
R
R
R
\
S
R
F H
contact point
[
MY M[
ψ
9
H
F
T
[\
R
R
[
XY 3
7RSYLHZRIW\UHFRQWDFWDUHDVKRZLQJLWVSRVLWLRQZLWKUHVSHFWWRWKHV\VWHPRID[HV
2R [R \R ]R &[\]
WULDG
3
[
\
F
&
\
)LJ
X
9
α
MX M\
[
2
β
ψ
Y
\S
R
R
[R \R
original straight grid line R
[
path of& V
R
[S
XY
IL[HGWRWKHURDGDQGLWVGHIRUPDWLRQV ZLWKUHVSHFWWRWKHPRYLQJ
' %
E
)RUWKHDQJOH WKHIROORZLQJUHODWLRQZLWK\ WKHODWHUDOGLVSODFHPHQWRI &KROGV
VLQ
'
\R GV
G
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
%$6,&7<5(02'(//,1*&216,'(5$7,216
S' F%T
F
T
ZKHUH LQGLFDWHVWKHSRVLWLRQRIWKHFRQWDFWFHQWUH&LQVSDFHDQG WKHSRVLWLRQ RIWKHPDWHULDOSRLQWZLWKUHVSHFWWRWKHFRQWDFWFHQWUH:HKDYHIRUWKHODWWHU YHFWRU
T ' [R %X H[ % \R %Y H\
ZLWK
H O DQGH W UHSUHVHQWLQJWKHXQLWYHFWRUVLQ[DQG\GLUHFWLRQV7KH
[
\
YHFWRURIWKHVOLGLQJYHORFLW\RIWKHPDWHULDOSRLQWUHODWLYHWRWKHURDGREYLRXVO\ EHFRPHV
9J ' S0 ' F0 % T0 ' 9F% [0R%X0 H [% \0R%Y0 H \% 0 ^[R%X H \& \ R%Y H [`
9 F0 GHQRWHVWKHYHFWRURIWKHVSHHGRISURSDJDWLRQRIFRQWDFWFHQWUH&
ZKHUH
F
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
[ ' &U \ @ R GW \0 ' G \ ' & [ VLQ @ GW [0 '
G
,I ò D_0 _ © U _ _
R
WKHHIIHFWRIWKHWLPHUDWHRIFKDQJHRIWKHFDPEHUDQJOHZKHHO
M M
SODQHURWDWLRQDERXWWKH[D[LV RQWKHSDUWLDOGHULYDWLYH \ WDQGWKXVRQ
\0
PD\
EHQHJOHFWHG7KLVHIIHFWLVGLUHFWO\FRQQHFWHGZLWKWKHVPDOOLQVWDQWDQHRXVRUVR FDOOHGQRQODJJLQJUHVSRQVHWRFDPEHUFKDQJHV6LPLODULQVWDQWDQHRXVUHVSRQVHV PD\RFFXUDVDUHVXOWRIQRUPDOORDGFKDQJHVZKHQWKHW\UHVKRZVFRQLFLW\RU SO\VWHHU 7KH WHUPV DVVRFLDWHG ZLWK
M\MW KDYH EHHQ QHJOHFWHG LQ WKH DERYH
HTXDWLRQDQGUHODWHGQHJOHFWLRQVZLOOEHSHUIRUPHGLQVXEVHTXHQWIRUPXODHIRU
\0R
%$6,&7<5(02'(//,1*&216,'(5$7,216
7KHUDGLXVRIWKHRXWHUVXUIDFHLQWKHXQGHIRUPHGVWDWHURPD\GHSHQGRQWKH ODWHUDOFRRUGLQDWH\,IWKHUROOLQJERG\VKRZVDWRXFKLQJVXUIDFHWKDWLVDOUHDG\ SDUDOOHOWRWKHURDGEHIRUHLWLVGHIRUPHGZHZRXOGKDYHLQWKHQHLJKERXUKRRG RI WKH FHQWUH RIFRQWDFWUR\
UR \ VLQ ,QFDVHRIDFDPEHUHGFDUW\UHD
GLVWRUWLRQRIEHOWDQGFDUFDVVLVQHHGHGWRHVWDEOLVKFRQWDFWRYHUDILQLWHDUHDZLWK WKHJURXQG7KHVKDSHRIWKHW\UHFURVVVHFWLRQLQWKHXQGHIRUPHGVWDWHJRYHUQV WKHGHSHQGHQF\RIWKHIUHHUDGLXVZLWKWKHGLVWDQFHWRWKHZKHHOFHQWUHSODQH ,Q )LJ DQ H[DPSOH LV JLYHQ RI WZR GLIIHUHQW FDVHV 7KH XSSHU SDUW FRUUHVSRQGVWRDUHDUYLHZDQGWKHORZHURQHWRDSODQYLHZRIDPRWRUF\FOHW\UH DQGRIDFDUW\UHSUHVVHGDJDLQVWDQDVVXPHGO\IULFWLRQOHVVVXUIDFH
,WPD\
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
[\
[ \ UHSUHVHQWLQJ WKH SDUWLDO
GHULYDWLYHVRIWKHVHQRUPDOORDGLQGXFHGORQJLWXGLQDODQGODWHUDOGHIRUPDWLRQV ZLWKUHVSHFWWR[:HKDYH
[
[\'
MXR M[
DQG
\
[\'
MYR
M[
7KHVHIXQFWLRQVGHSHQGRQWKHYHUWLFDOORDGDQGRQWKHFDPEHUDQJOH ,IWKHW\UHLVFRQVLGHUHGWRUROORQDIULFWLRQOHVVIODWVXUIDFHDWDFDPEHUDQJOH RUZLWKQRQ]HUR
¶VODWHUDODQGORQJLWXGLQDOVOLGLQJRIWKHFRQWDFWSRLQWVZLOO
RFFXU HYHQ ZKHQ WKH ZKHHO GRHV QRW H[KLELW ODWHUDO ORQJLWXGLQDO RU WXUQ VOLS 6LQFHKRUL]RQWDOIRUFHVGRQRWRFFXULQWKLVLPDJLQDU\FDVHXDQGYDUHGHILQHG WREH]HUR7KHFRRUGLQDWHVZLWKUHVSHFWRIWKHPRYLQJD[HVV\VWHP&[\] RI WKH FRQWDFW SRLQW VOLGLQJ RYHU WKH K\SRWKHWLFDO IULFWLRQOHVV VXUIDFH ZHUH GHQRWHGDV[RDQG\R7KHWLPHUDWHVRIFKDQJHRI[RDQG\RGHSHQGRQWKHSRVLWLRQ LQWKHFRQWDFWSDWFKRQWKHVSHHGRIUROOLQJ
M
DQGRQWKHFDPEHUDQJOH :H
M
ILQGDIWHUGLVUHJDUGLQJWKHHIIHFWRI \R W
[0R ' [0 % X0 R ' & ^ %
[\
U \
` @ R @
[
\0R ' \0 % Y0 R ' & ^[ R VLQ %
[\ U \
R ` @ \
%$6,&7<5(02'(//,1*&216,'(5$7,216
Ω
Ω
γ
γ
U
o
U
o
& 4o
µ =0
4
µ =0
\ ]
]
4o
\\
undeformed tyre peripheral line
& \
4
\
]
[
[
[
\o
\o
contact patch contact line at µ =0
[o
& 4o
µ =0
(a4)
[o \\
4o
µ =0 relatively stiff belt
&
\
4
undeformed tyre peripheral line
)LJ
$PRWRUF\FOHW\UHOHIW DQGDFDUW\UHULJKW LQFDPEHUHGSRVLWLRQWRXFKLQJWKH DVVXPHGO\ IULFWLRQOHVV URDG VXUIDFH 7KH IRUPHU ZLWKRXW DQG WKH ODWWHU ZLWK WRUVLRQDQGEHQGLQJRIWKHFDUFDVVDQGEHOW7RSUHDUYLHZERWWRPWRSYLHZ
%$6,&7<5(02'(//,1*&216,'(5$7,216
)RUKRPRJHQHRXVUROOLQJERGLHVHJDUDLOZD\ZKHHORUDUXEEHUEDOO ZLWK FRXQWHUVXUIDFHVDOUHDG\SDUDOOHOEHIRUHWRXFKLQJWRUVLRQDERXWWKHORQJLWXGLQDO D[LV DQG ODWHUDO EHQGLQJ GR QRW RFFXU SO\VWHHU DQG FRQLFLW\ DUH DEVHQW DQG KRUL]RQWDOFRPSUHVVLRQPD\EHQHJOHFWHG UR\
¶VYDQLVK 0RUHRYHUZHWKHQKDG
UR \VLQ VRWKDWIRUWKLVVSHFLDOFDVH UHGXFHVWR
[0R ' [0 ' & ^UR & \ VLQ \0R ' \0 ' &[ VLQ @
`@
)RUDW\UHWKHWHUPVZLWK DUHDSSURSULDWH&LUFXPIHUHQWLDOFRPSUHVVLRQ
[
GHFUHDVHVWKHHIIHFWLYHUROOLQJUDGLXVUHWKDWLVGHILQHGDWIUHHUROOLQJ$WFDPEHU GXH WR WKH VWUXFWXUH RI D FDU W\UH ZLWK D EHOW WKDW LV UHODWLYHO\ VWLII LQ ODWHUDO EHQGLQJWKHFRPSUHVVLRQH[WHQVLRQIDFWRU
[
ZLOOQRWEHDEOHWRFRPSHQVDWHIRU
WKHIDFWWKDWDFDUW\UHVKRZVRQO\DUHODWLYHO\VPDOOYDULDWLRQRIWKHIUHHUDGLXV UR\ DFURVV WKH ZLGWK RI WKH WUHDG ZKLOH IRU WKH VDPH UHDVRQ WKH ODWHUDO GLVWRUVLRQ IDFWRU
\
DSSHDUV WR EH FDSDEOH RI FRQVLGHUDEO\ FRXQWHUDFWLQJ WKH
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
[
JLYHV ULVH WR D GHFUHDVH RI WKH
HIIHFWLYHUROOLQJUDGLXV7KHFRPSUHVVLRQPD\QRWEHXQLIRUPDORQJWKH[D[LV )RURXUSXUSRVHVKRZHYHUZHZLOOGLVUHJDUGWKHUHVXOWLQJVHFRQGDU\HIIHFWV 7KHFRHIILFLHQWLQ WKDWUHODWHVWKHSDVVDJHYHORFLW\ UHYROXWLRQ
&[0R DQGWKHVSHHGRI
RIWKHZKHHOLVGHVLJQDWHGDVWKHORFDOHIIHFWLYHUROOLQJUDGLXVUHMDW
HJURZMRIWUHDGHOHPHQWV7KLVUDGLXVGHSHQGVRQWKHODWHUDOSRVLWLRQ\MDQG ZLOOFKDQJHZLWKWKHYHUWLFDOORDG)]DQGWKHFDPEHUDQJOH 7KHYHORFLW\WKURXJK WKHFRQWDFW]RQHRIWKHHOHPHQWVRIURZMLV
UHM '
^
%
\
U \ ' U H % UHM \M
M ` R M
[
&[0R ' UHM
:HPD\ZULWH
LQZKLFKXVHKDVEHHQPDGHRIWKHRYHUDOOW\UHHIIHFWLYHUROOLQJUDGLXVUHGHILQHG DFFRUGLQJWR(TV DQG
UHM\M WKHSRVVLEO\DQWLV\PPHWULFYDULDWLRQRI
WKHORFDOHIIHFWLYHUROOLQJUDGLXVRYHUWKHWUHDGZLGWKZLWKUHVSHFWWRUHGXHWR
%$6,&7<5(02'(//,1*&216,'(5$7,216
ORDGLQJ DW FDPEHU RU FRQLFLW\ 7KH DYHUDJH HIIHFWLYH UROOLQJ UDGLXV UH LV H[SHFWHGWRGHSHQGRQWKHFDPEHUDQJOHDVZHOO :HZLOOQRZWU\WRPDNHDGLVWLQFWLRQEHWZHHQWKHFRQWULEXWLRQVWRWKHDQWL V\PPHWULFYDULDWLRQRULJLQDWLQJIURPFRQLFLW\DQGIURPFDPEHU:HZULWHIRUWKH ORFDOSDVVDJHYHORFLW\
[0R ' &>^ %
\ U & ^ %
` H
[
FRQ
\ \
` R VLQ @ @
[
,QWKLVZD\ZHKDYHDFKLHYHGDVWUXFWXUHFORVHO\UHODWHGWRWKHILUVWHTXDWLRQRI )RU D UROOLQJ HODVWLF ERG\ OLNH D EDOO RU D PRWRUF\FOH W\UH
[
ZLOO EH
FORVH WR ]HUR ZKLOHDFDUWUXFNRUUDFLQJFDUW\UHLVH[SHFWHGWRKDYHD
[
FORVHUWRWKHYDOXH$ELDVSO\W\UHIHDWXULQJDPRUHFRPSOLDQWFDUFDVVDQG WUHDG EDQG DQG D URXQGHU FURVV VHFWLRQ SURILOH LV H[SHFWHG WR EHWWHU µUHFRYHU¶ IURPWKHWRUVLRQHIIHFWVUHVXOWLQJLQDYDOXHLQEHWZHHQWKHWZRH[WUHPHV,WLV WUXHWKDWLQJHQHUDOELDVSO\W\UHVVKRZFRQVLGHUDEO\ODUJHUVLGHIRUFHVDVDUHVXOW RIFDPEHUWKDQUDGLDOSO\FDUW\UHV :KHQWKHSURGXFWRI
\
DQGURUHLVFRQVLGHUHGWREHQHJOLJLEOHZHZULWHIRU
WKHODWHUDOYHORFLW\RIWKHSRLQWVRYHUDIULFWLRQOHVVVXUIDFHZLWKUHVSHFWWRWKH[ D[LVLQVWHDGRI
\0R ' & ^[ R VLQ %
[\ U
@ H` @
\
7KHUHVXOWLQJH[SUHVVLRQV PD\QRZEHVXEVWLWXWHGLQH[SUHVVLRQ IRUWKHVOLGLQJVSHHGFRPSRQHQWV )RUDJLYHQPDWHULDOSRLQWRIWKHW\UHRXWHUVXUIDFHWKDWQRZUROOVRQDURDG VXUIDFH ZLWK IULFWLRQ UHLQWURGXFHG WKH DVVRFLDWHG GHIOHFWLRQV X DQG Y ZKLFK RFFXURQWRSRIWKHLQLWLDOORDGLQGXFHGGHIOHFWLRQVXRDQGYR DUHIXQFWLRQVRILWV ORFDWLRQLQWKHFRQWDFWSDWFKDQGRIWKHWLPHX
X[R\RW DQGY
Y[R\RW
+HQFHZHKDYHIRUWKHWLPHUDWHVRIFKDQJH
X0 '
X' GW
G
M X G [R
G\ % MX R % MX M [R G W M \R G W MW
[ G\ Y0 ' G Y ' M Y % MY R % MY GW M [R G W M \R G W MW
G R
LQZKLFKWKHH[SUHVVLRQV DSSO\,IZHGLVUHJDUGVLGHZD\VUROOLQJDQG QHJOHFW D SRVVLEOH VPDOO HIIHFW RI WKH YDULDWLRQ RI \R WKDW RFFXUV DW FDPEHU FRQLFLW\RUSO\VWHHUWKHVHFRQGWHUPVRIWKHULJKWKDQGPHPEHUVGLVDSSHDU7KH UHPDLQLQJH[SUHVVLRQVDUHVXEVWLWXWHGLQ $FFRUGLQJ WR (T WKH OLQHDU VSHHG RI UROOLQJ DQG WKH YHORFLW\ RI WKH FRQWDFWFHQWUHDUHUHODWHGWKURXJKWKHVOLSVSHHG:HKDYHZKHQGLVUHJDUGLQJWKH
%$6,&7<5(02'(//,1*&216,'(5$7,216
JHQHUDOO\YHU\VPDOOGLIIHUHQFHEHWZHHQWKHYHORFLWLHVRI& DQG& WKHYHFWRU UHODWLRQVKLS
9F ' 9 V % 9 U H[
9U ' UH
ZLWK
$IWHU VLPSOLI\LQJ WKH HTXDWLRQV E\ QHJOHFWLQJ SURGXFWV RI DVVXPHGO\ VPDOO TXDQWLWLHVLHGHIOHFWLRQVDQGLQSXWVOLS TXDQWLWLHV DQGXVLQJWKHH[SUHVVLRQ IRU
]
ZKLFKLVWKHDEVROXWHDQJXODUYHORFLW\RIWKHZKHHOERG\DERXWWKH]D[LV
DVLQGLFDWHGLQ(T FRPSRQHQWVRIWKHVOLGLQJYHORFLW\EHFRPH
\ 9 % MX & \R ] % M[ MW 9 J\ ' 9V\ & MY % \[\ @9 U % MY % [R ] M[ MW 9J[ ' 9 V[ &
:LWKDOO
MX
%
FRQ
@ U
[
\ \
@ R VLQ @
[
¶VRPLWWHGZHZRXOGUHWXUQWRWKHEDVLFFDVHRIHJDEDOOUROOLQJDQG
VOLSSLQJRYHUDIODWULJLGVXUIDFH,Q&KDSWHUDSK\VLFDOW\UHPRGHOLVGHYHORSHG XVLQJWKHDERYHHTXDWLRQV7KHUHWKH
¶VDUHH[SUHVVHGLQWHUPVRIWKHFDPEHU
DQJOH DQG RI HTXLYDOHQW FDPEHU DQG VOLS DQJOHV
FRQ
DQG
SO\
WR DFFRXQW IRU
FRQLFLW\DQGSO\VWHHUFI(TV
/HWXVQRZFRQVLGHUWKHVSHFLDOFDVHRIDIUHHO\UROOLQJW\UHVXEMHFWHGWRRQO\
VPDOOODWHUDOVOLSDQGVSLQ
_ _ © _ _ © UH © _5_ZLWK5WKHLQVWDQWDQHRXV
UDGLXVRISDWKFXUYDWXUHDQGWKXV_
n _©UH DQGQHJOHFWHIIHFWVRILQLWLDOQRQ
W
SDUDOOHOOLW\RIWRXFKLQJVXUIDFHVDVZHOODVµEXLOWLQ¶ORDGDQGFDPEHULQGXFHG GHIRUPDWLRQHIIHFWVDOO ¶V
9V[ '
9V\ ' & 9F
:HWKHQKDYHDSSUR[LPDWHO\
9U ' 9 F
]
' 0&
VLQ
' &9F n
8VLQJ WKH WUDYHOOHG GLVWDQFH V LQVWHDG RI WKH WLPH W DV DQ LQGHSHQGHQW YDULDEOH ZH REWDLQ WKH IROORZLQJ H[SUHVVLRQV IRU WKH VOLGLQJ YHORFLWLHV RI D FRQWDFWSRLQWZLWKFRRUGLQDWHV[\ RIDIUHHO\UROOLQJW\UHDWVPDOOODWHUDOVOLS DQGVSLQ
9J[ ' 9 F \n & MX % MX M[ MV 9J\ ' 9 F & & [n & MY % MY M[ MV
,Q WKH FRQWDFW DUHD UHJLRQV RI DGKHVLRQ PD\ RFFXU DV ZHOO DV UHJLRQV ZKHUH VOLGLQJWDNHVSODFH,QWKHUHJLRQRIDGKHVLRQWKHW\UHSDUWLFOHVWRXFKLQJWKHURDG GR QRW PRYH DQG ZH KDYH 9J[
9J\
,Q WKLV SDUW RI WKH FRQWDFW DUHD WKH
IULFWLRQDOVKHDUIRUFHVDFWLQJIURPURDGWRW\UHRQDXQLWDUHDZLWKFRPSRQHQWV
%$6,&7<5(02'(//,1*&216,'(5$7,216
GHQRWHG E\ T[ DQG T\ QRW WR EH FRQIXVHG ZLWK WKH FRPSRQHQWV RI WKH SRVLWLRQ YHFWRU XVHGHDUOLHU GRQRWH[FHHGWKHPD[LPXPDYDLODEOHIULFWLRQDOIRUFH SHUXQLWDUHD7KHPD[LPXPIULFWLRQDOVKHDUVWUHVVLVJRYHUQHGE\WKHFRHIILFLHQW RI IULFWLRQ DQG WKH QRUPDO FRQWDFW SUHVVXUH T] ,Q WKH DGKHVLRQ UHJLRQ (TV UHGXFHWR
MX
& MX ' & \n M[ MV MY & MY ' & & [n M[ MV
)XUWKHUPRUHZHKDYHWKHFRQGLWLRQ
T[ % T\
T]
,QWKHUHJLRQRIVOLGLQJHTXDWLRQV KROG,IWKHGHIRUPDWLRQJUDGLHQWV ZHUHNQRZQWKHYHORFLW\FRPSRQHQWV9J[DQG9J\PLJKWEHREWDLQHGIURPWKHVH HTXDWLRQV)RUWKHIULFWLRQDOVWUHVVYHFWRUZHREWDLQ
T T ' &T] 9J[ 9J\ 9J
[ \
ZLWK
9J ' 9J[ % 9J\
n
)RUWKHVLPSOHUFDVHWKDWRQO\ODWHUDOVOLSRFFXUV
WKHHTXDWLRQVIRUWKH
ODWHUDOGHIRUPDWLRQVUHGXFHWR LQDGKHVLRQUHJLRQ
MY M[
&
*T\*
MY MV
'&
T]
LQVOLGLQJUHJLRQ
9 & MY ' & & J\ M[ MV 9F MY
T \ ' &T] VJQ 9 J\ 7KHHTXDWLRQV DSSO\LQJHQHUDO7KHLUVROXWLRQVFRQWDLQFRQVWDQWVRI
%$6,&7<5(02'(//,1*&216,'(5$7,216
LQWHJUDWLRQZKLFKGHSHQGRQWKHVHOHFWHGSK\VLFDOPRGHOGHVFULSWLRQRIWKHW\UH $SSOLFDWLRQVRIWKHGLIIHUHQWLDOHTXDWLRQVGHULYHGDERYHZLOOEHGHPRQVWUDWHGLQ VXEVHTXHQWFKDSWHUVERWKIRUVWHDG\VWDWHDQGQRQVWHDG\VWDWHFRQGLWLRQV,QWKH FDVHRIVWHDG\VWDWHPRWLRQRIWKHZKHHOWKHSDUWLDOGHULYDWLYHVZLWKUHVSHFWWR
M M
M M
WLPHWRUWUDYHOOHGGLVWDQFHVEHFRPHHTXDOWR]HUR Y V
X V
7KHQWKH
GHIRUPDWLRQJUDGLHQWVLQWKHDUHDRIDGKHVLRQ9J IROORZHDVLO\IURPWKHWKHQ RUGLQDU\GLIIHUHQWLDOHTXDWLRQV,QWKHODVWVLPSOHFDVH ZHZRXOGREWDLQ GYG[
ZKLFKPHDQVWKDWWKHFRQWDFWOLQHLVVWUDLJKWDQGUXQVSDUDOOHOWRWKH
VSHHGYHFWRU
9 F
([HUFLVH3DUWLDOGLIIHUHQWLDOHTXDWLRQVZLWKORQJLWXGLQDOVOLSLQFOXGHG (VWDEOLVKWKHGLIIHUHQWLDOHTXDWLRQVIRUWKHVOLGLQJYHORFLWLHVVLPLODUWRWKH(TV EXWQRZZLWKORQJLWXGLQDOVOLS LQFOXGHG
9 û 9 DQGWKDW9
WKDWFDVH
U
F
DQG
n UHPDLQVPDOO 1RWHWKDWLQ
PD\EHH[SUHVVHGLQWHUPVRI U
9 -9 DQG F[
F
$OVRILQG
WKHSDUWLDOGLIIHUHQWLDOHTXDWLRQVJRYHUQLQJWKHGHIOHFWLRQVLQWKHDGKHVLRQ]RQHVLPLODU WRWKH(TV
7\UH0RGHOV,QWURGXFWRU\'LVFXVVLRQ
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
) ) DQG0 FXUYHVEXW
IRUPXODZKLFKQRWRQO\SURYLGHVDQH[FHOOHQWILWIRUWKH
\
[
]
LQDGGLWLRQIHDWXUHVFRHIILFLHQWVZKLFKKDYHFOHDUUHODWLRQVKLSVZLWKW\SLFDOVKDSH
%$6,&7<5(02'(//,1*&216,'(5$7,216
from experimental data only
using similarity method
through simple physical model
through complex physical model
fitting full scale tyre test data by regression techniques
distorting, rescaling and combining basic characteristics
using simple mechanical representation, possibly closed form solution
describing tyre in greater detail, computer simulation, finite element method
degree of fit number of full scale tests
complexity of formulations effort insight in tyre behaviour number of special experiments
approach more empirical
theoretical
)LJ)RXUFDWHJRULHVRISRVVLEOHW\SHVRIDSSURDFKWRGHYHORSDW\UHPRGHO
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
%$6,&7<5(02'(//,1*&216,'(5$7,216
9J
)[
T
)
sliding
&
9V
9F
α
0] )\
T
adhesion T ]
friction coefficient
µR 0 )LJ
µ
pressure distribution
sliding speed 9 J
D 7KHEUXVKW\SHW\UHPRGHODWFRPELQHGORQJLWXGLQDOEUDNH VOLSDQGODWHUDOVOLS LQFDVHRIHTXDOORQJLWXGLQDODQGODWHUDOVWLIIQHVVHV E 7KHOLQHDUO\GHFD\LQJIULFWLRQFRHIILFLHQW
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µEUXVKPRGHO¶ RI)LJDUHHVSHFLDOO\XVHIXOWRJHWDEHWWHUXQGHUVWDQGLQJRIW\UHEHKDYLRXU 7KH EUXVK PRGHO ZLWK D SDUDEROLF SUHVVXUH GLVWULEXWLRQ ZLOO EH GLVFXVVHG DW OHQJWKLQ&KDSWHU 7KHULJKWPRVWJURXSRI)LJLVDLPHGSULPDULO\DWPRUHGHWDLOHGDQDO\VLV RIWKHW\UH7KHFRPSOH[ILQLWHHOHPHQWEDVHGPRGHOVEHORQJVWRWKLVFDWHJRU\
%$6,&7<5(02'(//,1*&216,'(5$7,216
9F α
&
carcass belt adhesion
tread rubber sliding
0] )\
pressure distribution T
]
)LJ7\UHPRGHOZLWKIOH[LEOHFDUFDVVDWVWHDG\VWDWHUROOLQJZLWKVOLSDQJOH
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µEUXVKPRGHO¶VXEMHFWHGWRFRPELQHGVOLSZLWKFDPEHUDFRQGLWLRQWKDW
%$6,&7<5(02'(//,1*&216,'(5$7,216
9V
ψ
α
wheel plane
9F
carcass
)[
belt
9J
contact line wheel spin axis
tread element
T
0] )\
)LJ &RPSXWHU VLPXODWLRQ W\UH PRGHO ZLWK IOH[LEOH FDUFDVV DUELWUDU\SUHVVXUH GLVWULEXWLRQ DQG IULFWLRQ FRHIILFLHQW IXQFWLRQV )RUFHV DFWLQJ RQ D VLQJOH WUHDG HOHPHQWPDVVGXULQJRQHSDVVDJHWKURXJKWKHFRQWDFWOHQJWKDUHLQWHJUDWHGWRREWDLQ
) )
WKHWRWDOIRUFHVDQGPRPHQW
[
\
0
DQG
]
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
%$6,&7<5(02'(//,1*&216,'(5$7,216
RFFXUVSRVVLEO\LQFOXGLQJZKHHOFDPEHUDQGWXUQVOLS ODUJHFDPEHURFFXUV ZKLFK PD\ QHFHVVLWDWH WKH FRQVLGHUDWLRQ RI WKH GLPHQVLRQV RI WKH W\UH FURVV VHFWLRQ WKH IULFWLRQ FRHIILFLHQW FDQ QRW EH DSSUR[LPDWHG DV D FRQVWDQW TXDQWLW\EXWPD\YDU\ZLWKVOLGLQJYHORFLW\DQGVSHHGRIWUDYHODVRFFXUVRQZHW RULF\VXUIDFHV WKHZDYHOHQJWKRIWKHSDWKRIFRQWDFWSRLQWVDWQRQVWHDG\ VWDWH FRQGLWLRQV FDQ QR ORQJHU EH FRQVLGHUHG ODUJH ZKLFK PD\ UHTXLUH WKH LQWURGXFWLRQRI WKH ODWHUDODQGORQJLWXGLQDOFRPSOLDQFHRIWKHFDUFDVV WKH ZDYHOHQJWKEHFRPHVUHODWLYHO\VKRUWZKLFKPD\QHFHVVLWDWHWKHFRQVLGHUDWLRQRI DILQLWHFRQWDFWOHQJWKUHWDUGDWLRQHIIHFW DQGSRVVLEO\WKHFRQWDFWZLGWKDWWXUQ VOLSDQGFDPEHU WKHVSHHGRIWUDYHOLVODUJHVRWKDWW\UHLQHUWLDEHFRPHVRI LPSRUWDQFH LQ SDUWLFXODU LWV J\URVFRSLF HIIHFW WKH IUHTXHQF\ RI WKH ZKHHO PRWLRQKDVUHDFKHGDOHYHOWKDWUHTXLUHVWKHLQFOXVLRQRIWKHILUVWRUHYHQKLJKHU PRGHVRIYLEUDWLRQRIWKHEHOW WKHYHUWLFDOSURILOHRIWKHURDGVXUIDFHFRQWDLQV YHU\VKRUWZDYHOHQJWKVZLWKDSSUHFLDEOHDPSOLWXGHVDVZRXOGRFFXULQWKHFDVH RI UROOLQJ RYHU D VKRUW REVWDFOH RU FOHDW WKHQ DPRQJVW RWKHU WKLQJV WKH W\UH HQYHORSLQJ SURSHUWLHV VKRXOG EH DFFRXQWHG IRU PRWLRQV EHFRPH VHYHUH ODUJH VOLS DQG KLJK VSHHG ZKLFK PD\ QHFHVVLWDWH PRGHOOLQJ WKH HIIHFW RI WKH ZDUPLQJ XS RI WKH W\UH LQYROYLQJ SRVVLEO\ WKH LQWURGXFWLRQ RI WKH W\UH WHPSHUDWXUHDVDPRGHOSDUDPHWHU$OOLWHPVPHQWLRQHGH[FHSWWKHODVWRQH ZLOO EHDFFRXQWHGIRULQWKHUHPDLQGHURIWKLVERRN
&KDSWHU 7+(25<2)67($'<67$7(6/,3)25&( $1'020(17*(1(5$7,21
,QWURGXFWLRQ
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µEUXVK¶PRGHORI)URPPDQGRI-XOLHQFI +DGHNHO IRUUHIHUHQFHV ZKRGLGQRWFRQVLGHUFDUFDVVFRPSOLDQFH)LDOD DQG )UHXGHQVWHLQ GHYHORSHG WKHRULHV LQ ZKLFK WKH FDUFDVV GHIOHFWLRQ LV DSSUR[LPDWHG E\ D V\PPHWULF SDUDEROD %|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
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
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¶V PRGHO V\PPHWULF SDUDEROLF FDUFDVV GHIOHFWLRQ G PRGHO RI )URPP EUXVKPRGHOZLWKULJLGFDUFDVV 7KHWUHDGHOHPHQWVWLIIQHVVLVWKHVDPHIRUHDFK PRGHO7KHSDUDPHWHUVLQWKHFDVHVDEDQGFKDYHEHHQFKRVHQLQVXFKDZD\
)\
d
2000
b
a,c
[N] 1000
0
0]
0
5
10
α
[deg]
0
[Nm]
d
b
c
-20
a
)LJ &RPSDULVRQRI FDOFXODWHG FKDUDFWHULVWLFVIRUIRXUGLIIHUHQWW\UHPRGHOVZLWKWUHDG HOHPHQWVDQGQRQV\PPHWULFDOSUHVVXUHGLVWULEXWLRQDWDJLYHQZKHHOYHUWLFDOORDG DVWULQJEEHDPF)LDODGEUXVKLHZLWKULJLGFDUFDVV IURP)UDQN
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
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¶VDQG)LDOD¶VPRGHOVDSSHDU WR RFFXU 'XH WR DSSUR[LPDWLRQV LQWURGXFHG E\ )LDOD WKH FRHIILFLHQWV LQ WKH H[SUHVVLRQIRUWKHVLGHIRUFHYHUVXVVOLSDQJOHLISDUDEROLFSUHVVXUHGLVWULEXWLRQ LVDGRSWHG EHFRPHHTXDOWRWKRVHREWDLQHGGLUHFWO\E\)URPP ,Q WKH FDOFXODWLRQV IRU )LJ )UDQN HPSOR\HG D FRQVWDQW FRHIILFLHQW RI IULFWLRQDQGDVOLJKWO\DV\PPHWULFYHUWLFDOSUHVVXUHGLVWULEXWLRQT][ IRXQG IURPPHDVXUHPHQWV7KHSRVLWLYHDOLJQLQJWRUTXHREWDLQHGDWODUJHUYDOXHVRIWKH VOLS DQJOH
DULVHV DV D UHVXOW RI WKLV DV\PPHWU\ 7KH SKHQRPHQRQ WKDW LQ
SUDFWLFHWKHDOLJQLQJWRUTXHLQGHHGYDULHVLQWKLVZD\LVGXHWRDFRPELQDWLRQRI VHYHUDO HIIHFWV7KH PDLQ FDXVH LV SUREDEO\ FRQQHFWHG ZLWK WKH DV\PPHWULF SUHVVXUHGLVWULEXWLRQRIWKHUROOLQJW\UHGXHWRK\VWHUHVLVRIWKHW\UHFRPSRXQG UHVXOWLQJLQDVPDOOIRUZDUGVKLIWRIWKHSRLQWRIDSSOLFDWLRQRIWKHQRUPDOORDG JLYLQJULVHWR UROOLQJUHVLVWDQFH DQGFRQVHTXHQWO\DWIXOOVOLGLQJDOVRRIWKH UHVXOWLQJVLGHIRUFH$QRWKHULPSRUWDQWIDFWRU FDXVLQJWKHPRPHQWWREHFRPH SRVLWLYHLVWKHIDFWWKDWWKHFRHIILFLHQWRIIULFWLRQLVQRWDFRQVWDQWEXWWHQGVWR GHFUHDVHZLWKVOLGLQJYHORFLW\$VPD\EHGHULYHGIURPHJ(T WKHVOLGLQJ YHORFLW\DWWDLQVLWVODUJHVWYDOXHVLQWKHUHDUSRUWLRQRIWKHFRQWDFWDUHDZKHUHWKH
M M
VORSH Y [EHFRPHVODUJHVW&RQVHTXHQWO\ZHH[SHFWWRKDYHODUJHUVLGHIRUFHV DFWLQJLQWKHIURQWKDOIRIWKHFRQWDFWDUHDDWIXOOVOLGLQJFRQGLWLRQVWKDQLQWKH UHDU KDOI 7KH UROOLQJ UHVLVWDQFH IRUFH WKDW GXH WR WKH ODWHUDO GLVWRUWLRQ DFWV VOLJKWO\EHVLGHWKHZKHHOSODQHPD\DOVRFRQWULEXWHWRWKHVLJQFKDQJHRI0]$ ZLWKVOLGLQJYHORFLW\GHFUHDVLQJQRWFRQVLGHUHGE\)UDQN FDXVHVWKHFUHDWLRQ RI D SHDN LQ WKH ) \
FXUYHV DQG D IXUWKHU VOLJKW GHFD\ 7KLV KDV RIWHQ EHHQ
REVHUYHGWRRFFXULQSUDFWLFHHVSHFLDOO\RQZHWDQGLF\URDGV,QWKHORQJLWXGLQDO IRUFHFKDUDFWHULVWLFWKHSHDNLVXVXDOO\PRUHSURQRXQFHG 7KH LQIOXHQFH RI GLIIHUHQW EXW V\PPHWULF VKDSHV IRU WKH YHUWLFDO IRUFH GLVWULEXWLRQ DORQJ WKH [D[LV KDV EHHQ WKHRUHWLFDOO\ LQYHVWLJDWHG E\ %RUJPDQ +HILQGVWKDWHVSHFLDOO\IRUW\UHVH[KLELWLQJDUHODWLYHO\ODUJHFDUFDVV FRPSOLDQFH WKH LQIOXHQFH RI WKH SUHVVXUH GLVWULEXWLRQ LV RI LPSRUWDQFH 1RQ V\PPHWULFPRUHJHQHUDOGLVWULEXWLRQVZHUHVWXGLHGE\*XR 0DQ\DXWKRUV DGRSWIRUWKHSXUSRVHRIPDWKHPDWLFDOVLPSOLFLW\WKHSDUDEROLFGLVWULEXWLRQ)LDOD )UHXGHQVWHLQ%HUJPDQ3DFHMND6DNDLDOVRQWKGHJUHH SDUDEROD 'XJRII HW DO XQLIRUP UHFWDQJXODU GLVWULEXWLRQ DQG
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
%HUQDUG HW DO WUDSH]LXP VKDSH 0RGHOV RI )LDOD DQG )UHXGHQVWHLQ IHDWXUHDIOH[LEOHFDUFDVVZKLOHWKHUHPDLQLQJDXWKRUVKDYHUHVWULFWHGWKHPVHOYHV WRDULJLGFDUFDVVRUDXQLIRUPO\GHIOHFWHGEHOW7KDWKRZHYHUHQDEOHGWKHPWR LQFOXGH WKH GHVFULSWLRQ RI WKH PRUH GLIILFXOW FDVH RI FRPELQHG VOLS 7KH LQWURGXFWLRQRIDQRQFRQVWDQWIULFWLRQFRHIILFLHQWKDVEHHQWUHDWHGE\DPRQJVW RWKHUV%|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
7\UH%UXVK0RGHO
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
PXOWLSOLHG ZLWK WKH HIIHFWLYH UROOLQJ
UDGLXVUHLVQRWHTXDOWRWKHIRUZDUGFRPSRQHQWRIWKHZKHHOVSHHG9[
9FRV
ZHKDYHIRUHDQGDIWVOLS8QGHUWKHVHFRQGLWLRQVGHSLFWHGLQ)LJKRUL]RQWDO
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
wheel slip speed
9
speed of travel 9 slip angle
9V\
α
α
Ω
SXUHVLGH VOLS
)\
)[
&
SXUHEUDNH
)[
&
)\
)]
9V[
VOLS
9V
9
α FRPELQHG
) W
VOLS
9
A
maximum deflection
)\ 9V\
)\
9
B
C sliding
)\
D
B
adhesion
α
A
C
α
0 full sliding
)\
0] W)\
D
SXUHVLGH
)\
)LJ
VOLS
7KHEUXVKW\UHPRGHO 7RSOHIWYLHZRIGULYHQDQGVLGHVOLSSLQJW\UH 7RSULJKWWKHW\UHDWGLIIHUHQWVOLSFRQGLWLRQV %RWWRPOHIWWKHW\UHDWSXUHVLGHVOLSIURPVPDOOWRODUJHVOLSDQJOH %RWWRPULJKWWKHUHVXOWLQJVLGHIRUFHDQGDOLJQLQJWRUTXHFKDUDFWHULVWLFV
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
GHIOHFWLRQVDUHGHYHORSHGDQGFRUUHVSRQGLQJIRUFHVDQGPRPHQWDULVH7KHWUHDG HOHPHQWVPRYHIURPWKHOHDGLQJHGJHDWWKHULJKWKDQGVLGHRIWKHSLFWXUHV WR WKH WUDLOLQJ HGJH 7KH WLS RI WKH HOHPHQW ZLOO DV ORQJ DV WKH DYDLODEOH IULFWLRQ DOORZVDGKHUHWRWKHJURXQGWKDWLVLWZLOOQRWVOLGHRYHUWKHURDGVXUIDFH $W WKH VDPH WLPH WKH EDVH SRLQW RI WKH HOHPHQW UHPDLQV LQ WKH ZKHHO SODQH DQG PRYHVEDFNZDUGVZLWKWKHOLQHDUVSHHGRIUROOLQJ9UWKDWLVHTXDOWR
UH ZLWK
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
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
WRSYLHZ 0]
wheel spin axis [
α
&
wheel plane
Y
α ) \ [ W
tread elements
\
9 [
YPD[
λ
D
D
D
[
road surface
T ]
VLGHYLHZ
)
]
]
)LJ%UXVKPRGHOPRYLQJDWSXUHVLGHVOLSVKRZQLQWRSDQGVLGHYLHZ
3XUHVLGHVOLS
7KHEUXVKPRGHOPRYLQJDWDFRQVWDQWVOLSDQJOHKDVEHHQGHSLFWHGLQJUHDWHU GHWDLO LQ )LJ ,W VKRZV D FRQWDFW OLQH ZKLFK LV VWUDLJKW DQG SDUDOOHO WR WKH YHORFLW\YHFWRU9LQWKHDGKHVLRQUHJLRQDQGFXUYHGLQWKHVOLGLQJUHJLRQZKHUH WKH DYDLODEOH IULFWLRQDO IRUFH EHFRPHV ORZHU WKDQ WKH IRUFH ZKLFK ZRXOG EH UHTXLUHGIRUWKHWLSVRIWKHWUHDGHOHPHQWVWRIROORZWKHVWUDLJKWOLQHIXUWKHU,Q WKHDGKHVLRQUHJLRQWKHOLQHDUYDULDWLRQRIWKHGHIRUPDWLRQLVLQDFFRUGDQFHZLWK WKHJHQHUDOHTXDWLRQ ZKHUHWDQ
M M
ZLWKDWVWHDG\VWDWH Y V
KDVEHHQDVVXPHGVPDOODQGUHSODFHGE\
)RUWKLVVLPSOHPRGHOWKHGHIRUPDWLRQRIWKH
WUHDGHOHPHQWDWWKHOHDGLQJHGJHYDQLVKHV&RQVHTXHQWO\WKHODWHUDOGHIRUPDWLRQ LQWKHDGKHVLRQUHJLRQUHDGV
Y ' D&[ WDQ ZKHUHDGHQRWHVKDOIWKHFRQWDFWOHQJWK ,QFDVHRIYDQLVKLQJVOLGLQJZKLFKZLOORFFXUIRU
6RUIRU64H[SUHVVLRQ
LVYDOLGIRUWKHHQWLUHUHJLRQRIFRQWDFW:LWKWKHODWHUDOVWLIIQHVVFS\RIWKH WUHDG HOHPHQWV SHU XQLW OHQJWK RI WKH DVVXPHGO\ UHFWDQJXODU FRQWDFW DUHD WKH IROORZLQJLQWHJUDOVDQGH[SUHVVLRQVIRUWKHFRUQHULQJIRUFH)\DQGWKHDOLJQLQJ WRUTXH0]KROG
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
) \ ' FS\
YG[ ' FS\ D
D
m&D
0 ] ' FS\
D
m&D
Y[G[ ' & FS\ D
&RQVHTXHQWO\ WKH FRUQHULQJ VWLIIQHVV DQG WKH DOLJQLQJ VWLIIQHVV EHFRPH UHVSHFWLYHO\
&) ' M) \ M
'
&0 ' & M0 ] M
' FS\ D
'
' FS\ D
1H[W ZH ZLOO FRQVLGHU WKH FDVH RI ILQLWH DQG D SUHVVXUH GLVWULEXWLRQ ZKLFK JUDGXDOO\GURSVWR]HURDWERWKHGJHV)RUWKHSXUSRVHRIVLPSOLFLW\ZHDVVXPH DSDUDEROLFGLVWULEXWLRQRIWKHYHUWLFDOIRUFHSHUXQLWOHQJWKDVH[SUHVVHGE\
T] '
) [ @ & D D
]
ZKHUH)]UHSUHVHQWVWKHYHUWLFDOZKHHOORDG+HQFHWKHODUJHVWSRVVLEOHVLGHIRUFH GLVWULEXWLRQEHFRPHV
T
* \ PD[* '
T] ' ) ] D & [ D
,Q)LJWKHPD[LPXPSRVVLEOHODWHUDOGHIRUPDWLRQYPD[
T\PD[FS\KDVEHHQ
LQGLFDWHG )RU WKH VDNH RI DEEUHYLDWLRQ WKH IROORZLQJ FRPSRVLWH W\UH PRGHO SDUDPHWHULVLQWURGXFHG
F D ' \ )] S\
7KHGLVWDQFHIURPWKHOHDGLQJHGJH WR WKH SRLQWZKHUHWKHWUDQVLWLRQIURPWKH DGKHVLRQWRWKHVOLGLQJUHJLRQRFFXUVLVZULWWHQDVD DQGLVGHWHUPLQHGE\WKH IDFWRU 7KHYDOXHRIWKLVQRQGLPHQVLRQDOTXDQWLW\LVIRXQGE\UHDOLVLQJWKDWDW WKLVSRLQWZKHUH[
[WWKHGHIOHFWLRQLQWKHDGKHVLRQUDQJHEHFRPHVHTXDOWRWKDW
RIWKHVOLGLQJUDQJH+HQFHZLWK(TV WKHIROORZLQJHTXDOLW\KROGV
T ' FS\ D & [W _WDQ _ ' _T\ PD[ _ '
_ \_
DQGWKXVIRU
' &
FS\ D &[ W D % [W D \
D[W DZHREWDLQWKHUHODWLRQVKLSZLWKWKHVOLSDQJOH
\ *WDQ *
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
)URP WKLV HTXDWLRQ WKH DQJOH
VO
ZKHUH WRWDO VOLGLQJ VWDUWV
FDQ EH
FDOFXODWHG
'
WDQ VO
\
$VWKHGLVWULEXWLRQRIWKHGHIOHFWLRQVRIWKHHOHPHQWVKDVQRZEHHQHVWDEOLVKHG WKH WRWDO IRUFH )\ DQG WKH PRPHQW 0] FDQ EH DVVHVVHG E\ LQWHJUDWLRQ RYHU WKH FRQWDFWOHQJWKOLNHLQ(T EXWQRZVHSDUDWHIRUWKHVOLGLQJUDQJHD[[W DQGWKHDGKHVLRQUDQJH[W[D )RUFRQYHQLHQFHZHLQWURGXFHWKHQRWDWLRQIRU WKHVOLS
\
'
WDQ
7KHUHVXOWLQJIRUPXODIRUWKHIRUFHUHDGV LI
* *#
VO
)\ ' )] & VJQ ' )] \ \ & \ \ % DQGLI
* *$
VO
\ \
EXWò
)\ ' )] VJQ DQGIRUWKHPRPHQW LI
* *#
D
VO
0 ] ' & )] D & VJQ ' & )] D \ \ & \ \ % ZLWKSHDNYDOXH )]DW
0] '
\
\ \
&
\ ,I
\ \
* *$
VO
EXWò D
7KHSQHXPDWLFWUDLOWZKLFKLQGLFDWHVWKHGLVWDQFHEHKLQGWKHFRQWDFWFHQWUH& ZKHUHWKHUHVXOWDQWVLGHIRUFH)\LVDFWLQJEHFRPHV LI
* *#
VO
W' &
0] & \ \ % \ \ & ' D )\ & \ \ % \ \
\ \
DQGLI
* *$
W'
VO
EXWò
D
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
)\
0]
Dµ)
] µ)] 0.1 1
0]
W
, 3W D
)\
.)\
1
W
1/(4θ\)
-1/θ\
tanα
1/θ\
arctan 3θ\ 27 256
γ =0
-0.1 -1 )LJ
O
&KDUDFWHULVWLFVRIWKHVLPSOHEUXVKPRGHOVLGHIRUFHDOLJQLQJWRUTXHDQGSQHXPDWLF WUDLOYVVOLSDQJOH
7KHVHUHODWLRQVKLSVKDYHEHHQVKRZQJUDSKLFDOO\LQ)LJ$WYDQLVKLQJVOLS DQJOHH[SUHVVLRQ UHGXFHVWR
W ' WR ' &
0] )\
' D
6
7KLVYDOXHLVVPDOOHUWKDQQRUPDOO\HQFRXQWHUHGLQSUDFWLFH7KHLQWURGXFWLRQRI DQHODVWLFFDUFDVVZLOOLPSURYHWKLVTXDQWLWDWLYHDVSHFW7KHQWKHPRUHUHDOLVWLF
. DPD\EHDFKLHYHGFI3DFHMNDIRUDVWUHWFKHGVWULQJPRGHO
YDOXHRIW
WRUHSUHVHQWWKHHODVWLFFDUFDVVDQG6HFWLRQIRUDPRUHJHQHUDOO\DSSOLFDEOH ZD\RIDSSURDFKWRDFFRXQWIRUWKHFDUFDVVFRPSOLDQFH $QRWKHUSRLQWLQZKLFK WKHVLPSOHPRGHOGHYLDWHVFRQVLGHUDEO\IURPH[SHULPHQWDOUHVXOWVFRQFHUQVWKH HIIHFW RI FKDQJLQJ WKH YHUWLFDO ZKHHO ORDG )] :LWK WKH DVVXPSWLRQ WKDW WKH FRQWDFWOHQJWKDFKDQJHVTXDGUDWLFDOO\ZLWKUDGLDOW\UHGHIOHFWLRQ DQGWKDW)] GHSHQGVOLQHDUO\RQ VRWKDWD
-)
]
LWFDQEHHDVLO\VKRZQWKDWIRUWKHEUXVK
PRGHO)\DQG0]YDU\SURSRUWLRQDOO\ZLWK)]DQG)]
UHVSHFWLYHO\([SHULPHQWV
KRZHYHUVKRZWKDW)\YDULHVOHVVWKDQOLQHDUO\ZLWK)],QPRVWFDVHVWKH)\YV )] FKDUDFWHULVWLF REWDLQHG DW D VPDOO YDOXH RI WKH VOLS DQJOH HYHQ VKRZV D PD[LPXP DIWHU ZKLFK WKH FRUQHULQJ IRUFH GURSV ZLWK LQFUHDVLQJ ZKHHO ORDG 2EYLRXVO\WKHVDPHKROGVIRUWKHFRUQHULQJVWLIIQHVV&) FI)LJ $OVRLQ WKLVUHVSHFWWKHLQWURGXFWLRQRIDQHODVWLFFDUFDVVLQSDUWLFXODUZKHQLWVODWHUDO
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
)\ µ)] 8
1
)
\
[N] 6
[deg]
8
4000
0.01
5
7
6
4
4
5
3
3000
2
3
2
2000 1000
-0.1
0]
0
µ
D )]
α
1
1 0
0.03 W[m]
2
1 0
0.02 4
3
0
0.005
5000
α
5
θ \ = 5, µ 1 D 2 = 13)] x10-5
) [kN] ]
20
[deg]
40
60
0] [Nm]
)LJ 7KHVRFDOOHG*RXJKSORWIRUWKHEUXVKPRGHOQRQGLPHQVLRQDODQGZLWKGLPHQVLRQ XVLQJDQDVVXPHGORDGYVFRQWDFWOHQJWKUHODWLRQVKLS
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
)\
)\
)[kN] ]
5
[N]
27.5 14 20000 25
10
15 12
)[kN] ]
[N]
20
25
10
20000
30
8
20
80 2
α [deg]
W
[mm]
0
20 10
15 10000
5
15
12
25 30
8
50
4
10000 10
10 14
20
40
6
15
5 25
6
10
4 W
5
2
40 50
[mm]
α [deg]
0 0
)LJ
250
[Nm]
500 0 ]
0
200
[Nm]
400 0 ]
/HIW*RXJKSORWRIWUXFNW\UHPHDVXUHGRQDGU\URDGVXUIDFH)UHXGHQVWHLQ
F
5LJKWEUXVKPRGHOZLWKWUHDGHOHPHQWVWLIIQHVV
S
)
GHFD\LQJOLQHDUO\ZLWK
]
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
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
3XUH/RQJLWXGLQDO6OLS
)RU WKH EUXVKW\SH W\UH PRGHO ZLWK WUHDG HOHPHQWV IOH[LEOH LQ ORQJLWXGLQDO GLUHFWLRQ WKH WKHRU\ IRU ORQJLWXGLQDO EUDNLQJ RU GULYLQJ IRUFH JHQHUDWLRQ GHYHORSVDORQJVLPLODUOLQHVDVWKRVHVHWRXWLQ6HFWLRQZKHUHWKHVLGHIRUFH DQG DOLJQLQJ WRUTXH UHVSRQVH WR VOLS DQJOH KDV EHHQ GHULYHG 7R VLPSOLI\ WKH GLVFXVVLRQZHUHVWULFWRXUVHOYHVKHUHWRQRQQHJDWLYHYDOXHVRIWKHIRUZDUGVSHHG 9[DQGRIWKHVSHHGRIUHYROXWLRQ
,Q)LJDVLGHYLHZRIWKHEUXVKPRGHOKDVEHHQVKRZQ$VZDVLQGLFDWHG EHIRUH WKH VRFDOOHG VOLS SRLQW 6 LV LQWURGXFHG 7KLV LV DQ LPDJLQDU\ SRLQW DWWDFKHG WR WKH ZKHHO ULP DQG LV ORFDWHG DW WKH LQVWDQW FRQVLGHUHG D GLVWDQFH HTXDOWRWKHHIIHFWLYHUROOLQJUDGLXVUHGHILQHGDWIUHHUROOLQJ EHORZWKHZKHHO FHQWUH$WIUHHUROOLQJE\GHILQLWLRQWKHVOLSSRLQW6KDVDYHORFLW\HTXDOWR]HUR
Ω
9U U H 9[
D
UH
[
U 6
9V[ X
∆W 9V[ ∆W 9[ ∆W 9U
6
9V[
D[ X
)LJ6LGHYLHZRIWKHEUXVKW\UHPRGHODWEUDNLQJQRVOLGLQJFRQVLGHUHG
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
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¶DOVRPRYHVZLWKWKHVDPH VOLSVSHHG9V[,WLVDVVXPHGWKDWWKHWUHDGHOHPHQWVDWWDFKHGDWWKHLUEDVHSRLQWV WRWKHFLUFXPIHUHQWLDOO\ULJLGFDUFDVVHQWHUWKHFRQWDFWDUHDLQYHUWLFDOSRVLWLRQ $WIUHHUROOLQJZLWKVOLSVSHHG 9 V[ RIERWKSRLQWV6DQG6¶ HTXDOWR]HURWKH RULHQWDWLRQ RI WKH HOHPHQWV UHPDLQV YHUWLFDO ZKLOH PRYLQJ IURP IURQW WR UHDU WKURXJK WKH FRQWDFW ]RQH &RQVHTXHQWO\ QR ORQJLWXGLQDO IRUFH LV EHLQJ WUDQVPLWWHGDQGZHKDYHDZKHHOVSHHGRIUHYROXWLRQ
'
R'
9[ UH
+HUHLWLVDVVXPHGWKDWWKHORQJLWXGLQDOFRPSRQHQWRIWKHVSHHGRISURSDJDWLRQ RIWKHFRQWDFWFHQWUH&LVHTXDOWRWKHORQJLWXGLQDOFRPSRQHQWRIWKHVSHHGRIWKH ZKHHOFHQWUH9F[
9[ $VKDVEHHQVHHQLQWKHSUHYLRXVFKDSWHUWKLVZLOORFFXU
RQ D IODW URDGVXUIDFHDWYDQLVKLQJ UROOLQJ
0 :KHQ
GLIIHUVIURPLWVYDOXHDWIUHH
WKHZKHHOLVEHLQJEUDNHGRUGULYHQDQGWKHORQJLWXGLQDOVOLSVSHHG9V[ R
EHFRPHV
9V[ ' 9[ & U H
,QWKHPRGHOWKHEDVHSRLQWVRIDOOWKHWUHDGHOHPHQWVPRYHZLWKWKHVDPHVSHHG 9V[$EDVHSRLQWSURJUHVVHVEDFNZDUGVWKURXJKWKHFRQWDFW]RQHZLWKDVSHHG 9UFDOOHGWKHOLQHDUVSHHGRIUROOLQJ$SSDUHQWO\ZHKDYH
9U '
UH ' 9 [ & 9V[
$QHOHPHQWWKHWLSRIZKLFKDGKHUHVWRWKHJURXQGDQGWKHEDVHSRLQWLVPRYHG WRZDUGVWKHUHDURYHUDGLVWDQFHD[IURPWKHOHDGLQJHGJHIRUZKLFKDWLPHVSDQ W
D[ 9ULVQHHGHG KDVGHYHORSHGDGHIOHFWLRQLQORQJLWXGLQDOGLUHFWLRQ
X ' & 9V[ D & [ 9U
7KHVDPHH[SUHVVLRQPD\EHREWDLQHGE\LQWHJUDWLRQRIWKHIXQGDPHQWDOHTXDWLRQ DQGQRWLQJWKDW9J[ FRQGLWLRQX
DW[
D
]
MXMW
¶V
DQGILQDOO\XVLQJWKHERXQGDU\
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
:H PD\ ZULWH WKH ORQJLWXGLQDO GHIOHFWLRQ X LQ WHUPV RI WKH µSUDFWLFDO¶ ORQJLWXGLQDOVOLS
X ' & D & [
9V[9[
9V[ ' D & [ 9[ & 9V[ %
,QWHUPVRIWKHDOWHUQDWLYHGHILQLWLRQRIORQJLWXGLQDOVOLSWKHµWKHRUHWLFDO¶VOLSWR EHXVHGLQWKHVXEVHTXHQWVHFWLRQDQGGHILQHGDV
['
&
9V[ ' 9U %
QRWHZHUHVWULFWRXUVHOYHVWRQRQQHJDWLYHVSHHGVRIUROOLQJ9U
$ DQG $
ZHREWDLQ
X ' D & [
[
,Q6HFWLRQOZHIRXQGZLWK(TV DQG IRUWKHODWHUDOGHIOHFWLRQDW SXUHVLGHVOLS
\
Y ' D & [
WDQ
\
&RPSDULVRQ RI WKH HTXDWLRQV DQG VKRZV WKDW WKH ORQJLWXGLQDO GHIRUPDWLRQVXZLOOEHHTXDOLQPDJQLWXGHWRWKHODWHUDOGHIRUPDWLRQVYLI WDQ HTXDOV
[
IULFWLRQFRHIILFLHQWV[
)RU HTXDO WUHDG HOHPHQW VWLIIQHVVHV FS[
\
FS\ DQG
\ LQODWHUDODQGORQJLWXGLQDOGLUHFWLRQVWKHVOLSIRUFH
FKDUDFWHULVWLFVLQERWKGLUHFWLRQVDUHLGHQWLFDOZKHQWDQ
DQG
DUHXVHG
DVDEVFLVVDFI)LJ$OVR(T KROGVIRUWKHORQJLWXGLQDOIRUFH)[LIWKH VXEVFULSWV\DUHUHSODFHGE\[DQGWDQ
E\
2EYLRXVO\WRWDOVOLGLQJZLOOVWDUWDW
[
[
RULQWHUPVRIWKH
SUDFWLFDOVOLSDW
)[
)\ µ)]
&)κ σ[ =
κ 1+κ
&)α
µ)] σ\ = tanα
)LJ (TXDOLW\ RI WKH WZR SXUH VOLS FKDUDFWHULVWLFV IRU DQ LVRWURSLF W\UH PRGHO LISORWWHG DJDLQVWWKHWKHRUHWLFDOVOLS
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
'
VO '
&
[
ZLWK
['
FS[D )]
/LQHDUL]DWLRQIRUVPDOOYDOXHVRIVOLS \LHOGVDGHIOHFWLRQDWFRRUGLQDWH[
X ' D & [
DQGDIRUHDQGDIWIRUFH
)[ ' FS[D
ZLWK FS[ WKH ORQJLWXGLQDO WUHDG HOHPHQW VWLIIQHVV SHU XQLW OHQJWK 7KLV UHODWLRQ FRQWDLQVWKHORQJLWXGLQDOVOLSVWLIIQHVV
&) ' M)[ M
'
' FS[ D
)RU HTXDO ORQJLWXGLQDO DQG ODWHUDO VWLIIQHVVHV FS[ VWLIIQHVVHV&)
FS\ ZH REWDLQ HTXDO VOLS
&) ,QUHDOLW\KRZHYHUDSSUHFLDEOHGLIIHUHQFHVEHWZHHQWKH
PHDVXUHGYDOXHVRI&) DQG&) PD\RFFXUVD\&) DERXWODUJHUWKDQ&) ZKLFKLVGXHWRWKHODWHUDOWRUVLRQDO FRPSOLDQFH RI WKH FDUFDVVRIWKHDFWXDO W\UH 6WLOO LW LV H[SHFWHG WKDW TXDOLWDWLYH VLPLODULW\ RI ERWK SXUH VOLS FKDUDFW HULVWLFVUHPDLQV
,QWHUDFWLRQEHWZHHQ/DWHUDODQG/RQJLWXGLQDO6OLSFRPELQHGVOLS
)RUWKHDQDO\VLVRIWKHLQIOXHQFHRIORQJLWXGLQDOVOLSRUORQJLWXGLQDOIRUFH RQWKH ODWHUDO IRUFH DQG PRPHQW JHQHUDWLRQ SURSHUWLHV ZH VKDOO IRU WKH VDNH RI PDWKHPDWLFDOVLPSOLFLW\UHVWULFWRXUVHOYHVWRWKHFDVHRIHTXDOORQJLWXGLQDODQG ODWHUDOVWLIIQHVVHVRIWKHWUHDGHOHPHQWVLVRWURSLFPRGHO LH
FS' FS[' F S\
DQGHTXDODQGFRQVWDQWIULFWLRQFRHIILFLHQWV
' [' \
$JDLQDSDUDEROLFSUHVVXUHGLVWULEXWLRQLVFRQVLGHUHG )LJXUHGHSLFWVWKHGHIRUPDWLRQVZKLFKDULVHZKHQWKHW\UHPRGHOZKLFK UXQVDWDJLYHQVOLSDQJOH LVGULYHQRUEUDNHG'XHWRWKHHTXDOVWLIIQHVVLQDOO
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
9U
9V
D
D
9
λ
2
D
9
[
σ
α
[
XY
]
δ
2
)
α
[
)
σ[
0
’inclined’ parabola
D
)
\
)
GULYLQJ
\
9V
9U
2
9
α
[
)
[
α
σ
]
0
-2
)
9
λ
D
δ
σ[
D
)
EUDNLQJ
\
)
\
V[
9 6
[
)
] )LJ
)]
[
T]
9HFWRUGLDJUDPDQGGHIRUPDWLRQRIWKHEUXVKPRGHOUXQQLQJDWDJLYHQVOLSDQJOHIRU WKHFDVHVRIGULYLQJDQGEUDNLQJ
KRUL]RQWDO GLUHFWLRQV DQG WKH LVRWURSLF IULFWLRQ SURSHUWLHV WKH GHIOHFWLRQV DUH GLUHFWHGRSSRVLWHWRWKHVOLSVSHHGYHFWRU
9V
DOVRLQWKHVOLGLQJUHJLRQ,QWKLV
ODWWHUUHJLRQWKHWLSVRIWKHHOHPHQWVVOLGHRYHUWKHURDGZLWKVOLGLQJVSHHG
T
9J
GLUHFWHGRSSRVLWHWRWKHORFDOIULFWLRQIRUFH SHUXQLWFRQWDFWOHQJWK 7KHZKROH GHIRUPDWLRQKLVWRU\RIDWUHDGHOHPHQWZKLOHUXQQLQJWKURXJKWKHFRQWDFWDUHDLV DRQHGLPHQVLRQDOSURFHVVDORQJWKHGLUHFWLRQRI
9V
7KHYHORFLW\RISURJUHVVLRQRIDEDVHSRLQWWKURXJKWKHFRQWDFWOHQJWKLVWKH UROOLQJVSHHG9UDJDLQDVVXPHGQRQQHJDWLYH 7KHGHIOHFWLRQUDWHRIDQHOHPHQW LQWKHDGKHVLRQUHJLRQLVHTXDOWRWKHVOLSVSHHG9V7KHWLPHZKLFKHODSVHVIURP WKHSRLQWRIHQWUDQFHWRWKHSRLQWDGLVWDQFH[LQIURQWRIWKHFRQWDFWFHQWUHHTXDOV
W ' D&[ 9U
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
,QWKLVSRVLWLRQWKHGHIOHFWLRQRIDQHOHPHQWWKDWLVVWLOOLQDGKHVLRQEHFRPHVLQ YHFWRULDOIRUP
X 9 ' 9 V W ' & V D & [ 9U Y
H'
,W VHHPV QDWXUDO DW WKLV VWDJH WR LQWURGXFH WKH DOWHUQDWLYH WKHRUHWLFDO VOLS TXDQWLW\DJDLQEXWQRZLQYHFWRULDOIRUP
[
'
'&
\
9V
9U
'&
9V[
9U 9V\
ZLWKWKHOLQHDUVSHHGRIUROOLQJ
9U ' 9[ & 9 V[
7KHUHODWLRQVRIWKHVHWKHRUHWLFDOVOLSTXDQWLWLHVZLWKWKHSUDFWLFDOVOLSTXDQWLWLHV
9V[9[ DQGWDQ
[
\
' '
9V\9[ DUH
%
WDQ
%
7KHGHIOHFWLRQRIDQHOHPHQWLQWKHDGKHVLRQUHJLRQQRZUHDGV
H ' D & [
IURPZKLFKLWLVDSSDUHQWWKDWORQJLWXGLQDODQGODWHUDOGHIOHFWLRQVDUHJRYHUQHG E\
[
DQG
\
UHVSHFWLYHO\DQGLQGHSHQGHQWRIHDFKRWKHU7KLVZRXOGQRWEHWKH
FDVHLIH[SUHVVHGLQWHUPVRIWKHSUDFWLFDOTXDQWLWLHV DQG
7KHORFDOKRUL]RQWDOFRQWDFWIRUFHDFWLQJRQWKHWLSVRIWKHHOHPHQWVSHUXQLW FRQWDFWOHQJWK UHDGV
T ' FS D & [
DGKHVLRQUHJLRQ
$VVRRQDV
T ' *T* ' T[ % T\
!
T]
WKHVOLGLQJUHJLRQLVHQWHUHG7KHQWKHIULFWLRQIRUFHYHFWRUEHFRPHV
T' &
9V 9V
T] '
T]
VOLGLQJUHJLRQ
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
ZKHUH
9V ' 9V[ % 9V\
DQG
[
'
\
%
6LPLODUO\WKHPDJQLWXGHRIWKHGHIOHFWLRQRIDQHOHPHQWEHFRPHV
H ' _H* ' X % Y
7KH SRLQW RI WUDQVLWLRQ IURP DGKHVLRQ WR VOLGLQJ UHJLRQ LV REWDLQHG IURP WKH FRQGLWLRQ
FS H ' T] RU
D & [W F S D & [W ' ) ] D
ZKLFK\LHOGV
FD [W ' S & D ' D ) ]
&
RULQVLPLODUWHUPVDV(T
' &
ZKHUH DQDORJRXV WR H[SUHVVLRQV DQG IRU WKH LVRWURSLF PRGHO SDUDPHWHU UHDGV
'
\
'
[
FS D )]
'
)URP(T WKHVOLS
VO
DWZKLFKWRWDOVOLGLQJVWDUWVFDQEHFDOFXODWHG:HJHW
DQDORJRXVWR
VO
'
7KHPDJQLWXGHRIWKHWRWDOIRUFH)
) ' )] & ) ' )]
)
* *QRZHDVLO\IROORZVLQDFFRUGDQFHZLWK
' )] ^ &
%
`
IRU IRU
# $
VO VO
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
DQG REYLRXVO\ IROORZV WKH VDPH FRXUVH DV WKRVH VKRZQ LQ )LJ 7KH IRUFH YHFWRU
)
DFWVLQDGLUHFWLRQRSSRVLWHWR
9
V
RU
+HQFH
)' )
IURPZKLFKWKHFRPSRQHQWV)[DQG)\PD\EHREWDLQHG 7KHPRPHQW0]LVREWDLQHGE\PXOWLSOLFDWLRQRI)\ZLWKWKHSQHXPDWLFWUDLO W7KLVWUDLOLVHDVLO\IRXQGZKHQZHUHDOL]HWKDWWKHGHIOHFWLRQGLVWULEXWLRQRYHU WKHFRQWDFWOHQJWKLVLGHQWLFDOZLWKWKHFDVHRISXUHVLGHVOLSLIWDQ
HT
FI)LJ
&RQVHTXHQWO\ WKH IRUPXOD UHSUHVHQWV WKH SQHXPDWLF WUDLO DW FRPELQHGVOLSDVZHOOLI
\
\
LVUHSODFHGE\
:HKDYHZLWK
0] ' & W @)\
,Q)LJVDQGWKHGUDPDWLFUHGXFWLRQRIWKHSXUHVOLSIRUFHVWKHVLGH IRUFH DQG WKH ORQJLWXGLQDO IRUFH UHVSHFWLYHO\ WKDW RFFXUV DV D UHVXOW RI WKH VLPXOWDQHRXVLQWURGXFWLRQRIWKHRWKHUVOLSFRPSRQHQWWKHORQJLWXGLQDOVOLSDQG WKH VLGH VOLS UHVSHFWLYHO\ KDYH EHHQ LQGLFDWHG :H REVHUYH DQ DOPRVW V\PPHWULFVKDSHRIWKHVHLQWHUDFWLRQFXUYHV7KHSHDNRIWKHVLGHIRUFHYHUVXV ORQJLWXGLQDOVOLSFXUYHVDWFRQVWDQWYDOXHVRIWKHVOLSDQJOHDSSHDUVWREHVOLJKWO\ VKLIWHGWRZDUGVWKHEUDNLQJVLGH7KLVSKHQRPHQRQZLOOEHIXUWKHUGLVFXVVHGLQ FRQQHFWLRQZLWKWKHDOWHUQDWLYHUHSUHVHQWDWLRQRIWKHVDPHUHVXOWVDFFRUGLQJWR )LJ$WYHU\ODUJHORQJLWXGLQDOVOLSWKDWLVZKHQ_9V[_9 [ 64WKHVLGHIRUFH DSSURDFKHV]HURDQGWKHVDPHRFFXUVIRUWKHORQJLWXGLQDOIRUFHZKHQWKHODWHUDO VOLS WDQ
JRHV WR LQILQLW\
6
E DW D JLYHQ YDOXH RI WKH ORQJLWXGLQDO VOLS
EHFDXVHREYLRXVO\LQWKDWFDVHZLWK9[6DOVRWKHORQJLWXGLQDOVOLSVSHHGPXVW YDQLVK)RUDORFNHGZKHHOZLWK9V[
)]FRV
9[DQG
ZHKDYH)\
)]VLQ
DQG)[
,QWKHGLDJUDPRI)LJWKHFDOFXODWHGYDULDWLRQVRI)\DQG0]ZLWK)[KDYH EHHQSORWWHGIRUVHYHUDOIL[HGYDOXHVRI
W
αHT α
)\
$OVRWKHFXUYHVIRUFRQVWDQW KDYH
9
& )[
α
σ tan α
)
1
)LJ(TXLYDOHQWVLGHVOLSDQJOHSURGXFLQJWKHVDPHSQHXPDWLFWUDLOW
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
EHHQGHSLFWHG)RUFODULILFDWLRQRIWKHQDWXUHRIWKH)\)[GLDJUDPWKHGHIOHFWLRQ RI DQ HOHPHQW QHDU WKH OHDGLQJ HGJH KDV EHHQ VKRZQ LQ )LJ 6LQFH WKH GLVWDQFHIURPWKHOHDGLQJHGJHKDVEHHQGHILQHGIRUWKLVRFFDVLRQWREHHTXDOWR
H
7KHUDGLXVRIWKHFLUFOH
XQLW\WKHGHIOHFWLRQ RIWKHHOHPHQWHTXDOVWKHVOLS GHQRWLQJ PD[LPXP SRVVLEOH GHIOHFWLRQV LV HTXDO WR
3000
7KH WZR SRLQWV RQ WKH
α =0.1rad
θ=5
0.075 0.05 0.025
[N]
2000
)\ 1000
side force
VO
0 -0.025
-1000
EUDNLQJ
GULYLQJ
-2000 -3000 -0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
longitudinal slip ratio κ [-] )LJ5HGXFWLRQRIVLGHIRUFHGXHWRWKHSUHVHQFHRIORQJLWXGLQDOVOLS
3000
θ =5
κ
GULYLQJ
[N]
2000
longitudinal force
)[
= 0.1 0.075 0.05 0.025
1000 0 -1000 -2000
EUDNLQJ -3000 -25
-20
-15
-10
κ -5
0
5
slip angle α [deg]
10
-0.025 -0.05 -0.075 = -0.1 15
20
)LJ5HGXFWLRQRIORQJLWXGLQDOIRUFHGXHWRWKHSUHVHQFHRIVLGHVOLS
25
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
α
VOPD[
µ)
α =0.1 rad
]
)
µ)
]
δVOG δVOE
9
GULYLQJ
0.025
0.05
0.1
)[
0]
)[
total sliding
0.025
κ=
δ
total sliding )[
VO
0.05
0]
-0.05
VO
σ σ
)\
-0.025
σ σ
-0.1
θ =5
)\
EUDNLQJ
wheel lock
µ)]
α =0.1 rad
0
0.1
0.025
α =0.05 rad )LJ
-0.1µ)D ]
&RUQHULQJIRUFHDQGDOLJQLQJWRUTXHDVIXQFWLRQVRIORQJLWXGLQDOIRUFHDWFRQVWDQW VOLSDQJOH RUORQJLWXGLQDOVOLS
FLUFOHZKHUHDWWKH
FRQVLGHUHGWKHVOLGLQJERXQGDU\
VO
LVDWWDLQHGFRUUHVSRQG
WRWKHSRLQWVRQWKH FXUYHRIWKHIRUFHGLDJUDPRI)LJ7KLVDOVRH[SODLQV WKHVOLJKWO\LQFOLQHGQDWXUHRIWKH FXUYHV$WEUDNLQJ)\DSSHDUVWREHDOLWWOH ODUJHUWKDQDWGULYLQJ7KLVLVDWOHDVWWUXHIRUWKHWZRFDVHVRI)LJRQHDW GULYLQJ DQG WKH RWKHU DW EUDNLQJ VKRZLQJ WKH VDPH VOLS DQJOH DQG WKH VDPH PDJQLWXGH RI WKH GHYLDWLRQ DQJOH
RI WKH VOLS YHORFLW\ YHFWRU DQG WKXV RI WKH
IRUFH YHFWRU ZLWK UHVSHFW WR WKH \ D[LV 7KHQ WKH VOLS VSHHGV 9V DUH HTXDO LQ PDJQLWXGH EXW DW EUDNLQJ WKH VSHHG RI UROOLQJ 9U LV REYLRXVO\ VPDOOHU :KHQ FRQVLGHULQJ WKH GHILQLWLRQ RI WKH WKHRUHWLFDO VOLS
DQG WKH IXQFWLRQDO
UHODWLRQVKLS ZLWKWKHIRUFH)LWEHFRPHVFOHDUWKDW)PXVWEHODUJHUIRUWKH FDVHRIEUDNLQJEHFDXVHRIWKHWKHQODUJHUPDJQLWXGHRI )LQDOO\WKHFDVHRI ZKHHOORFNLVSRLQWHGDW7KHQWKHIRUFHYHFWRU
)
ZKLFKLQPDJQLWXGHLVHTXDO
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
α
VOPD[
begin of contact line
σ
VO
δ
δ δ
VOE
VOG
α leading edge
VO
VO
σ
σ
α
α
wheel lock
α
α
tread element near leading edge
9
1
)LJ 7KHVLWXDWLRQQHDUWKHOHDGLQJHGJH9DULRXVGHIOHFWLRQVRIDQHOHPHQWDWDGLVWDQFH HTXDOWRIURPWKHOHDGLQJHGJHDUHVKRZQFRUUHVSRQGLQJZLWKSRLQWVLQWKHXSSHU GLDJUDPRI)LJ
\
GULYLQJ
EUDNLQJ
6
5 5’ 4 3
,,,,99
)
)
,,
IRUFH
2
YHFWRUV
[
,
1
2
α
,,
3
1
,
9
9 9 U
V
2
,,
3
,,,
,,,
9 9 U
,9
YHFWRUV
9
5’
5 5’ 5
9 U )LJ
V
,9
4
4
YHORFLW\
1
,
0
9
6 6
) ) GLDJUDPH[WHQGHGZLWKWKHFRUUHVSRQGLQJYHORFLW\GLDJUDP7KHEUDNH
7KH
[
\
SHGDO IRUFHLVNHSWFRQVWDQWZKLOHWKHVOLSDQJOHLVLQFUHDVHG7KHVDPHLVGRQHDW DFRQVWDQWGULYLQJHIIRUW,QWKHFDVHRIEUDNLQJWKHVSHHGRIUROOLQJ
9 GHFUHDVHV U
XQWLOWKHZKHHOJHWVORFNHG,QWKHVSHFLDOFDVH WKHZKHHOLVURWDWHGEDFNZDUGVWR
9
NHHSWKHVOLSVSHHG
V
)
DQGWKXVWKHIRUFHYHFWRU
LQWKHRULJLQDOGLUHFWLRQ
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
WR )] LV GLUHFWHG RSSRVLWH WR WKH VSHHG YHFWRU RI WKH ZKHHO FRLQFLGHVZLWKWKHVOLSVSHHGYHFWRU
9
9 ZKLFK WKHQ
V
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
$
VO
/DWHU RQ ZH ZLOO VHH WKDW WKHVH
FRPSXWHGPRPHQWFKDUDFWHULVWLFVPD\DSSUHFLDEO\GHYLDWHIURPH[SHULPHQWDOO\ REWDLQHGFXUYHV $WWKLVVWDJHZHZLOOILUVWDSSO\WKHNQRZOHGJHJDLQHGVRIDUWRWKHDQDO\VLV RI D SUDFWLFDO VLWXDWLRQ WKDW RFFXUV ZLWK D ZKHHO WKDW LV EUDNHG RU GULYHQ DW FRQVWDQWEUDNHSUHVVXUHDQGWKURWWOHUHVSHFWLYHO\ ZKLOHLWVVOLSDQJOHLVYDULHG ,Q)LJWKHIRUFHGLDJUDPLVVKRZQLQFRPELQDWLRQZLWKWKHFRUUHVSRQGLQJ
)
)\ µ)]
[
1
EUDNLQJ
0
3
56
2
sinα
VO
)
)\ 2 1µ)]
1
α
5 3,4,5’
Ω <0 5’
0
6
4
2 1
)[ µ)
1
0
-1
]
)
)\ µ)]
[
1
GULYLQJ
0
,,,,9 ,,,
,9
,,
1,
0
α
)\ 2 µ)]
,,
)
,
VO
sinα
1
)[ µ)
]
1
0
)LJ 7\UH FKDUDFWHULVWLFV DW FRQVWDQW EUDNH SHGDO IRUFH DQGGULYLQJIRUFH1XPHUDOV FRUUHVSRQGWRWKRVHRI)LJ
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
YHORFLW\GLDJUDP$WDJLYHQEUDNLQJIRUFH)[DQGZKHHOVSHHGRIWUDYHO9WKH VOLS DQJOH
LV FKDQJHG IURP ]HUR WR
E 7KH YDULDWLRQ RI VOLS VSHHG 9 DQG V
UROOLQJVSHHG9UPD\EHIROORZHGIURPFDVHZKHUH
WRFDVHZKHUHWRWDO
VOLGLQJVWDUWVDQGIXUWKHUWRFDVHZKHUH9UDQGWKXV
YDQLVKHVDQGWKHZKHHO
EHFRPHVORFNHG$IXUWKHULQFUHDVHRI DWFRQVWDQWEUDNHSHGDOIRUFH DVLQFDVH ZLOO QHFHVVDULO\ OHDG WR D UHGXFWLRQ LQ EUDNLQJ IRUFH )[ XQOHVV WKH ZKHHO LV URWDWHGLQRSSRVLWHGLUHFWLRQ
DVUHSUHVHQWHGE\FDVH ,QWKHFDVHV RI
GULYLQJ LQGLFDWHG E\ 5RPDQ QXPHUDOV WKH GULYLQJ IRUFH )[ FDQ EH PDLQWDLQHG ZLWK*
LUUHVSHFWLYHRIWKHYDOXHRI
7KHQDWXUHRIWKHUHVXOWLQJ)\
*E
FKDUDFWHULVWLFVDWJLYHQGULYLQJRUEUDNLQJ
HIIRUWLVVKRZQLQ)LJ3ORWWLQJRI)\YHUVXVVLQ
LVDGYDQWDJHRXVEHFDXVH
WKHSRUWLRQZKHUHZKHHOORFNRFFXUVLVWKHQUHSUHVHQWHGE\DVWUDLJKWOLQH $QRWKHU LPSRUWDQW DGYDQWDJH RI SXWWLQJ VLQ
LQVWHDG RI WDQ
DORQJ WKH
DEVFLVVDLVWKDWDIWHUKDYLQJFRPSOHWHGWKHGLDJUDPIRUQHJDWLYHYDOXHVRIVLQ UHVXOWLQJLQDQRGGO\V\PPHWULFJUDSK WKHFRPSOHWHUDQJHRI
LVFRYHUHGWKH
9PD\VZLQJDURXQGRYHUWKHZKROHUDQJHRIE$QDSSOLFDWLRQ
VSHHGYHFWRU
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
HVWDEOLVKHGE\WKHEUXVKPRGHODQGWKRVHGXHWRWKHIRUFHV)\DQG)[ZKLFKVKRZ OLQHVRIDFWLRQVKLIWHGZLWKUHVSHFWWRWKHFRQWDFWFHQWUH&RYHUWKHGLVWDQFHVXF DQGYRYFUHVSHFWLYHO\7KHVHOIDOLJQLQJWRUTXHQRZUHDGV 0
]
' '
& & ]
0N
]
0
)
[
Y
%
R
F )
[
% & \
Y
)
F
) X
\
) Y
[ R
F
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
)\
radius 1.1)]
12o o 8 o
6 4
o
)[
)]
1000N
0
-1000N
EUDNLQJ
GULYLQJ
-0.08
-0.04
κ= -0.02
0.02
0.04
0.06
0.08
α =2o
)]
) ) FKDUDFWHULVWLFVIRUW\UHPHDVXUHGRQGU\LQWHUQDOGUXPZLWKGLDPHWHU
)LJ
\
[
RIPIURP+HQNHU
K)\
0]
GULYLQJ
EUDNLQJ
α =8o α =+4 oo α =-4
)[
2000N
α =8o 0
2000N
α =4 o
-100Nm
)LJ
&RPELQHG VOLS VLGH IRUFH DQG PRPHQW YHUVXV ORQJLWXGLQDO IRUFHFKDUDFWHULVWLFV PHDVXUHGDWWZRYDOXHVRIWKHVOLSDQJOHIRUDW\UHRQDGU\IODWVXUIDFHIURP 1RUGHHQDQG&RUWHVH
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
YR YF
0
]
&
XF
α
9 [
[
)
α
0
]
\
\
) )LJ
)
([WHQGHGW\UHEUXVKPRGHOVKRZLQJRIIVHWDQGGHIOHFWLRQRIFDUFDVVOLQHVWUDLJKW DQGSDUDOOHOWRZKHHOSODQH
ZKHUHWKHFRPSOLDQFHFRHIILFLHQWFKDVEHHQLQWURGXFHGWKDWLVGHILQHGE\
F
'
g\ &
&
F\
g[
&
F[
+HUH &F[ DQG &F\ GHQRWH WKH ORQJLWXGLQDO DQG ODWHUDO FDUFDVV VWLIIQHVVHV UHVSHFWLYHO\ DQG
g
[
DQG
g
\
WKH HIIHFWLYH IUDFWLRQV RI WKH DFWXDO GLVSODFHPHQWV
7KHVH IUDFWLRQV PXVW EH FRQVLGHUHG EHFDXVH RI FRQFXUUHQW ODWHUDO DQG ORQJLWXGLQDOUROOLQJRIWKHW\UHORZHUVHFWLRQDJDLQVWWKHURDGVXUIDFHXQGHUWKH DFWLRQ RI D ODWHUDO DQG IRUH DQG DIW IRUFH UHVSHFWLYHO\ ZKLFK ZLOO FKDQJH WKH QRUPDO IRUFH GLVWULEXWLRQ LQ WKH FRQWDFW SDWFK DQG WKHUHE\ UHGXFH WKH DFWXDO GLVSODFHPHQWV RI WKH OLQHV RI DFWLRQ RI ERWK KRUL]RQWDO IRUFHV 7KH UHVXOWLQJ FDOFXODWLRQVFDQEHSHUIRUPHGLQDGLUHFWVWUDLJKWIRUZDUGPDQQHUEHFDXVHWKHVOLS DQJOHRIWKHH[WHQGHGPRGHOLVWKHVDPHDVWKHRQHIRUWKHLQWHUQDOEUXVKPRGHO /DWHURQLQ6HFWLRQWKHHIIHFWRIWKHLQWURGXFWLRQRIDWRUVLRQDODQGEHQGLQJ VWLIIQHVVRIWKHFDUFDVVDQGEHOWZLOOEHGLVFXVVHG7KHUHVXOWLQJPRGHOKRZHYHU LV D ORW PRUH FRPSOH[ DQG FORVHG IRUP VROXWLRQV DUH QR ORQJHU SRVVLEOH ,Q 6HFWKHWHFKQLTXHRIWKHWUHDGHOHPHQWIROORZLQJPHWKRGZLOOEHHPSOR\HG LQWKHWUHDGVLPXODWLRQPRGHOWRGHWHUPLQHWKHUHVSRQVH 7KHFRPELQHGVOLSUHVSRQVHRIWKHVLPSOHH[WHQGHGPRGHORI)LJLVJLYHQ LQ WKH )LJV DQG ,W LV REVHUYHG WKDW LQ )LJ WKH DOLJQLQJ WRUTXH FKDQJHVLWVVLJQLQWKHEUDNLQJUDQJH7KLVLVGXHWRWKHWHUPLQ(T ZLWK WKHFRPSOLDQFHFRHIILFLHQW F7KHUHVXOWLQJTXDOLWDWLYHVKDSHLVTXLWHVLPLODUWR WKHH[SHULPHQWDOO\IRXQGFXUYHVRI)LJ,Q)LJWKHHIIHFWRIDQLQLWLDO RIIVHWRIYR
PPKDVEHHQGHSLFWHG
+HUH FDUFDVV FRPSOLDQFH KDV EHHQ GLVUHJDUGHG DQG RQO\ WKH ODVW WHUP RI KDVEHHQDGGHG:HVHHWKDWDPRPHQWLVJHQHUDWHGDOUHDG\DW]HURVOLS
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
DQJOH7KHFXUYHVIRXQGLQ)LJIRUQRQ]HURVLGHVOLSDUHWKHQVLPSO\DGGHG WRWKHLQFOLQHGVWUDLJKWOLQHEHORQJLQJWR
7KHW\SHRIFXUYHVWKDWUHVXOWDUH
RIWHQIRXQGH[SHULPHQWDOO\7KHHIIHFWRIODWHUDOFRPSOLDQFHPD\WKHQEHYHU\
0]
θ =5
µ)]D
0.1
F = 2.5 x e-6m/N Yo=0
-0.025
)[
µ)]
1
0
0
-1
α= 0.1rad
0.075 0.05
GULYLQJ
0.025
EUDNLQJ
-0.1
)LJ 7KHLQIOXHQFH RIODWHUDOFDUFDVVFRPSOLDQFHRQWKHDOLJQLQJWRUTXHIRUDEUDNHGRU GULYHQZKHHODFFRUGLQJWRWKHH[WHQGHGEUXVKPRGHO
θ =5
0] µ)]D
0.1
F=0 Yo=0.005m
-0.025
)[
µ)]
0
1
-1
0
GULYLQJ
α= 0.1rad
0.075
0.025
0.05
EUDNLQJ
-0.1
Y
)LJ 7KHLQIOXHQFHRIDQLQLWLDOODWHUDORIIVHW
R
) RQWKHDOLJQLQJWRUTXHIRUDEUDNHG
RI
[
RUGULYHQZKHHODFFRUGLQJWRWKHH[WHQGHGEUXVKPRGHO
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
VPDOORUFDQFHOOHGRXWE\WKHHIIHFWRIWKHIRUHDQGDIWFRPSOLDQFHRIWKHFDUFDVV VHFRQGWHUPRIULJKWKDQGPHPEHURI
([HUFLVH&KDUDFWHULVWLFVRIWKHEUXVKPRGHO &RQVLGHU WKH EUXVK W\UH PRGHO DV WUHDWHG LQ 6HFWLRQ (ODVWLF DQG IULFWLRQDO SURSHUWLHVDUHWKHVDPHLQDOOKRUL]RQWDOGLUHFWLRQVDQG(T KROGV )RU
# #ò
) )
\
\
WKHVLGHIRUFHFKDUDFWHULVWLFLVGHVFULEHGE\DFFRUGLQJWR(T
) )
]
WDQ
WDQ
) &
]
)
WDQ
]
&DOFXODWHWKHYDOXHRI DQGWDQ
VO
# $
VO
VO
IRU
1
1UDG
6NHWFK WKH
)
\
WDQ
FKDUDFWHULVWLF DQG DOVR WKH
0
]
WDQ
DQG
W
WDQ
FXUYHV
DFFRUGLQJWR(TV IRU
D
P
5HSODFHLQWKH
)
\
WDQ
GLDJUDPWKHRUGLQDWH
)
\
E\
)
DQGWKHDEVFLVVDWDQ
E\
WKHUHE\DVVHVVLQJWKHWRWDOIRUFHYVVOLSGLDJUDP&DOFXODWHWKHVOLSYDOXHV DQG XVLQJHTXDWLRQV DQG IRURQHYDOXHRIWDQ RIYDOXHVRI
LQDVXLWDEOHUDQJHHJIURP WR
)IRUHDFKRIWKH
YHFWRU
)
WKHIULFWLRQFLUFOHZLWKUDGLXV WZRSRLQWVZKHUH
VO
) )
YDOXHVDQGVNHWFKWKH
WDQ
PD[ VO
)
\
[
W
'UDZ
LQZKLFKWKHFXUYHZLOODSSHDU1RWHWKH ]
ZKHUHWKHFXUYHWRXFKHVWKHFLUFOH,QGLFDWHWKH
0¶ W)
]
F[
WKHUHE\ HVWDEOLVKLQJ WKH
IRUWKHVDPHVHULHVRI YDOXHVDQGWDQ \
0
1RZXVH(T DQGFDOFXODWHWKHWRUTXH
& 64
W
GLDJUDP WKH DEVFLVVD E\
GLDJUDP'HWHUPLQHWKHYDOXHVRI
1P
\
'HWHUPLQH WKH IRUFH
FXUYHIRUWDQ [
SRLQWZKHUHZKHHOORFNRFFXUV 5HSODFH LQ WKH WDQ
DQGDQXPEHU
]
&
IRUDODWHUDOFDUFDVVVWLIIQHVV
DQGGLVUHJDUGWKHFRUUHFWLRQIDFWRU
g
\
F\
'UDZIRUWKHFDVHVPHQWLRQHGLQTXHVWLRQZKHUHWKH FXUYHWRXFKHVWKHIULFWLRQ FLUFOH
VO
WKHIRUFHDQGYHORFLW\GLDJUDPDFFRUGLQJWR)LJ'RWKHVDPH
IRUWKHFDVHRIZKHHOORFN 6NHWFKWKH
)
VLQ
\
) )
FKDUDFWHULVWLFV)LJ IRU
EUDNHSHGDOIRUFHFRUUHVSRQGLQJWRWKHYDOXHRI WRXFKHVWKHIULFWLRQFLUFOH
VO
[
DQGDOVRIRUWKDWFRQVWDQW
ZKHUHWKHWDQ [
FXUYH
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
&DPEHUDQG7XUQLQJ6SLQ
)RUWKHVWXG\RIKRUL]RQWDOFRUQHULQJRQHVKRXOGQRWRQO\FRQVLGHUVLGHVOLSEXW DOVR WKH LQIOXHQFH RI WZR RWKHU HIIHFWV ZKLFK LQ PRVW FDVHV H[FHSW IRU WKH PRWRUF\FOH DUHRIPXFKOHVVLPSRUWDQFHWKDQVLGHVOLS7KHLQWURGXFWLRQRIWKHVH WZRLQSXWYDULDEOHVZKLFKFRPSOHWHVWKHGHVFULSWLRQRIWKHRXWRISODQHW\UHIRUFH DQGPRPHQWJHQHUDWLRQDUHILUVWO\WKHZKHHOFDPEHURUWLOWDQJOH EHWZHHQWKH ZKHHOSODQHDQGWKHQRUPDOWRWKHURDGFI)LJ DQGVHFRQGO\WKHWXUQVOLS
0 9[ %RWK DUH FRPSRQHQWV RI WKH WRWDO VSLQ n )LUVW ZH ZLOO DQDO\VH WKH VLWXDWLRQLQWKHDEVHQFHRIODWHUDODQGORQJLWXQDOZKHHOVOLS
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
γ
wheel spin axis
contact area of imaginary ball
Ω
imaginary ball
ω]F
9
ω]W
. ψ (=ω ]
[ path
\
]W )
)LJ
2D
wheel plane
5
&
Y
X
[
γ
\
vertical projection of horizontally undeformed peripheral line
:KHHOUROOLQJDWDFDPEHUDQJOHZKLOHWXUQLQJDORQJDFLUFXODUSDWKZLWKRXWVLGHVOLS $WWKHULJKWDWRSYLHZRIWKHSHULSKHUDOOLQHRIWKHQRQUROOLQJZKHHOFRQVLGHUHGDV EHLQJDSDUWRIDQLPDJLQDU\EDOOSUHVVHGDJDLQVWDIODWVXUIDFH
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
LQWHUVHFWLRQRIZKHHOSODQHDQGEDOORXWHUVXUIDFHIRUPVWKHSHULSKHUDOOLQHRIWKH W\UH:KHQORDGHGYHUWLFDOO\WKHEDOODQGFRQVHTXHQWO\WKHSHULSKHUDOOLQHDUH DVVXPHGWRVKRZQRKRUL]RQWDOGHIRUPDWLRQVZKLFKLQUHDOLW\ZLOODSSUR[LPDWHO\ EHWKHFDVHIRUDKRPRJHQHRXVEDOOVKRZLQJDUHODWLYHO\VPDOOFRQWDFWDUHD :HDSSO\WKHWKHRU\RIDUROOLQJDQGVOLSSLQJERG\DQGFRQVLGHUHTXDWLRQV DQGUHVWULFWRXUVHOYHVWRWKHFDVHRIVWHDG\VWDWHSXUHVSLQWKDWLV ZLWK
7KHQZLWKWXUQVOLSYHORFLW\
9V[' 9V\' 9U' 9 F
]' 0
&
0 LQ)LJZHILQG
]W
VLQ
' 9F
5
&
UH
VLQ
)XUWKHUPRUHWKHGLIIHUHQFHEHWZHHQ[\ DQG[R\R ZLOOEHQHJOHFWHGDQGW\UH FRQLFLW\DQGSO\VWHHUGLVUHJDUGHG7KHFRUUHFWLRQIDFWRUV DWWULEXWHGWRFDPEHU PD\EHDSSUR[LPDWHGE\
\
[ '
&g \ [
[\
&g [
\ '
UH
g
VLQ
ZLWK ERWK UHGXFWLRQ IDFWRUV HTXDO RU FORVH WR ]HUR IRU D UDLOZD\ ZKHHO RU D PRWRUF\FOHW\UHDQGH[SHFWHGWREHFORVHUWRXQLW\IRUDVWHHOEHOWHGFDURUWUXFN W\UH FI GLVFXVVLRQ EHORZ (T :LWK ERWK IDFWRUV DVVXPHG HTXDO DQG GHQRWHGDV
n'
g ZHPD\GHILQHDWRWDODFWXDOVSLQ
&
9F
^0
& & g
VLQ `
'&
5
%
&g
UH
VLQ
7KHODVWWHUPUHSUHVHQWVWKHFXUYDWXUH5 RIWKHW\UHSHULSKHUDOOLQHWRXFKLQJ D IULFWLRQOHVV VXUIDFH DW D FDPEHUHG SRVLWLRQ :KHQ GLVUHJDUGLQJ D SRVVLEOH XQLIRUPRIIVHWRIWKLVOLQHZHREWDLQZKHQLQWHJUDWLQJ E\XVLQJ DQG DSSUR[LPDWLQJ E\WDNLQJ [ 0
\R ' \ R ' &
&g
U
H
R
D &[
' [0 ' UH
VLQ
ZKLFKUHGXFHVH[SUHVVLRQV IRUWKHVOLGLQJYHORFLWLHVZLWK WR
9 J[ ' &
MX M[
& \ n 9F
9J\ ' &
MY M[
% [ n 9F
,Q WKH UDQJH RI DGKHVLRQ ZKHUH WKH VOLGLQJ YHORFLWLHV YDQLVK WKH GHIOHFWLRQ JUDGLHQWVEHFRPH
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
X ' \n G[
G
Y ' &[n G[ G
,QWHJUDWLRQ\LHOGVWKHIROORZLQJH[SUHVVLRQVIRUWKHKRUL]RQWDOGHIRUPDWLRQVLQWKH FRQWDFWDUHD
X ' \[ n % & Y ' &ò [ n % &
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
[' D Y' X'
:LWKWKHXVHRI WKHIRUPXODHIRUWKHGHIRUPDWLRQVLQWKHDGKHVLRQ]RQH VWDUWLQJDWWKHOHDGLQJHGJHUHDG
X ' & \ D & [ n Y ' ò D & [ n
GHQRWLQJWKHVWLIIQHVVRIWKHWUHDGUXEEHUSHUXQLW $IWHULQWURGXFLQJFS[
DQGFS\ DUHD LQ [ DQG \ GLUHFWLRQ UHVSHFWLYHO\ DQG DVVXPLQJ VPDOO VSLQ DQG KHQFH YDQLVKLQJVOLGLQJZHFDQFDOFXODWHWKHODWHUDOIRUFHDQGWKHPRPHQWDERXWWKH YHUWLFDOD[LVE\LQWHJUDWLRQRYHUWKHFRQWDFWDUHD:HREWDLQ
)\ ' FNS\ D E n ' &)n n
0] ' FNS[ D E
n'
&0n n
RULQWHUPVRISDWKFXUYDWXUHLI
)\ ' &&)n 0 ] ' &&0n
5
5
&
&
/ RUFRQVWDQW DQGFDPEHUDQJOH
&g
' &)n & % &)
&g
& % &
UH
UH
5
' &0n
5
VPDOO
0
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
WXUQVOLSFDPEHU
5γ
5
ϕ
tyre peripheral line
9F
contact line 0] \
[
)\
5
)LJ7RSYLHZRIFDPEHUHGW\UHPRGHOUROOLQJLQDFXUYHZLWKUDGLXV
,QFDVHRISXUHWXUQLQJWKHIRUFHDFWLQJRQWKHW\UHLVGLUHFWHGDZD\IURPWKH SDWKFHQWUHDQGWKHPRPHQWDFWVRSSRVLWHWRWKHVHQVHRIWXUQLQJ&RQVHTXHQWO\ ERWKWKHIRUFHDQGWKHPRPHQWWU\WRUHGXFHWKHFXUYDWXUH_5_,QFDVHRISXUH FDPEHUWKHIRUFHRQWKHZKHHOLVGLUHFWHGWRZDUGVWKHSRLQWRILQWHUVHFWLRQRIWKH ZKHHOD[LVDQGWKHURDGSODQHZKLOHWKHPRPHQWWULHVWRWXUQWKHUROOLQJZKHHO WRZDUGV WKLV SRLQW RI LQWHUVHFWLRQ 1R UHVXOWLQJ IRUFH RU WRUTXH LV H[SHFWHG WR RFFXUZKHQ
g VLQ
g
UH5)RUWKHVSHFLDOFDVHWKDW
WKLVZLOORFFXUZKHQ
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
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
6 47KH]RQHVRIVOLGLQJJURZZLWKLQFUHDVLQJVSLQDQGZLOOWKHUHE\FDXVH
WKDW
n7KHFDVHRIILQLWHFRQWDFW
DOHVVWKDQSURSRUWLRQDOYDULDWLRQRI)\DQG0]ZLWK
ZLGWKLVWRRGLIILFXOWWRKDQGOHE\DVLPSOHDQDO\VLV,WZLOOEHGHDOWZLWKODWHURQ LQ6HFZKHQWKHWUHDGHOHPHQWIROORZLQJVLPXODWLRQPHWKRGLVLQWURGXFHG )RUQRZZHDVVXPHDWKLQW\UHPRGHOZLWKE
,IDVEHIRUHDSDUDEROLF
SUHVVXUH GLVWULEXWLRQ LV DVVXPHG ZLWK D VLPLODU YDULDWLRQ RI WKH PD[LPXP SRVVLEOH ODWHUDO GHIOHFWLRQ YPD[ LW LV REYLRXV IURP (T VKRZLQJ WKDW WKH ODWHUDOGHIOHFWLRQLVDOVRDSSUR[LPDWHO\ SDUDEROLFWKDWQRVOLGLQJZLOORFFXUXS WR D FHUWDLQ FULWLFDO YDOXH RI VSLQ
n
VO
ZKHUH WKH DGKHVLRQ OLPLW LV UHDFKHG
nDQG0
WKURXJKRXWWKHFRQWDFWOHQJWK8SWRWKLVSRLQW)\YDULHVOLQHDUO\ZLWK
]
UHPDLQVHTXDOWR]HUR $FFRUGLQJWR(T ZLWK
g DVVXPHGWRWDNH D YDOXHWKDWLVPLQLPDOO\
HTXDO WR ]HUR VSLQ GXH WR FDPEHU WKHRUHWLFDOO\ FDQQRW H[FHHG WKH YDOXH UH &RQVHTXHQWO\DWODUJHUYDOXHVRIVSLQWXUQVOLSPXVWEHLQYROYHG%H\RQGWKH FULWLFDOYDOXH
n
VO
WKHVLWXDWLRQEHFRPHVTXLWHFRPSOH[7KHGLVFXVVLRQPD\EH
VLPSOLILHGE\FRQVLGHULQJDWXUQWDEOHRQWRSRIZKLFKWKHZKHHOUROOVZLWKLWV VSLQ D[LV IL[HG 7KH FRQGLWLRQ RI DGKHVLRQ LV VDWLVILHG ZKHQ WKH GHIOHFWLRQV UHPDLQ ZLWKLQ WKH ERXQGDULHV JLYHQ E\ WKH SDUDEROD¶V Y PD[ DV LQGLFDWHG LQ )LJD ,Q WKH VDPH ILJXUH WKH FRUUHVSRQGLQJ VLWXDWLRQ DW FDPEHU KDV EHHQ LQGLFDWHGDWWKHVDPHYDOXHRIVSLQZLWKFXUYDWXUH5 RIWKHGHIOHFWHGSHULSKHUDO OLQH\
R
(T RQD
VXUIDFHHTXDOWRWKHSDWKFXUYDWXUH5
6OLGLQJ RFFXUV DV VRRQ DV WKH SRLQWV RI WKH WDEOH VXUIDFH FDQ FURVVWKH
DGKHVLRQERXQGDU\YPD[7KLVRFFXUVVLPXOWDQHRXVO\IRUDOOWKHSRLQWVLQWKH IURQWKDOIRIWKHFRQWDFWOLQHZKHQWKHSDWKFXUYDWXUHH[FHHGVWKHFXUYDWXUHRI WKH DGKHVLRQ ERXQGDU\ 2QFH WKH FRQWDFW SRLQW DUULYHV LQ WKH UHDU KDOI RI WKH FRQWDFW]RQHDW[
WKHSRLQWFDQPDLQWDLQDGKHVLRQEHFDXVHQRZWKHSRLQW
RQ WKH WDEOH PRYHV WRZDUGV WKH LQVLGH RI WKH DGKHVLRQ ERXQGDULHV 7KH SRLQW [VFI)LJEF
IROORZVDFLUFOHXQWLOWKHRSSRVLWHERXQGDU\LVUHDFKHGDW[
ZKHUHWKHGHIRUPDWLRQYLVRSSRVLWHLQVLJQDQGUHDFKHVLWVPD[LPXPYDOXHYPD[
n
DIWHUZKLFKVOLGLQJRFFXUVDJDLQ:LWKLQFUHDVLQJWXUQVOLS
5 WKLVODWWHU
VOLGLQJ]RQHJURZV$WWKHVDPHWLPHWKHVLGHIRUFH)\GHFUHDVHVDQGWKHWRUTXH
n!n
0]WKDWDULVHVIRU
VO
LQFUHDVHVXQWLOWKHVLWXDWLRQLVUHDFKHGZKHUH5DQG)\
DSSURDFK]HURDQG0]DWWDLQVLWVPD[LPXPYDOXHW\UHVWDQGLQJVWLOODQGURWDWLQJ DERXWLWVYHUWLFDOD[LV ,Q)LJFWKHUDGLXV5KDVEHHQFRQVLGHUHGODUJHZLWK UHVSHFW WR KDOI WKH FRQWDFW OHQJWK D 7KHQ WKH FLUFOH VHJPHQWV PD\ EH DSSUR[LPDWHGE\SDUDEROD¶VZKLFKPDNHVWKHDQDO\VLVDORWVLPSOHU,QWKHORZHU SDUW RI WKH ILJXUH D FDPEHU HTXLYDOHQW JUDSK KDV EHHQ GHSLFWHG 7KH FXUYHG FRQWDFWOLQHLQWKHDGKHVLRQUDQJHLVWKHQFRQYHUWHGWRDVWUDLJKWKRUL]RQWDO OLQH 7KHJUDSKLVREWDLQHGIURPWKHWXUQVOLSJUDSKE\VXEWUDFWLQJIURPDOOWKHFXUYHV WKHGHIOHFWLRQYn4WKDWZRXOGRFFXULIIXOODGKHVLRQFDQEHPDLQWDLQHGHJDW
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
6 4 :HXVHWKLVJUDSKDVWKHEDVLVIRUWKHFDOFXODWLRQRIWKHIRUFHDQGPRPHQW
UHVSRQVHWRVSLQ)LUVWIRUWKHFDVHRISXUHVSLQDQGWKHQIRUWKHFDVHRIFRPELQHG VSLQDQGVLGHVOLS
7KHSDUDPHWHU
\
GHILQHGE\(T LVXVHGDJDLQDQGWKHPD[LPXPSRVVLEOH
ODWHUDOGHIOHFWLRQUHDGVDFFRUGLQJWR
YPD[ ' D & [ D \
7KHGHIOHFWLRQDWDVVXPHGIXOODGKHVLRQZRXOGEHFRPH
Yn 4 '
ò
n D & [
(TXDWLQJWKHGHIOHFWLRQDWIXOOVOLGLQJWRWKHRQHDWIXOODGKHVLRQ\LHOGVWKHVSLQ DWWKHYHUJHRIVOLGLQJ
nVO '
VJQ
D
n
\
7KHORFDWLRQVRIWKHWUDQVLWLRQSRLQWVLQGLFDWHGLQ)LJFDUHIRXQGWREHJLYHQ E\WKHFRRUGLQDWHV
&D
[V' [ V'
D \ _n _ %
ò
\ V'\V'&
D \ _n _ & \
D VJQn
7KHGHIOHFWLRQVEHFRPHLQWKHILUVWDQGVHFRQGVOLGLQJUHJLRQVDQGLQWKHDGKHVLRQ UHJLRQUHVSHFWLYHO\
Y ' &Y ' YPD[ VJQ n
Y ' \ V % Yn 4
,QWHJUDWLRQRYHUWKHFRQWDFWOHQJWK\LHOGVIRUWKHVLGHIRUFH
) \ ' )]
n D \ _n _ % VJQ
DQGIRUWKHPRPHQW
0 ] ' )] D
n n
$W
VO
D \ _n _ & VJQ n D \ _n _ %
WKHIRUFHUHGXFHVWR)]VJQ
nDQGWKHPRPHQWWR]HUR7KHVDPHFDQ
EH REWDLQHG IURP WKH H[SUHVVLRQV KROGLQJ IRU WKH FDVH RI IXOO DGKHVLRQ ZKHQFS\
ELVUHSODFHGE\FS\DQGWKHZLGWKELVWDNHQHTXDOWR]HUR
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
WXUQVOLSJUDSK
ϕ
5
wheel plane
9F
contact line
[ Yϕ
Y
Y PD[
\ FDPEHUJUDSK
YPD[
Y
peripheral line on frictionless surface
\γο
contact line
\
5γ
[
9F
D
’large’WXUQVOLS
ϕ
YPD[
wheel plane
5
[
contact line
YPD[
4 (as if µ 64)
Y
Y ϕ 0]
(contact line if
)\
µ 64) E
\
’large’WXUQVOLS wheel plane
(|5| >>D)
ϕ
YPD[
[
contact line
YPD[
\
upward shifted Yϕ 4
γ
4 (as if µ 64)
Y ϕ Y
0]
\ PD[/
HTXLYDOHQWJUDSK
Y
\γο
contact line
(
\ PD[5
\
V
[V
)\
=0
[ V
\
\γο Y PD[
ϕ4 )
Y
γο
PD[
\ Y
[
F
)LJ 7KHW\UHEUXVKPRGHOZLWK]HURZLGWKUROOLQJZKLOHWXUQLQJRUDWDFDPEHUDQJOHD DWIXOODGKHVLRQ7XUQLQJDWODUJHVSLQVKRZLQJVOLGLQJDWWKHIURQWKDOIDQGDWWKH UHDUHQGEF ZLWKSDUDEROLFDSSUR[LPDWLRQRIWKHFLUFXODUSDWKLQF
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
0 ] )\ Dµ)] µ)] 0.33
θ\ = 5 asymptote
)\
1
0] 0
0
1 θ\
1
Dϕ
D5
)LJ)RUFHDQGPRPHQWYHUVXVQRQGLPHQVLRQDOVSLQIRUVLQJOHURZEUXVKPRGHO
6WKHIRUFH) YDQLVKHV
)RUVSLQDSSURDFKLQJLQILQLW\WKDWLVZKHQWKHUDGLXV5
\
ZKLOHWKHPRPHQWUHDFKHVLWVPD[LPXPYDOXH
)\ '
0] ' 0] PD[ '
)] D VJQn
,Q)LJWKHSXUHVSLQFKDUDFWHULVWLFVKDYHEHHQSUHVHQWHGDFFRUGLQJWRWKH DERYHH[SUHVVLRQV,WLVRILQWHUHVWWRQRWHWKDWDOWKRXJKWKHSDUDERODGRHVQRW UHVHPEOHWKHFLUFXODUSDWKVRZHOODWVPDOOHUUDGLLWKHUHVXOWLQJUHVSRQVHVHHPV WREHDFFHSWDEOHDWOHDVWLIWKHGHIOHFWLRQVUHPDLQVXIILFLHQWO\VPDOO
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
R
OLQH
7KH VKDGHG DUHD FRUUHVSRQGV WR WKH GHIRUPDWLRQ RI WKH EUXVK PRGHO ZKHQ VXEMHFWHG VROHO\ WR WKH VOLS DQJOH ZLWK WKH PD[LPXP SRVVLEOH GHIOHFWLRQ YPD[ UHSODFHGE\LQWKLVFDVHZLWKQHJDWLYHVOLSDQJOH \PD[/7KLVZRXOGFRUUHVSRQG WR WKH LQWURGXFWLRQ RI DQ DGDSWHG IULFWLRQ FRHIILFLHQW RU SDUDPHWHU
\
$SSDUHQWO\WKHDFWXDOGHIOHFWLRQRIWKHHOHPHQWVDWFDPEHUDQGVLGHVOLSLVWKHQ REWDLQHGE\DGGLQJWKHGHIOHFWLRQYn4WKDWZRXOGRFFXUZKHQWKHPRGHOZRXOGEH VXEMHFWHGWRVSLQRQO\DQGIXOODGKHVLRQLVDVVXPHG7KHDGDSWHGSDUDPHWHU
\
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
’small’WXUQVOLS ZLWKVOLSDQJOH
ϕ
4
Yϕ
YPD[
α
wheel plane
α
[
contact line
YPD[
9F \
γ
HTXLYDOHQWJUDSK
\ PD[/
[VR
\γR
contact line
α
(
α
\ PD[5
\
ϕ4 )
Y
[ 9F
)LJ 7KHEUXVKPRGHOWXUQLQJDWµVPDOO¶VSLQRUVXEMHFWHGWRFDPEHUDWWKHVDPHOHYHO RIVSLQZKLOHUXQQLQJDWDVOLSDQJOH
WXUQVRXWWRUHDG
(
\ '
\
& Dn
\
\'
ZLWK
VJQ
FS\ D
) ]
ZLWKWKHFRQGLWLRQIRUWKHVSLQWREHµVPDOO¶
n _ nO '
_
D
\
7KH VLGH IRUFH EHFRPHV VLPLODU WR (T ZLWK WKH FDPEHU IRUFH DW IXOO DGKHVLRQDGGHG
*
*
LI WDQ
_
\
#
_
)\ ' )]
*
DQGLI WDQ
*
_
\VO
\ \ \
_!
\
&
(
\ \
%
(
\ \
% F S\D n
\VO
)\ ' )] VJQ )RUWKHPRPHQWZHREWDLQ
D
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
*
*
LI WDQ
_
\
#
_
0 ] ' & )] D
*
*
DQGLI WDQ
0] '
_
\
\VO
\ \
_!
\
(
&
\ \
(
%
\ \
(
&
\ \
\VO
D
:HPD\LQWURGXFHDSQHXPDWLFWUDLOW WKDWPXOWLSOLHGZLWKWKHIRUFH)\ GXHWR WKHVOLSDQJOHZLWKWKHODVWWHUPRI RPLWWHG SURGXFHVWKHPRPHQW0]
*
LI WDQ
*
_
\
#
_
\VO
\
( 0 & % ( W '& ' D ( ) & % ]
\
\
\
\
\
(
&
(
\
DQGHOVHW
\
\
\
\
\
7KHJUDSKRI)LJFODULILHVWKHFRQILJXUDWLRQRIWKHYDULRXVFXUYHVDQG WKHLU PXWXDO UHODWLRQVKLS )RU GLIIHUHQW YDOXHV RI WKH FDPEHU DQJOH
WKH
FKDUDFWHULVWLFV IRU WKH IRUFH DQG WKH PRPHQW YHUVXV WKH VOLS DQJOH KDYH EHHQ FDOFXODWHG ZLWK WKH DERYH HTXDWLRQV DQG SUHVHQWHG LQ )LJ 7KH FRUUHVSRQGLQJ *RXJK SORW KDV EHHQ GHSLFWHG LQ )LJ 7KH UHODWLRQVKLS EHWZHHQ DQGWKHVSLQ
nIROORZVIURP(T
7KH FXUYHV HVWDEOLVKHG VKRZ JRRG TXDOLWDWLYH DJUHHPHQW ZLWKPHDVXUHG
0
)\ Wα , ] D Dµ)] µ)]
θ\ = 5 (1-ε γ ) D = 1 UH 3
α
0.2
W 0
]
1
W )
α \α
)\ )
\α
)\γ
0
1/θ \
1/θ\*-0.2
1/θ\*+ -1
tanα
γ = 10
)LJ%DVLFFRQILJXUDWLRQRIWKHFKDUDFWHULVWLFVYVVLGHVOLSDWFDPEHUDQJOH
O
GHJ
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
)\ µ)]
θ\ = 5
(1-εγ ) D = 1 UH 3
20
10
γ
0
[deg]
1 -10
0] -20
-20
0
-10
α
20 10
)\
0 -10 -20
-1 -0.2
0
γ
20 [deg]
[deg]
]
Dµ)] )LJ7KHFDOFXODWHGVLGHIRUFHDQGPRPHQWFKDUDFWHULVWLFVDWYDULRXVFDPEHUDQJOHV
FKDUDFWHULVWLFV6RPHGHWDLOVLQWKHLUIHDWXUHVPD\EHGLIIHUHQWZLWKUHVSHFWWR H[SHULPHQWDO HYLGHQFH ,Q WKH QH[W VHFWLRQ ZKHUH WKH VLPXODWLRQ PRGHO LV LQWURGXFHGWKHHIIHFWRIYDULRXVRWKHUSDUDPHWHUVOLNHWKHZLGWKRIWKHFRQWDFW SDWFKDQGWKHSRVVLEO\FDPEHUGHSHQGHQWDYHUDJHIULFWLRQFRHIILFLHQWRQWKHSHDN VLGHIRUFHZLOOEHGLVFXVVHG 7KHQH[WLWHPWREHDGGUHVVHGLVWKHUHVSRQVHWRODUJHVSLQLQWKHSUHVHQFHRI VLGH VOLS )LJXUHV DE UHIHU WR WKLV VLWXDWLRQ /DUJH VSLQ ZLWK WZR VOLGLQJ UDQJHVRFFXUVZKHQWKHIROORZLQJWZRFRQGLWLRQVDUHIXOILOOHG
θ\ = 5
(1-εγ ) D UH
)\
µ)]
1 =3
1
α 20
-1
10
-2 0 -3
0
-4
-10
-5 -6
-20
3 2 1
0
[deg]
6 5 4
γ
[deg]
-1 )LJ7KHFRUUHVSRQGLQJ*RXJKSORW
0.2 0]
Dµ)]
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
n_ $ n '
_
D
O
DQG WDQ_
_
# D _n _ &
\
!
\
,IWKHVHFRQGFRQGLWLRQLVQRWVDWLVILHGZHKDYHDUHODWLYHO\ODUJHVOLSDQJOHDQG WKHHTXDWLRQV KROGDJDLQ7KLVVLWXDWLRQLVLOOXVWUDWHGLQ)LJF )RUWKHGHYHORSPHQWRIWKHHTXDWLRQVIRUWKHGHIOHFWLRQVZHUHIHUWR)LJD ZLWKWKHFDPEHUHTXLYDOHQWJUDSK)LUVWWKHGLVWDQFHV\ZLOOEHHVWDEOLVKHGDQG WKHQWKHGHIOHFWLRQVYn4ZLOOEHDGGHGWRREWDLQWKHDFWXDOGHIOHFWLRQVY$GKHVLRQ RFFXUVLQEHWZHHQWKHWZRVOLGLQJUDQJHV7KHVWUDLJKWOLQHUXQVSDUDOOHOWRWKH VSHHGYHFWRUDQGWRXFKHVWKHERXQGDU\\PD[5 7KHWDQJHQWSRLQWIRUPVWKHILUVW WUDQVLWLRQ SRLQW IURP VOLGLQJ WR DGKHVLRQ 0RUH WR WKH UHDU WKH VWUDLJKW OLQH LQWHUVHFWVWKHRWKHUERXQGDU\\PD[/:LWKWKHIROORZLQJWZRTXDQWLWLHVLQWURGXFHG
$ '
D _n _ &
$ '
D _n _ %
\
\
ZHGHULYHIRUWKH[FRRUGLQDWHVRIWKHWUDQVLWLRQSRLQWV
[ ' D WDQ V
DQG
n
VJQ
$
[ ' & T % VJQn T % SU S
V
ZLWK
S'
$ D
n
VJQ
T'
WDQ
U ' D$
n%\ %[
VJQ
V
WDQ
V
7KHGLVWDQFHV\LQWKHILUVWVOLGLQJUHJLRQ[V[D UHDG
\ ' &$ D &[ D
n
VJQ
LQWKHDGKHVLRQUHJLRQ[V[[V
\ ' \ % [ & [ WDQ
V
V
ZLWK
\ ' \ [ V
V
DQGLQVHFRQGVOLGLQJUDQJHD[[V
\ ' &$ D & [ D
n
VJQ
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
’large’WXUQVOLS ZLWKVOLSDQJOH
wheel plane
α
contact line
α
YPD[
upward shifted Y ϕ4
γ
ϕ
YPD[
Y
HTXLYDOHQWJUDSK
\ PD[/
Y
\γο
contact line
α ϕ
range for to be ’large’
α
\ V
\ V [V
\
(
ϕ4 )
Y
\ PD[5
α
[ V
’large’VSLQ
γ
9F
Yϕ4
\
[
[ 9F
D
α <0
HTXLYDOHQWJUDSK
\ PD[/
α ϕ
range for to be ’large’
\γο
α
[ \ PD[5
α
\
9F
\PD[/ α ϕ
9F
range for to be ’large’
α
α \ PD[5
\
’small’ VSLQ
γ
[ V
HTXLYDOHQWJUDSK
α
α >0
E
α <0
α ϕ
[
\ PD[/ \γο
range for to be ’large’ \ PD[5
[
α
\
9F
\PD[/ α ϕ
α
range for to be ’large’
9F
[
α
\ PD[5 \
α >0 F
)LJ
7KHPRGHOUXQQLQJDWODUJHVSLQWXUQLQJDQGHTXLYDOHQWFDPEHU DWDUHODWLYHO\VPDOO DE RUODUJHSRVLWLYHRUQHJDWLYHVOLSDQJOHF
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
,QWHJUDWLRQRYHUWKHFRQWDFWOHQJWKDIWHUDGGLWLRQRIYn4DQGPXOWLSOLFDWLRQZLWK WKHVWLIIQHVVSHU XQLWOHQJWKFS\JLYHVWKHVLGHIRUFHDQGDIWHUILUVWPXOWLSO\LQJ ZLWK[WKHDOLJQLQJWRUTXH:HREWDLQWKHIRUPXODH
) \ ' FS VJQn $ D [V & [V & $ D [V & [V % D % FS \V % [V WDQ [V & [V & WDQ [V & [V
0 ] ' & FS VJQn $ D & D [V % [V & $ D & D [V % [V % D % F S \V % [V WDQ [V & [V & WDQ [V & [V
7KH UHVXOWLQJ FKDUDFWHULVWLFV KDYH EHHQ SUHVHQWHG LQ )LJV DQG 7KH JUDSKVIRUPDQH[WHQVLRQRIWKHGLDJUDPRI)LJZKHUHWKHOHYHORIFDPEHU FRUUHVSRQGWRµVPDOO¶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
EURZDQG\5
EURZIURPWKHZKHHO
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
)\ µ)]
θ\ = 5
1
Dϕ 0.8 0.7 0.6
0
-10
α
10
0.5 0.4
0.3
0.2
0.1 0
[deg]
-.1 -.2 -0.3
-1
)LJ6LGHIRUFHFKDUDFWHULVWLFVRIWKHVLQJOHURZEUXVKPRGHOXSWRODUJHOHYHOVRIVSLQ
Dn
VLQ
&RPSDUHZLWK)LJZKHUHVSLQLVVPDOODQG
0
θ\ = 5
]
Dµ)]
Dϕ
0.33
0.6 0.5 0.4 0.3 0.2
0.1
-10
0.8 0.7
0
0
α
10
0.1 0
[deg]
-0.1 -0.2 -0.3 )LJ $OLJQLQJWRUTXHFKDUDFWHULVWLFVRIWKHVLQJOHURZEUXVKPRGHOXSWRODUJHOHYHOVRI
Dn
VSLQ&RPSDUHZLWK)LJZKHUHVSLQLVVPDOODQG
VLQ
FHQWUHSODQH,QIDFWZHPD\DVVXPHWKDWZHGHDOZLWKWZRZKHHOVDWWDFKHGWR HDFKRWKHURQWKHVDPHVKDIWDWDGLVWDQFHEURZIURPHDFKRWKHU7KHZKHHOVDUH VXEMHFWHGWRWKHVDPHVLGHVOLSDQGWXUQVOLSYHORFLWLHV9V\DQG
0 DQGVKRZWKH
VDPHFDPEHUDQJOH +RZHYHUWKHORQJLWXGLQDOVOLSYHORFLWLHVDUHGLIIHUHQWIRU WKHFDVHRIFDPEHUEHFDXVHRIDGLIIHUHQFHLQHIIHFWLYHUROOLQJUDGLL:HKDYHIRU
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
WKHORQJLWXGLQDOVOLSYHORFLW\RIWKHOHIWRUULJKWZKHHOSRVLWLRQHGDWDGLVWDQFH\/5 IURPWKHFHQWUHSODQH
9
' 9 &\
V[/5
V[
^0
/5
& & g
VLQ `
7KLV H[SUHVVLRQ LV REWDLQHG E\ FRQVLGHULQJ (T LQ ZKLFK FRQLFLW\ LV
g
GLVUHJDUGHGVWHDG\VWDWHLVDVVXPHGWRRFFXUDQGWKHFDPEHUUHGXFWLRQIDFWRU LVLQWURGXFHG7KHIDFWRUV DUHGHILQHGDVOLNHLQ
\ ' &g
[
[ \ ' & g [ VLQ U
\
H
)URP(TV XVLQJ WKHVOLGLQJYHORFLW\FRPSRQHQWVDUHREWDLQHG
9
J[/5
'9
V[/5
&
MX
/5
M[
9
U
9 ' 9 & MY 9 % [ ^ 0 & & g M[ J\
V\
U
VLQ `
$IWHULQWURGXFLQJWKHWKHRUHWLFDOVOLSTXDQWLWLHVIRUWKHWZRDWWDFKHGZKHHOV
[/5
'&
9
V[/5
9
\
'&
U
9 9
V\
0
'&
9
U
U
ZHILQGIRUWKHJUDGLHQWVRIWKHGHIOHFWLRQVLQWKHDGKHVLRQ]RQHZKHUH9J
LIVPDOOVSLQLVFRQVLGHUHGVOLGLQJRQO\DWWKHUHDU
MX
/5
M[ MY M[
'&
V[/5
' & &[ \
% & g
U
VLQ
H
ZKLFK\LHOGVDIWHULQWHJUDWLRQIRUWKHGHIOHFWLRQVLQWKHDGKHVLRQ]RQH[[W
X ' D& [ /5
Y ' D&[
[/5
% ò D &[
\
% & g
U
7KHWUDQVLWLRQSRLQWIURPDGKHVLRQWRVOLGLQJDW[
VLQ
H
[WFDQEHDVVHVVHGZLWKWKH
DLGRIWKHFRQGLWLRQ
F H ' T S
/5
]
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
ZLWKWKHPDJQLWXGHRIWKHGHIOHFWLRQ
H ' _H * ' X % Y
/5
/5
/5
6ROYLQJIRU[WDQGSHUIRUPLQJ WKHLQWHJUDWLRQRYHUWKHDGKHVLRQUDQJHPD\EH FDUULHGRXWQXPHULFDOO\,QWKHVOLGLQJUDQJHWKHGLUHFWLRQRIWKHGHIOHFWLRQVYDU\
H
ZLWK [ $V DQ DSSUR[LPDWLRQRQHPD\DVVXPHWKDWWKHVHGHIOHFWLRQV IRUDQ
9
LVRWURSLFPRGHO DUHDOOGLUHFWHGRSSRVLWHWRWKHVOLSVSHHG
V/5
5HVXOWVRIVXFK
LQWHJUDWLRQV\LHOGLQJWKHYDOXHVRI)[)\DQG0]ZLOOQRWEHVKRZQKHUH:HUHIHU WR 6DNDL IRU DQDO\WLFDO VROXWLRQV RI WKH VLQJOH URZ EUXVK PRGHO DW FRPELQHGVOLSZLWKFDPEHU
7KH7UHDG6LPXODWLRQ0RGHO
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
9V
ψ
α
wheel plane 6
carcass
9E
9J
wheel spin axis
)[
%
contact line
)LJ
[
&
belt
0] \
9F
tread element
3
T
)\
(QKDQFHGPRGHOZLWKGHIOHFWHGFDUFDVVDQGWUHDGHOHPHQWWKDWLVIROORZHGIURPIURQW WRUHDU
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
%
L
9
∆VL
%
L
H
EL
H
L
L
3 L
J
9
JL
L
3
T
L
L
H
)LJ 7KHLVRWURSLFWUHDGHOHPHQWZLWKGHIOHFWLRQ LQWZRVXFFHVVLYHSRVLWLRQVLDQGL
9E
,WVEDVHSRLQW%PRYHVZLWKVSHHG
9J
DQGLWVWLS3VOLGHVZLWKVSHHG
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
H
)LJWKHWUHDGHOHPHQWGHIOHFWLRQYHFWRU KDVEHHQVKRZQ7KHWUHDGHOHPHQW LVDVVXPHGWREHLVRWURSLFWKXVZLWKHTXDOVWLIIQHVVHVLQ[DQG\GLUHFWLRQ7KHQ
9
ZKHQWKHHOHPHQWLVVOLGLQJWKHVOLGLQJVSHHGYHFWRU
T
WRWKHIULFWLRQIRUFHYHFWRU
WKDWKDVDVHQVHRSSRVLWH
J
H
LVGLUHFWHGRSSRVLWHWRWKHGHIOHFWLRQYHFWRU 7KH
ILJXUHGHSLFWVWKHGHIOHFWHGHOHPHQWDWWKHHQGVRIWZRVXFFHVVLYHWLPHVWHSVL
J
DQGL7KHILUVWREMHFWLYHLVQRZWRILQGDQH[SUHVVLRQIRUWKHGLVSODFHPHQW RI WKHWLSRIWKHHOHPHQWZKLOHVOLGLQJRYHUWKHJURXQG
)
7KHFRQWDFWOHQJWKLVGLYLGHGLQWRQLQWHUYDOV2YHUHDFKWLPHVWHS
WWKHEDVH
SRLQW%PRYHVRYHUDQLQWHUYDOOHQJWKWRZDUGVWKHUHDU:LWKJLYHQOHQJWK
[ ' D Q
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
DQGWKHOLQHDUVSHHGRIUROOLQJ9U
W'
UH
WKHWLPHVWHS
WLVREWDLQHG
[ 9
U
:LWKWKHYHORFLW\YHFWRU
9
E
V
RISRLQW%WKHGLVSODFHPHQWYHFWRU
RIWKLVSRLQW
RYHUWKHWLPHVWHSEHFRPHV
V' 9 W
E
7KHEDVHSRLQW%PRYHVDORQJWKHEHOWSHULSKHUDOOLQHRUDOLQHSDUDOOHOWRWKLV OLQH:LWKWKHNQRZQODWHUDOFRRUGLQDWH\ERIWKLVOLQHRIEDVHSRLQWVZLWKUHVSHFW
M M
WRWKHZKHHOFHQWUHSODQHWKHORFDOVORSH \E [FDQEHDVVHVVHG7KHQZLWKWKH
9
VOLSYHORFLW\
V
RIWKHVOLSSRLQW6WKH\DZUDWHRIWKHOLQHRILQWHUVHFWLRQ
WKHUROOLQJVSHHG9UWKHFRPSRQHQWVRI
9
E
0 DQG
FDQEHIRXQG
7KHYHORFLW\RISRLQW%PD\EHFRQVLGHUHGDVWKHVOLGLQJYHORFLW\RIWKLVSRLQW ZLWKUHVSHFWWRWKHJURXQGDQGZHPD\HPSOR\WKHHTXDWLRQV IRULWV DVVHVVPHQW,QWKHVHHTXDWLRQVDWVWHDG\VWDWHWKHWLPHGHULYDWLYHVRIXDQGY
M M
M M
YDQLVKWKHVORSH X [LVUHSODFHGE\]HURDQGIRU Y [ZHWDNHWKHJUDGLHQWRI WKH EHOW GHIOHFWLRQ FDXVHG E\ WKH H[WHUQDO IRUFH DQG PRPHQW :H KDYH ZLWK DYHUDJH[SRVLWLRQ[E
9 ' 9 &\ E[
V[
ER
[
J
DQG
9 ' 9 %[ E\
V\
J
E
&
&
[
VLQ
M\ 9 M[
&
FRQ[
\9%
U
\ \
[
VLQ
ER
E
U
E
ZLWKWKHVOLSDQGUROOYHORFLWLHV
9 ' &9 V[
F[
9 'U U
' 9 &9
H
F[
V[
9 ' &9
V\
F[
WDQ
7KH ODWHUDO GLVSODFHPHQW \E RI WKH EHOW DW WKH FRQWDFW FHQWUH LV DWWULEXWHG WR FDPEHUFRQLFLW\DQGWKHODWHUDOH[WHUQDOIRUFHWKURXJKWKHODWHUDOFRPSOLDQFH RI WKH FDUFDVV 7KH JUDGLHQW
M\ M[ PD\ EH DSSUR[LPDWHO\ DVVHVVHG E\ E
E
DVVXPLQJDSDUDEROLFEDVHOLQH\E[E H[KLELWLQJDQDYHUDJHVORSHFVLQIOXHQFHG E\ WKH DOLJQLQJ WRUTXH WKURXJK WKH \DZ FRPSOLDQFH DQG SO\VWHHU DQG D FXUYDWXUH FF LQIOXHQFHG E\ WKH VLGH IRUFH WKURXJK WKH EHQGLQJ VWLIIQHVV DQG FDPEHUDQGFRQLFLW\FI :HKDYHIRUWKHODWHUDOFRRUGLQDWH
\ '&D U
E
H
&g
\
) % VLQ % F %F [ % F [ \
FRQ
V
ODW
E
F
E
E
URZ
DQGIRULWVDSSUR[LPDWLRQXVHGLQ
\ ' E ER
URZ
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
DQGWKHVORSH
M\ 0N [ ) [ ' F %F [ ' & % F F U M[ E
]
V
F
E
\
E
EHQG
H
E
\DZ
E
%
VLQ
[ \
\
E
E
&RQLFLW\DQGSO\VWHHUZLOOEHLQWHUSUHWHGKHUHWREHFDXVHGE\µEXLOWLQ¶FDPEHU DQG VOLS DQJOHV 7KHVH HTXLYDOHQW FDPEHU DQG VOLS DQJOHV
FRQ
DQG
SO\
DUH
LQWURGXFHGLQWKHH[SUHVVLRQVRIWKHTXDQWLWLHV :HGHILQH
\ ' &g
[
E
[
DQG
[ \ ' & g
\
E
E
g
7KHFRHIILFLHQWV
[
\
FRQ[
[ U g
DQG
\ ' & E
\ U [ U
E
&g
[
FRQ
H
E
VLQ
% &g
H
\
\
E FRQ
%
SO\
H
PD\EHWDNHQHTXDOWRHDFKRWKHU7KHILUVWWHUPRIWKH
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
\E\U ZLWK\U
g
\U
EVLQ ZLWKDQXSSHUOLPLWRI
LWVPDJQLWXGHHTXDOWRE7KHPRPHQW0¶FDXVHVWKHWRUVLRQRIWKHFRQWDFWSDWFK ] DQGLVDVVXPHGWRDFWDURXQGDSRLQWFORVHUWRLWVFHQWUHOLNHGHSLFWHGLQ)LJ
g
LV XVHG IRU WKLV SXUSRVH 0RUH UHILQHPHQWV PD\ EH $ UHGXFWLRQ SDUDPHWHU ¶ \ LQWURGXFHG)RUGHWDLOVZHUHIHUWRWKHFRPSOHWHOLVWLQJRIWKHSURJUDP7UHDG6LP LQFOXGHGLQWKH$SSHQGL[
V
HVWDEOLVKHGZHFDQGHULYHWKHFKDQJH
:LWKWKHGLVSODFHPHQWYHFWRU
H
LQGHIOHFWLRQ RYHURQHWLPHVWHS%\NHHSLQJWKHGLUHFWLRQV RI PRWLRQRIWKH SRLQWV % DQG 3 LQ )LJ FRQVWDQW GXULQJ WKH WLPH VWHS DQ DSSUR[LPDWH H[SUHVVLRQIRUWKHQHZGHIOHFWLRQYHFWRULVREWDLQHG$IWHUWKHEDVHSRLQW%KDV PRYHGDFFRUGLQJWRWKHYHFWRU
H ' H & %J & V L
L
L
L
V
ZHKDYH
ZLWK
J
J ' & H H& & L
L
L
L
+HUHHLGHQRWHVWKHDEVROXWHYDOXHRIWKHGHIOHFWLRQDQGJLWKHGLVWDQFH3KDV VOLGHG LQ WKHGLUHFWLRQRI
H
L
,QFDVHRIDGKHVLRQWKHVOLGLQJGLVWDQFHJL
:KHQWKHWLSVOLGHVWKHGHIOHFWLRQEHFRPHV
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
H' L
T F L
]L
S
)URP DQDSSUR[LPDWHH[SUHVVLRQIRUJLFDQEHHVWDEOLVKHGFRQVLGHULQJ WKDWJLVVPDOOZLWKUHVSHFWWRWKHGHIOHFWLRQH:HREWDLQ
J ' H& L
L
H
& &
&
H
[L
H
V % H & & V &H & V % H & H & & V
[L
[L
&
[L
\L
[L
\L
\L
L
\L
\L
LQZKLFKH[SUHVVLRQ LVWREHVXEVWLWXWHG,IJLLVSRVLWLYHVOLGLQJUHPDLQV ,IQRWDGKHVLRQFRPPHQFHV,QWKHFDVHRIVOLGLQJWKHGHIOHFWLRQFRPSRQHQWVFDQ EHIRXQGIURP XVLQJ 7KHQWKHIRUFHYHFWRUSHUXQLWOHQJWKLV
H
T ' T L
H
]L
L
L
,IDGKHVLRQRFFXUVJL
DQGZLWK WKHGHIOHFWLRQLVGHWHUPLQHGDJDLQ
7KHIRUFHSHUXQLWOHQJWKQRZUHDGV
T'FH L
S
L
$VVRRQDVWKHFRQGLWLRQIRUDGKHVLRQ
T ' _T _ # L
L
T F L
]L
S
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
'
%D 9 R
E
'XULQJWKHSDVVDJHRIWKHHOHPHQWWKURXJKWKHFRQWDFW]RQHWKHIRUFHV
T
FDOFXODWHGE\PXOWLSO\LQJ WKHWLPHVWHS
L
ZLWKWKHSDUWRIWKHFRQWDFWOHQJWK
)
L
DUH
[FRYHUHGRYHU
W
6XEVHTXHQWO\WKHWRWDOIRUFHFRPSRQHQWVDQGWKHDOLJQLQJPRPHQWDUHIRXQG
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
E\ DGGLQJ WRJHWKHU DOO WKH FRQWULEXWLRQV
)[L
)\L DQG [EL
)\L \ELHII
) [L
UHVSHFWLYHO\$FRUUHFWLRQWRWKHPRPHQWDUPPD\EHLQWURGXFHGWRDFFRXQWIRU WKHVLGHZD\VUROOLQJRIWKHW\UHFURVVVHFWLRQZKLOHEHLQJFDPEHUHGDQGGHIOHFWHG FDXVLQJ ODWHUDO VKLIW RI SRLQW RI DFWLRQ RI WKH UHVXOWLQJ QRUPDO ORDG )] DQG VLPLODUO\RIWKHORQJLWXGLQDOIRUFH)[ $OVRWKHFRXQWHUHIIHFWRIWKHORQJLWXGLQDO GHIOHFWLRQXFPD\FRQWULEXWHWRWKLVFRUUHFWLRQIDFWRUFI(T ,QWKHVHTXHODQXPEHURIH[DPSOHUHVXOWVRIXVLQJWKHWUHDGVLPXODWLRQPRGHO KDYHEHHQSUHVHQWHG7KHIROORZLQJFDVHVKDYHEHHQLQYHVWLJDWHG
6OLGLQJ YHORFLW\ GHSHQGHQW IULFWLRQ FRHIILFLHQW ULJLG FDUFDVV EXW SDUDPHWHUFRI(T LVLQFOXGHG )LJD
)OH[LEOHFDUFDVVZLWKRXWDQGZLWKFDPEHU)LJE
)LQLWHWUHDGZLGWKWZRURZVRIWUHDGHOHPHQWVIOH[LEOHFDUFDVV ZLWK DQGZLWKRXWFDPEHU)LJF
&RPELQHGODWHUDODQGWXUQVOLSWZRURZVIOH[LEOHFDUFDVV )LJG
3QHXPDWLFWUDLODWSXUHVLGHVOLSIOH[LEOHDQGULJLGFDUFDVV )LJ
7KHFRPSXWDWLRQVKDYHEHHQFRQGXFWHGZLWKWKHVHWRISDUDPHWHUYDOXHVOLVWHG LQ7DEOH 7DEOH3DUDPHWHUYDOXHVXVHGLQWKHWUHDGVLPXODWLRQPRGHO)LJV
D
P
)]
1
FODW
N1P
&)
)]
E
P
9F
PV
FEHQG
N1P
\R
YR(T
EURZ
P
R
F\DZ
N1PUDG
UH
P
D
VP
F
FODW
g
RWKHU ¶V
g
FS
\U
FRQ
&) D QURZ
,QFDVHRIDµULJLG¶FDUFDVVSDUDPHWHUF F ODW FEHQG F\DZ
SO\
&) R)]
PN1 (T LVXVHGZKLOH
64 $V LQGLFDWHG TXDQWLWLHV F DQG S
IROORZ IURP WKH PRGHO
SDUDPHWHUV 7KHJUDSKVRIWKHORZHUKDOIRI)LJDUHODWLYHWRWKHXSSHUKDOIVKRZWKH PRVWSURPLQHQWHIIHFWRIDZLWKVOLGLQJYHORFLW\GHFUHDVLQJIULFWLRQFRHIILFLHQW $WD
WKHVLGHIRUFH)\H[KLELWVDFOHDUSHDNDWDVOLSDQJOHRIDERXWVHYHQ
GHJUHHVZKHQWKHZKHHOLVUROOLQJIUHHO\
$OVRWKHIRUHDQGDIWIRUFH) [
WHQGVWRGHFUHDVHDIWHUKDYLQJUHDFKHGLWVSHDNYDOXH7KHLQZDUGHQGLQJVRIWKH FXUYHVDWFRQVWDQWYDOXHVRIVOLSDQJOHVKRZQLQWKHERWWRPGLDJUDPDUHW\SLFDO HVSHFLDOO\IRUDW\UHUXQQLQJRQZHWURDGVXUIDFHV7KHSHDNYDOXHVWKHPVHOYHV ZLOOGHFUHDVHZKHQWKHVSHHGRIWUDYHOLVLQFUHDVHGQRWVKRZQ ZKLOHWKHLQLWLDO
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
VORSHVVOLSVWLIIQHVVHV UHPDLQXQFKDQJHG7KHFKDQJHLQVLJQRIWKHDOLJQLQJ WRUTXH LQ WKH EUDNLQJ UDQJH RI LWV GLDJUDP RFFXUV EHFDXVH RI WKH UHWDLQHG SDUDPHWHU F WKDW SURGXFHV WKH HIIHFW RI WKH IOH[LEOH FDUFDVV DFFRUGLQJ WR (TV 7KHH[SHFWHGFKDQJHLQVLJQRIWKHDOLJQLQJWRUTXHFXUYHDWSXUH VLGHVOLSDWKLJKHUOHYHOVRIVLGHVOLSGRHVQRWRFFXUGXHWRWKHOLPLWDWLRQRIWKH HVWLPDWLRQRIWKHVOLGLQJVSHHGIRUZKLFKWKHVSHHGRIDSRLQWRIWKHEHOWLVWDNHQ KHUHULJLG ,Q)LJEZLWKWKHFDUFDVVFRQVLGHUHGIOH[LEOHWKLVVLJQFKDQJH GRHVVKRZXSDVLOOXVWUDWHGLQWKHXSSHUULJKWGLDJUDP $PRUHLPSRUWDQWHIIHFWRIWKHIOH[LEOHFDUFDVVSUHVHQWHGLQ)LJELVWKH GHFUHDVHG FRUQHULQJ VWLIIQHVV ZKLOH WKH IRUH DQG DIW VOLS VWLIIQHVV UHPDLQV WKH VDPH7KHORZHUYDOXHRI)\DW
E
LQWKH)\YV)[GLDJUDPFOHDUO\GHPRQ
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
E
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
E
DVSLQWRUTXHLVJHQHUDWHGWKDWDSSHDUVWRURWDWHWKHORZHUSDUWRIWKHEHOWLQVXFK DZD\WKDWDQDSSDUHQWVOLSDQJOHDULVHVWKDWLQFUHDVHVWKHFDPEHUWKUXVW7KH XSSHUOHIWKDQGGLDJUDPVKRZVWKHLQFUHDVHLQVLGHIRUFH7KHULJKWKDQGGLDJUDP LQGLFDWHVWKHFRQVLGHUDEOHULVHLQWKHDOLJQLQJWRUTXHDVDUHVXOWRIWKHVSLQWRUTXH LQWKHUDQJHRIVPDOOORQJLWXGLQDOIRUFH)[
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
κ =0
3 single row
)\
[kN]
rigid carcass
0
-0.2
0
W
-20
-10
3 1 row
)\
0
8
rigid
o
10
α
-40
o
20
-20
0
0
[Nm]
γ =ϕ = 0
α = -2o -3
0
1 row
)\
W
)[
-50
3
[kN]
Dµ =0.03
κ =0
3 rigid
0
10
-2 0
[Nm]
0
W
o
o
-10
3 1 row rigid
6
0
o
10
α
o
-40
20
Dµ =0.03
8
o
4 6 8
γ =ϕ = 0 W
)[
0
-3
4
κ=0 γ =ϕ = 0 W
o
o
-20
-10
o
0
10
0]
0
γ =ϕ = 0
α = -2o -3
0
)
[kN] [
W
3
[Nm]
-50
o
-2
0
0
α
20
Dµ =0.03
40 1 row, rigid
2
[kN]
3
[kN]
Dµ =0.03
-0.2
γ =ϕ = 0 -20
α =2
1 row
0
-3
o
20
Dµ =0
40 rigid
0
α
]
-0.2
[kN]
-2
-10
o
0] 0
)\
o
2
[kN]
-2
γ =ϕ = 0 W
o
50 1 row, rigid
Dµ =0
6 4
κ=0 -0.2
γ =ϕ = 0 o
rigid carcass
]
[Nm]
0
o
Dµ =0
40 single row
Dµ =0
-3
α=2
4 6
0 o 8
γ =ϕ = 0 W
-3
0
)[
)LJD&KDUDFWHULVWLFVFRPSXWHGZLWKWUHDGVLPXODWLRQPRGHO
[kN]
3
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
1 row
Dµ =0.03
κ =0
3 flexible
)\
0
]
-0.2
[kN]
[Nm]
0
0
γ =ϕ = 0 -3
-20
o
-10
3 1 row
)\
o
0
10
[kN]
0
20
1 row
γ =ϕ = 0 W
)[
3
[kN]
Dµ =0.03
κ =0
)\
-20
-10
0
-0.2
[kN]
10
0
0
α=
[Nm]
-50
2 4 6 8
γ =ϕ = 0 W
)[
0
-3
3
[kN]
Dµ =0.03
1 row
3 flexible
)\
o
8
γ =0
0
α =-2o
γ =10o,ϕ = 0 W
-3
o
-20
o
-10
o
0
3 1 row
10
8
flexible
α
o
Dµ =0.03
α = -2o 0
W
-0.8
-0.4
0
0.4
κ
0.8
Dµ =0.03
200 1 row, flexible
γ =10o,ϕ=0 )[ [kN] 3
[Nm]
-150
-2 8o 02
4
0
W
-3
γ =10o,ϕ=0
0]
0
0
-3
20
4 2
[kN]
-2
o
[kN]
0
)\
20
Dµ =0.03
-2
o
α
]
o
0
flexible
0
3 flexible
o
0
-3
o
40 1 row
α = -2o -2
γ =ϕ = 0 W
-40
o
Dµ =0.03
8 6 4 2
flexible
α
κ=0 -0.2
W
o
Dµ =0.03
1 row
40 flexible
α= 6
γ =10o,ϕ = 0 W
-3
0
)[
)LJE&KDUDFWHULVWLFVFRPSXWHGZLWKWUHDGVLPXODWLRQPRGHO
[kN]
3
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
3 2 rows
)\
0]
α = -2
γ =10o,ϕ=0 )[ [kN] 3
0
[Nm]
Dµ =0.03
2 rows
3 flexible
-150
o
2
γ =10o,ϕ = 0 W
)[
0
-3
0
20
-5
Dµ =0.03
o
-20
o
-10
γ =0
2 rows
0
0
0
o
o
10
)\
α
-100
20
Dµ =0.03
0
γ =ϕ
0
o
o
W
-10
0
3 2 rows
10
4 2
[kN]
α = -2
0
o
α
20
o
8
α =-2o W
-0.8
-0.4
0
0.4
κ
0 0.8
Dµ =0.03
40 2 rows, flexible
0
α =-2o
]
0
0 4 6 8
[Nm]
γ =ϕ
o
W
-3
o
10
Dµ =0.03
0
0
0
γ =ϕ -3
20
Dµ =0.03
8
flexible
α
o
-10
)\
[kN]
-20
o
-20
2 rows
0
o
0
W
3 flexible
-0.2
o
κ =0,ϕ
[kN]
-3
0
γ =-5o
[Nm]
κ =0,ϕ
κ =0
3 flexible
o
20
o
W
-3
3
[kN]
flexible
]
o
0
-2
0
150 2 rows
)\
[kN]
)\
-2
0
o
W
-3
α= 8
0
0
Dµ =0.03
200 2 rows, flexible
4 2
[kN]
-2
Dµ =0.03
8
flexible
)[
[kN]
0
-50
3
2
γ =ϕ = 0 W
-3
0
)[
[kN]
)LJF&KDUDFWHULVWLFVFRPSXWHGZLWKWUHDGVLPXODWLRQPRGHO
3
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
2 rows
Dµ =0.03
Dϕ = 0.1
3 flexible
)\
W
[kN]
-0.3
o
o
-20
-10
o
0
10
α
Dµ =0
-200
20
W
flexible
[Nm]
o
Dϕ = 0.1
3 2 rows
0
200
0
-0.3
-3
-0.1 o -10 0
o
-20
3
γ =κ = 0 o
o
-20
-10
Dϕ = -0.3 o
0
10
Dϕ = 0.8
2 rows flexible
W
α
o
20
Dµ =0
W
o
8
γ =κ = 0
[Nm]
o
-200
10
Dµ =0
α
o
20
0 -0.1 -0.2
γ =κ = 0 o
-20
o
-10
0
10
-0.3 o 20
α
Dµ =0
200 flexible
o
0
2 rows
2 rows flexible
6 4 2
0.1
0
0 -0.2
[kN]
0.1 0 -0.1 -0.2
]
0.2
0.8
)\
0.8
0
γ =κ = 0
-0.1
-3
[kN]
flexible
0 -0.2
)\
0
Dµ =0.03
2 rows
]
0.2
0 0.8
200
α =-2
o
]
0
[Nm]
4 6
2
8o
0
0
γ =κ = 0
α =-2
o
-1
0
0.1
0.2
0.3
0.4
0.5
Dϕ D5 W
0
γ =κ = 0
-50 0
0.1
0.2
0.3
0.4
0.5
Dϕ D5
W
)LJG&KDUDFWHULVWLFVFRPSXWHGZLWKWUHDGVLPXODWLRQPRGHO
0.5 2 rows
κ =γ =ϕ
flexible WD
W
0
0.5 1 row
Dµ =0
[-]
κ =γ =ϕ
rigid WD
W
0
Dµ =0
[-]
0
0
-0.1
-0.1
o
-20
o
-10
0
o
10
α
o
20
o
-20
o
-10
0
o
10
α
o
20
)LJ3QHXPDWLFWUDLOYDULDWLRQDVFRPSXWHGZLWKWUHDGVLPXODWLRQPRGHO
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
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
DQGPD\EHFRPSDUHGZLWK)LJV$QLQWHUHVWLQJ
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
)
0 )\ D µ) µ)
0
]
]
)]=20000N
\ 12000
2 rows
]
8000
]
4000
asymptote 0
0
0.33 1 1 row
0]
]
asymptote
1200
)
\
)
800
2 rows
\
400
0
0
1θ
D
ϕ
W
1
0
D5
0 0
4 2
[m] [1/m]
5 5
8 4 1/
)LJ 3XUHWXUQVOLSFKDUDFWHULVWLFVDFFRUGLQJWRWKHPRGHOOHIWULJLGFDUFDVVZLWKRQHRU WZRURZVRIHOHPHQWV DQGUHVXOWVIURPH[SHULPHQWVZLWKDELDVSO\WUXFNW\UHULJKW
S
H+'
L
9
EDU
0RGHOSDUDPDWHUV
D
NPKRQGU\URDGIURP)UHXGHQVWHLQ
E
URZ
D
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
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
P)]
1DQG
DYHU\
UHDVRQDEOHDOVRTXDQWLWDWLYHFRUUHVSRQGDQFHIRUERWKWKHIRUFHDQGWKHPRPHQW FKDUDFWHULVWLFVRI)LJFDQEHREWDLQHG)UHXGHQVWHLQVXJJHVWVWKHIROORZLQJ IRUPXODIRUWKHSHDNPRPHQWJHQHUDWHGDWSXUHWXUQLQJDWZKHHOVSHHG9
*0]PD[* '
0]n 4 . )] D % E
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
g LQ(T
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
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
α =2o, κ = -0.1, γ =0, 5=4 α =2o, κ = -0.1, γ =10o, 5=4 α =2o, κ = -0.1, γ =0 , 5=D
flexible carcass
0.02m
α =4 , κ = 0, γ =0, o
5=4
0.02m
)LJ ([DPSOHVRIGHIOHFWLRQSDWWHUQVDWYDULRXVFRPELQDWLRQVRIVOLSIRUWKHVLQJOHURZ PRGHOZLWKULJLGDQGIOH[LEOHFDUFDVV
5 9
F
α
Erow D
0.02m 0.02m
α =4o, κ = 0, γ =0, 5=D
)LJ 'HIOHFWLRQVRIWKHWUHDGHOHPHQWVDQGRIWKHEHOWIRUWKHWKUHHURZPRGHODWVLGHVOLS DQGWXUQLQJ
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
$SSOLFDWLRQ%UDNLQJXSWR:KHHO/RFN
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
P X0 & YU ' )[ % )[ P Y0 % XU ' )\ % )\ PN U0 ' D)\ & E)\
ZLWK
WDQ WDQ
' & Y % DU X ' & Y & EU X
DQG
)\ ' ) \ )\ ' )\
)] )]
)RUDSURSHUVLPXODWLRQRIWKHPRWLRQWKHVOLSUDWLR¶V ZKHHOVSHHGVRIUHYROXWLRQ
VKRXOGUHVXOWIURPWKH
L
ZKLFKZRXOGUHTXLUHDGGLWLRQDOGHJUHHVRIIUHHGRP L
)RU WKLV RFFDVLRQ ZH ZLOO HPSOR\ DQ DOWHUQDWLYH DSSURDFK WKDW LQYROYHV WKH LQWURGXFWLRQRIIXQFWLRQVIRUWKHVLGHIRUFHLQZKLFKLWVGLUHFWGHSHQGHQFHRQWKH EUDNLQJHIIRUWRUEUDNHSUHVVXUH LVLQFOXGHG7KLVLVSRVVLEOHZKHQµGU\¶IULFWLRQ LV DVVXPHG WR RFFXU EHWZHHQ W\UH DQG URDG DQG WKH FXUYHV RI WKH ULJKWPRVW
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
locked wheel
)2
O
E
)[2
α2
D
U X
)\2
Y
rolling wheel
)[1
\
α1
)\1
[
)LJµ%LF\FOH¶PRGHOZLWKRQHZKHHOORFNHG
GLDJUDPRI)LJGRQRWVKRZLQZDUGHQGLQJVDQGWKXVGRXEOHYDOXHGIXQFWLRQV DUH DYRLGHG 7KH FKDUDFWHULVWLFV RI )LJ ZLOO EH XVHG IRU RXU DQDO\VLV ,Q )LJWKHFKDUDFWHULVWLFVKDYHEHHQUHSURGXFHGRQHFXUYHIRUWKHIUHHO\UROOLQJ ZKHHO DQRWKHU RQH IRU WKH ZKHHO WKDW LV EHLQJ EUDNHG ZLWK D EUDNLQJ HIIRUW FRUUHVSRQGLQJWRDEUDNHIRUFH%
)[DWVWUDLJKWDKHDGUROOLQJ
VPDOO DQG
ILQDOO\IRUDZKHHOWKDWLVORFNHGFRPSOHWHO\7KH VHFRQG FXUYHVKRZVWKDWDW KLJKHUVOLSDQJOHVDORZHUKRUL]RQWDOSODWHDXLVIROORZHGXQWLOWKHVWUDLJKWLQFOLQHG OLQHLVUHDFKHGZKHUHDOVRWKLVZKHHOJHWVORFNHGQRWHWKDWVLQ
KDVEHHQXVHGDV
DEVFLVVDZKLFKPDNHVWKHUHODWLRQVKLSOLQHDU $IXOOGLVFXVVLRQRIWKLVEHKDYLRXU KDVEHHQJLYHQLQ6XEVHFWLRQLQFRQQHFWLRQZLWKWKHWUHDWPHQWRIWKHW\UH EUXVKPRGHO $VVKRZQLQ)LJZHDVVXPHWKDWWKHZKHHORQD[OHLVORFNHGDQGWKDW RQD[OHWKHZKHHOPD\EHEUDNHG7KHFKDUDFWHULVWLFVRI)LJDSSO\)LUVW WKH VLWXDWLRQ QHDU WKH XQGLVWXUEHG VWUDLJKW DKHDG PRWLRQ ZLOO EH VWXGLHG /LQHDULVDWLRQZLWKERWKVOLSDQJOHVDVVXPHGVPDOO\LHOGVIRUWKHKRUL]RQWDOZKHHO IRUFHV
)\1,2 µ)]1,2
)\1,2
1 1
1 1
braked 2
0
µ)]1,2
free rolling
2
(locked wheel)
sinα 1,2
1
)[1,2
%1
µ)]1,2
µ)]1
0
)LJ7\UHVLGHIRUFHFKDUDFWHULVWLFVIRUIUHHO\UROOLQJDQGEUDNHGZKHHOV
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
)[ )\ )[ )\
' &%
' )\
% 6 & % ' & )] FRV 6 & )] ' )] VLQ 6 )] ' &
)RUWKHORFNHGZKHHOZHKDYHQRZDQHIIHFWLYHFRUQHULQJVWLIIQHVV
& ' )] ' D PJ O
$VDUHVXOWWKH(TV UHGXFHWR
P X0 ' & % & )]
P Y0 %
X PN U0 %
_ _
& % & Y % PX %
X
_ _
X
_ _
D & % E & U %
X
_ _
D& & E&
U'
D& & E& Y '
)XUWKHUPRUHZHKDYHLQWURGXFHGWKHDEVROXWHYDOXHRIWKHIRUZDUGVSHHGXLQWKH GHQRPLQDWRUVWRDOORZIRUWKHFRQVLGHUDWLRQRIQHJDWLYHYDOXHVRIXZKLOHDWWKH UHPDLQV XQFKDQJHG FI WKH FRUUHVSRQGLQJ HTXDWLRQV
VDPH WLPH WKH VLJQ RI
IRUWKHQRQGULYHQRUEUDNHGYHKLFOH 1HJDWLYHYDOXHVRIXFRUUHVSRQGWR WKHFDVHRIORFNHGIURQWZKHHOV (OLPLQDWLRQRIWKHODWHUDOYHORFLW\YIURPHTXDWLRQV \LHOGV
P N X U % P &N % % D % N & % E & _X_U0% % O & & & PX_X_D& & E& U '
:KHQFRPSDUHGZLWK(T FRQVLGHULQJ LWLVQRWHGWKDWLQWKHVHFRQG
FRHIILFLHQW RI WKH WHUP N & KDV GLVDSSHDUHG DQG WKDW N % KDV EHHQ DGGHG7KLVLVGXHWRWKHGLIIHUHQWLDWLRQRIXLQWKHHOLPLQDWLRQSURFHVVRIY:H KDGRULJLQDOO\LQWKHVHFRQGFRHIILFLHQWRI WKHWHUP
PN X0 VJQ X ' &N D PJ % % ' &N & % % O
ZKLFKH[SODLQVWKHFKDQJHVREVHUYHG,QRWKHUUHVSHFWV(T LVVLPLODUWR WKH KRPRJHQHRXV YHUVLRQ RI (T +RZHYHU DQ LPSRUWDQW GLIIHUHQFH DSSHDUVLQWKHFRHIILFLHQWVZKLFKDUHQRZGHSHQGHQWRQWKHWLPHEHFDXVHRIWKH SUHVHQFHRIWKHOLQHDUO\ ZLWKWLPHGHFUHDVLQJVSHHGX:HKDYHWKHDGGLWLRQDO HTXDWLRQIRUXE\LQWHJUDWLRQRIWKHILUVWRI(TV
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
X ' XR &
D PJ % % P O
X @W
VJQ
7KHH[DFWVROXWLRQRI(T FDQEHIRXQGEHFDXVHRIWKHIRUWXQDWHIDFWWKDW WKHHTXDWLRQFDQEHUHGXFHGWRWKHGLIIHUHQWLDOHTXDWLRQRI%HVVHOWKHVROXWLRQRI ZKLFKLVNQRZQLQWDEXODUIRUP%HIRUHJLYLQJDQH[DPSOHRIVXFKDFRPSOHWH VROXWLRQ ZH ZLOO DQDO\VH WKH PRWLRQ MXVW DIWHU WKH DSSOLFDWLRQ RI D VOLJKW GLVWXUEDQFHZKLFKDOORZVXVWRDSSUR[LPDWHWKHHTXDWLRQWRRQHZLWKFRQVWDQW SDUDPHWHUV LQ ZKLFK X LV UHSODFHG E\ LWV LQLWLDO YDOXH XR )RU WKLV VXEVWLWXWLYH HTXDWLRQWKHVROXWLRQFDQEHIRXQGHDVLO\:HREWDLQ
U' ' H W%' H
W
ZLWK'GHQRWLQJWKHFRQVWDQWVRILQWHJUDWLRQJRYHUQHGE\WKHLQLWLDOYDOXHVRI WKHVWDWHYDULDEOHVYDQGU DQG
UHSUHVHQWLQJWKHHLJHQYDOXHVWKDWLVWKHURRWV
RIWKHFKDUDFWHULVWLFHTXDWLRQRIWKHVXEVWLWXWLYHGLIIHUHQWLDOHTXDWLRQ
$ % % % & & XR_XR_ ' '
&RPSDULVRQZLWKWKHFRHIILFLHQWVRI(T UHYHDOVWKHH[SUHVVLRQVIRUWKH TXDQWLWLHV$%&DQG'6LQFHDSSDUHQWO\WKHILUVWWKUHHDUHDOZD\VSRVLWLYH DQGLQWKHFDVHFRQVLGHUHG & LV PXFKVPDOOHUWKDQ&ZKLFKPDNHV'DOVRD SRVLWLYH TXDQWLW\ ZH H[SHFW WKDW WKH VXEVWLWXWLYH V\VWHP FDQ RQO\ EHFRPH XQVWDEOHEHFDXVHRIDSRVVLEO\QHJDWLYHWKLUGFRHIILFLHQW7KHQRQHRIWKHURRWV OHWVVD\
EHFRPHVSRVLWLYHZKLOHUHPDLQLQJUHDO 2EYLRXVO\WKLVFDQRQO\
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
XFULW '
JO )] & &
)URPWKLVH[SUHVVLRQLWFDQEHVHHQWKDWZKHQDULJLGIURQWW\UHLVFRQVLGHUHGVR
6 4 ZHREWDLQWKHVLPSOHIRUPX
WKDW&
FULW
%JO 7KHLQIOXHQFHRIDQHODVWLF
IURQWW\UHLVQRWYHU\JUHDWDVWKHIURQWFRUQHULQJVWLIIQHVVPD\UDQJHIURPWR
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
WLPHV WKH W\UH QRUPDO ORDG &RQVLGHULQJ D GU\ URDG ZLWK
ZH ILQG D
FULWLFDOVSHHGRIDERXWNPK ,Q YLHZ RI WKLV YHU\ ORZ VSHHG DERYH ZKLFK WKH YHKLFOH ZLWK UHDU ZKHHOV ORFNHGEHFRPHVXQVWDEOHDQGWKHIDFWWKDWQRUPDOO\WKHVSHHGZLOODOUHDG\EHD ORWODUJHUWKDQWKHFULWLFDOVSHHGZKHQWKHEUDNHVDUHEHLQJDSSOLHGLW LV PRUH XVHIXOWRFRQVLGHUWKHGHJUHHRILQVWDELOLW\DVDIXQFWLRQRIVSHHG:HZLOODGRSW WKH URRW
WR UHSUHVHQW WKH GHJUHH RI LQVWDELOLW\ )LJXUH LOOXVWUDWHV WKH
PDQQHULQZKLFKWKHURRWVYDU\ZLWKVSHHGRIWUDYHO7KHGLDJUDPLQFOXGHVWKH FDVHRIIURQWZKHHOVORFNHGX ZKHUHREYLRXVO\WKHURRWVUHPDLQQHJDWLYH DQGEHFRPHFRPSOH[DWKLJKHUVSHHGVRIWUDYHOUHDOSDUWGRWWHGFXUYH $VD SUDFWLFDOLQGLFDWLRQIRUWKHGHJUHHRILQVWDELOLW\ZHPD\HPSOR\WKHKHLJKWRIWKH KRUL]RQWDODV\PSWRWHWRZKLFK
/
X64
WHQGVZKHQX
6 4 )RUWKDWZHILQG
J @ DE @ & & O N )]
'
ZKLFKVKRZVWKDWLQVWDELOLW\GHFUHDVHVZKHQWKHIURQWW\UHFRUQHULQJVWLIIQHVVLV ORZHU DQG EHFRPHV PRUH LQ EDODQFH ZLWK WKH KHUH YHU\ ORZ UHDU µFRUQHULQJ VWLIIQHVV¶ ,WDOVREHFRPHVFOHDUWKDWZKHQWKHIURQWZKHHOVDUHORFNHGDV )] WKHGHJUHHRILQVWDELOLW\UHGXFHVWR]HURUHSUHVHQWLQJLQGLIIHUHQW
ZHOO&
VWDELOLW\ 7KHFRPSOHWHVROXWLRQRI(T FDQEHIRXQGE\DSSO\LQJWKH/RPPHO WUDQVIRUPDWLRQ RI WKH %HVVHO GLIIHUHQWLDO HTXDWLRQ DQG XVLQJ WKH PRGLILHG
λ1,2 front wheels locked
rear wheels locked
λ1 -2
0
Xcrit
2
λ1,X 4
X µJO
λ1 λ2
λ2
)LJ 5HDO SDUW RI WKH HLJHQYDOXHV RI WKH VXEVWLWXWLYH OLQHDUV\VWHPZLWKFRQVWDQW FRHIILFLHQWV
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
IXQFWLRQV RI %HVVHO LQ LWV VROXWLRQ 7KHVH IXQFWLRQV DUH DYDLODEOH LQ WDEXODWHG IRUP DQG FDQ EH IRXQG LQ WKH ERRN RI $EUDPRZLW] DQG 6WHJXQ FI (TIRUWKHWUDQVIRUPHGHTXDWLRQDQGSIRUWKHVROXWLRQ ,QWKHSDSHURI.RLWHUDQG3DFHMND WKHH[DFWVROXWLRQKDVEHHQJLYHQ WRJHWKHUZLWKH[DPSOHVRIWKHQXPHULFDOO\FRPSXWHGVROXWLRQIRUWKHFRPSOHWH QRQOLQHDUV\VWHP)RUWKHVSHFLDOFDVHZLWKSDUDPHWHUVD
E
N%
&)]
WKHVROXWLRQRIWKHOLQHDUHTXDWLRQ LVSUHVHQWHGLQ)LJ7KHH[DFW
VROXWLRQ LV FRPSDUHG ZLWK WKH DSSUR[LPDWH RQH RI WKH VXEVWLWXWLYH HTXDWLRQ7KHVROXWLRQLQWKHIRUPRIDVWDEOHDQGDQXQVWDEOHEUDQFKKDVEHHQ GUDZQ LQ WKH X U SODQH FRQVWLWXWLQJ WKH SURMHFWLRQ RI WKH FRPSOHWH GLPHQVLRQDOWUDMHFWRU\LQWKHXYU VSDFH$ORQJWKHXD[LVZHPD\LQWURGXFH WKHWLPHD[LVFRQVLGHULQJWKHUHODWLRQVKLS ,QWKHFDVHFRQVLGHUHGWKHQRQ GLPHQVLRQDOHLJHQYDOXHV
%OJ WDNHWKHYDOXHVDQGUHVSHFWLYHO\
)RUWKHVDNHRIFRPSDULVRQWKHLQLWLDOYDOXHVKDYHEHHQWDNHQWKHVDPHIRUERWK SDLUV RI VROXWLRQV 1R YLVLEOH GLIIHUHQFH FDQ EH GHWHFWHG EHWZHHQ WKH WZR GHFD\LQJVWDEOHEUDQFKHV7KHµXQVWDEOH¶EUDQFKRIWKHH[DFWVROXWLRQKRZHYHU VKRZVIRUJUHDWHUYDOXHVRIWLPHDQLQFUHDVLQJGLIIHUHQFHZLWKWKHFRUUHVSRQGLQJ DSSUR[LPDWHH[SRQHQWLDOVROXWLRQ$WWKHVWDUWRIWKHPRWLRQZKHUHWKHUHDUD[OH WHQGV WR EUHDN DZD\ WKH DJUHHPHQW LV YHU\ JRRG ,W FDQ EH VKRZQ WKDW ZKHQ H[SDQGHGLQDVHULHVRISRZHUVRIWKHWLPHWWKHVROXWLRQVDUHLGHQWLFDOXSWRDQG
LQFOXGLQJWKHWKLUGWHUPZLWKW 7KLVVXSSRUWVRXUFKRLFHRIGHILQLQJWKHGHJUHH RILQVWDELOLW\ $ORQJ WKH YHUWLFDO D[LV WKH \DZ UDWH LV SORWWHG $IWHU KDYLQJ DWWDLQHG D
U
substitutive Bessel
XFULW 0
1
2 W
3
X
µJO
0
)LJ 6ROXWLRQV RI WKH H[DFW DQG WKH DSSUR[LPDWHG OLQHDUV\VWHPVUHSUHVHQWLQJWKH YDULDWLRQRIWKH\DZUDWHUZLWKIRUZDUGYHORFLW\XRUZLWKWLPHW
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
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
[
)\
1
sin α1
%1 = 0
0.5
\
XR=
3
UR=
0.064
)LJ
JO
µ = 0.25 JO
'HYHORSPHQWRIWKHYHKLFOHPRWLRQDIWHUWKHUHDUZKHHOVJHWORFNHGZKLOH WKHIURQWZKHHOVUHPDLQUROOLQJIUHHO\
[ 1
sin α1
0.5
\
)\1
%1= 0.66µ)]1 µ = 0.25
XR=
3
UR=
0.064
in this range: front wheels also locked
JO JO
)LJ 'HYHORSPHQWRIWKHYHKLFOHPRWLRQDIWHUWKHUHDUZKHHOVJHWORFNHGZKLOHDOVRWKH IURQWZKHHOVDUHEHLQJEUDNHG
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
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
ZH LQGHHG
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
&KDSWHU 6(0,(03,5,&$/7<5(02'(/6
,QWURGXFWLRQ
,Q WKH SUHFHGLQJ FKDSWHU WKH WKHRU\ RI WKH W\UH IRUFH DQG PRPHQW JHQHUDWLQJ SURSHUWLHV KDYH EHHQ GHDOW ZLWK EDVHG RQ SK\VLFDO W\UH PRGHOV 7KH SUHVHQW FKDSWHUWUHDWVPRGHOVWKDWKDYHEHHQVSHFLILFDOO\GHVLJQHGWRUHSUHVHQWWKHW\UH DV D YHKLFOH FRPSRQHQW LQ D YHKLFOH VLPXODWLRQ HQYLURQPHQW 7KH PRGHOOLQJ DSSURDFKLVWHUPHGµVHPLHPSLULFDO¶EHFDXVHWKHPRGHOVDUHEDVHGRQPHDVXUHG GDWD EXW PD\ FRQWDLQ VWUXFWXUHV WKDW ILQG WKHLU RULJLQ LQ SK\VLFDO PRGHOV OLNH WKRVH WUHDWHG LQ WKH SUHFHGLQJ FKDSWHU 7KH PDWKHPDWLFDO GHVFULSWLRQV DUH UHVWULFWHG WR VWHDG\VWDWH VLWXDWLRQV 7KH QRQVWHDG\VWDWH EHKDYLRXU ZLOO EH GLVFXVVHGLQVXEVHTXHQWFKDSWHUV ,Q WKH SDVW VHYHUDO W\SHV RI PDWKHPDWLFDO IXQFWLRQV KDYH EHHQ XVHG WR GHVFULEHWKHFRUQHULQJIRUFHFKDUDFWHULVLF([SRQHQWLDODUFWDQJHQWSDUDEROLFXS WRLWVPD[LPXP DQGK\SHUEROLFWDQJHQWIXQFWLRQVGLIIHUHQFHRIWZR KDYHEHHQ WULHGZLWKPRUHDQGOHVVVXFFHVV2IWHQRQO\YHU\FUXGHDSSUR[LPDWLRQVFRXOG EHDFKLHYHG7RLPSURYHWKHDFFXUDF\WDEOHVRIPHDVXUHGGDWDSRLQWVKDYHEHHQ XVHGWRJHWKHUZLWKLQWHUSRODWLRQVFKHPHV$OVRKLJKHURUGHUSRO\QRPLDOVZHUH SRSXODUEXWSURYHGQRWDOZD\VVXLWDEOHLQWHUPVRIDFFXUDF\DQGWKHYHU\ODUJH GHYLDWLRQV WKDW RFFXU RXWVLGH WKH UDQJHV RI VOLS FRYHUHG E\ WKH RULJLQDO PHDVXUHPHQWGDWDXVHGLQWKHILWWLQJSURFHVV0DWKHPDWLFDOUHSUHVHQWDWLRQVRI ORQJLWXGLQDOIRUFHDQGDOLJQLQJWRUTXHFDPHODWHUDQGRQO\UHODWLYHO\UHFHQWO\WKH FRPELQHG VOLS FRQGLWLRQ ZDV LQFOXGHG LQ WKH HPSLULFDO GHVFULSWLRQ 7KH ORQJLWXGLQDOVOLSUDWLRZDVLQWURGXFHGDVDQLQSXWYDULDEOHLQVWHDGRIWKHEUDNLQJ RU GULYLQJ IRUFH ZKLFK ZDV FRPPRQ SUDFWLFH LQ WKH HDUO\ GD\V RI YHKLFOH G\QDPLFV DQDO\VLV 7KLV ODWWHU PHWKRG KRZHYHU LV VWLOO LQ XVH IRU FHUWDLQ DSSOLFDWLRQV ,QWKHIROORZLQJVHFWLRQVRIWKLVFKDSWHUILUVWDUHODWLYHO\VLPSOHDSSURDFK ZLOOEHGLVFXVVHGWKDWLVEDVHGRQWKHVLPLODULW\FRQFHSW DQGDIWHUWKDWLQWKH UHPDLQGHU RI WKH FKDSWHU D GHWDLOHG GHVFULSWLRQ ZLOO EH JLYHQ RI WKH 0DJLF )RUPXODW\UHPRGHO7KHWZRPRGHODSSURDFKHVEHORQJWRWKHVHFRQGDQGILUVW FDWHJRULHVRI)LJUHVSHFWLYHO\
6(0,(03,5,&$/7<5(02'(/6
7KH6LPLODULW\0HWKRG
7KHPHWKRGWREHGLVFXVVHGLQWKLVVHFWLRQLVEDVHGRQWKHREVHUYDWLRQWKDWWKH SXUHVOLSFXUYHVUHPDLQDSSUR[LPDWHO\ VLPLODULQVKDSHZKHQWKHW\UHUXQVDW FRQGLWLRQVWKDWDUHGLIIHUHQWIURPWKHUHIHUHQFHFRQGLWLRQ7KHUHIHUHQFHFRQGLWLRQ LVGHILQHGKHUHDVWKHVWDWHZKHUHWKHW\UHUXQVDWLWVUDWHGQRPLQDO ORDG)]R DWFDPEHUHTXDOWR]HUR WR]HUR
DWHLWKHUIUHHUROOLQJ
RUDWVLGHVOLSHTXDO
DQGRQDJLYHQURDGVXUIDFHR $VLPLODUVKDSHPHDQVWKDWWKH
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¶&¶'¶DQG(¶IRUWKHQRQGLPHQVLRQDO
)] = 1.0
)\ = )\
0
200 lb. 600 lb. 1000 lb. 1400 lb. 1800 lb.
µ )] Magic Formula fit
-1.0 -10
= 0.714, & \ = 1.40, ' 1.000, ( = -0.20 \ \ % \
-5
0
α= )LJ
&)α()] ) µ )]
5
10
tanα
5HVXOWRI5DGW¶VQRQGLPHQVLRQDOLVDWLRQRIW\UHFKDUDFWHULVWLFVVKRZLQJWKDWWKHW\UH VLGHIRUFHFKDUDFWHULVWLFVPHDVXUHGDWGLIIHUHQWORDGVUHGXFHWRYLUWXDOO\WKHVDPH FXUYHZKHQWKHIRUFHDQGWKHVOLSDQJOHDUHQRUPDOL]HGDVLQGLFDWHG
6(0,(03,5,&$/7<5(02'(/6
Magic Formula fit
= 0.852, & ] = 2.30, ' ] 0.510, ( ] = -2.75 % ]
0.4 0] =
0 0]
µ )] W
-0.4
)] = 200 lb. 600 lb. 1000 lb. 1400 lb. 1800 lb.
-10
-5
0
α= )LJ
&)α()] ) µ )]
5
10
tanα
5HVXOWRI5DGW¶VQRQGLPHQVLRQDOLVDWLRQRIWKHW\UHDOLJQLQJWRUTXHFKDUDFWHULVWLFDIWHU QRUPDOLVLQJWKHPRPHQWDQGWKHVOLSDQJOHDVLQGLFDWHGWUHSUHVHQWVWKHSQHXPDWLF WUDLODWYDQLVKLQJVOLSDQJOH
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
3XUH6OLS&RQGLWLRQV
7KH IXQFWLRQV UHSUHVHQWLQJ WKH UHIHUHQFH FXUYHV ZKLFK DUH IRXQG DW SXUH VOLS FRQGLWLRQV DUH GHVLJQDWHG ZLWK WKH VXEVFULSW R :H KDYH IRU LQVWDQFH WKH UHIHUHQFH IXQFWLRQ )\
)\R
WKDW UHSUHVHQWV WKH VLGH IRUFH YV VOLS DQJOH
UHODWLRQVKLSDWQRPLQDOORDG)]RZLWKORQJLWXGLQDOVOLSDQGFDPEHUHTXDOWR]HUR DQGWKHIULFWLRQOHYHOUHSUHVHQWHGE\R:HPD\QRZWU\WRFKDQJHWKHFRQGLWLRQ WRDVLWXDWLRQDWDGLIIHUHQWZKHHOORDG)]7ZREDVLFFKDQJHVZLOORFFXUZLWKWKH FKDUDFWHULVWLF D FKDQJH LQ OHYHO RI WKH FXUYH ZKHUH VDWXUDWLRQ RI WKH VLGH
6(0,(03,5,&$/7<5(02'(/6
IRUFHWDNHVSODFHSHDNOHYHO DQG DFKDQJHLQVORSHDWYDQLVKLQJVLGHVOLS 7KHILUVWPRGLILFDWLRQFDQEHFUHDWHGE\PXOWLSO\LQJWKHFKDUDFWHULVWLFERWK
LQ YHUWLFDO DQG KRUL]RQWDO GLUHFWLRQ ZLWK WKH UDWLR )] )]R 7KH KRUL]RQWDO PXOWLSOLFDWLRQLVQHHGHGWRQRWGLVWXUEWKHRULJLQDOVORSH:HREWDLQIRUWKHQHZ IXQFWLRQ
)\ '
)] ) ) ]R \R
HT
ZLWKWKHHTXLYDOHQWVOLSDQJOH
)]R )]
HT '
%HFDXVHREYLRXVO\WKHGHULYDWLYHRI)\RZLWKUHVSHFWWRLWVDUJXPHQW LVHTXDOWRWKHRULJLQDOFRUQHULQJVWLIIQHVV&)
M)\ M
M
HT
M
'
)
G \R G HT
HT
R
WKHVDPHYDOXHIRUWKHVORSHDW
)] G)\R ) ]R G HT
'
DW
ZHILQGIRUWKHGHULYDWLYHRI
)\ZLWKUHVSHFWWR
HT
' &) R
ZKLFKSURYHVWKDWWKHVORSHDWWKHRULJLQRIWKHFKDUDFWHULVWLFLVQRWDIIHFWHGE\ WKH VXFFHVVLYH PXOWLSOLFDWLRQV 7KH VHFRQG VWHS LQ WKH PDQLSXODWLRQ RI WKH
&α & α
)\
) R
3 [kN]
1
)
)
)
2
0
[kN/rad]
]
&α
1 2
)
2
1
)
)
1
0
\
15
0
0
2
]
-1
1
[kNm]
0
0
0.3 [rad]
α
&α
10
0
[kNm/rad]
]R
&α
]
0 R
-3
)LJ
&α
0
0
)
8VLQJWKHVLPLODULW\PHWKRGWRDGDSW
0 FXUYHVWRDQHZOHYHORIORDG
DQG
]
0 2
-2
300
)
&α
\R
0 1
2
&α
30
]R
\
2
4 [kN]
)
]
)LJ &RUQHULQJ DQGDOLJQLQJVWLII QHVVYHUVXVZKHHOORDG
6(0,(03,5,&$/7<5(02'(/6
RULJLQDOFXUYHLVWKHDGDSWDWLRQRIWKHVORSH7KLVLVDFFRPSOLVKHGE\DKRUL]RQWDO PXOWLSOLFDWLRQRIWKHQHZO\REWDLQHGFKDUDFWHULVWLF7KLVLVGRQHE\PXOWLSO\LQJ WKH DUJXPHQW ZLWK WKH UDWLR RI WKH QHZ DQG WKH RULJLQDO FRUQHULQJ VWLIIQHVV &RQVHTXHQWO\WKHQHZDUJXPHQWUHDGV
HT '
&) )] )]R &) R )]
7RJHWKHUZLWK(T ZHKDYHWKHQHZIRUPXODWLRQIRUWKHVLGHIRUFHYVVOLS DQJOHUHODWLRQVKLSDWWKHQHZORDG)],Q)LJWKHWZRVWHSVWDNHQWRREWDLQWKH QHZFXUYHKDYHEHHQLOOXVWUDWHG+HUHWKHQRPLQDOORDG)]R 1DQGWKHQHZ ORDG)]
1
,QWKHVDPHILJXUHWKHFKDUDFWHULVWLFRIWKHDOLJQLQJWRUTXHKDVEHHQDGDSWHG WRWKHQHZFRQGLWLRQ)RUWKLVZHXVHWKHNQRZOHGJHJDLQHGZKHQZRUNLQJZLWK WKH EUXVK PRGHO FI )LJ 0RUH VSHFLILFDOO\ ZH ZLOO REH\ WKH UXOH WKDW DFFRUGLQJWRWKHWKHRU\REYLRXVO\WKHSRLQWZKHUHWKH0]FXUYHUHDFKHVWKH D[LVOLHVEHORZWKHSHDNRIWKH) \FXUYH,Q)LJWKLVRFFXUVDWWDQ
\
7KLVUHTXLUHPHQWPHDQVWKDWWKHVDPHHTXLYDOHQWVOLSDQJOH PXVWEHXVHG DVIRUWKHDUJXPHQWRIWKH)\IXQFWLRQ 6HFRQGO\ZHZLOOXVHLQIRUPDWLRQ RQWKHQHZYDOXHRIWKHDOLJQLQJVWLIIQHVV&0 7KHUHIHUHQFHFXUYHIRU0]RWKDW LV XVHG LV PRUH UHDOLVWLF WKHQ WKH WKHRUHWLFDO FXUYH RI )LJ 7\SLFDOO\ WKH PRPHQWFKDQJHVLWVVLJQLQWKHODUJHUVOLSDQJOHUDQJHZKHUH)\UHDFKHVLWVSHDN :LWKWKHVDPHHTXLYDOHQWVOLSDQJOHDQGWKHQHZDOLJQLQJVWLIIQHVVZHREWDLQ WKHH[SUHVVLRQIRUWKHQHZYDOXHRIWKHDOLJQLQJWRUTXHDFFRUGLQJWRWKHVLPLODULW\ FRQFHSW
0] '
)] &0 )] &) R 0 )]R &0 R &) )] ]R
HT
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
6(0,(03,5,&$/7<5(02'(/6
6LGHIRUFHDWQRPLQDOORDG)]R
)\R ' '\R
&\
VLQ>
%\R & (\ %\R &
DUFWDQ^
%\R
DUFWDQ
`@
ZLWKVWLIIQHVVIDFWRU
% \R ' &) R &\ ' \R
SHDNIDFWRUIRUWKHVLGHIRUFHZKLFKLQJHQHUDOLVGLIIHUHQWIURPWKHRQHIRUWKH ORQJLWXGLQDOIRUFHFRQVHTXHQWO\ZHLQWURGXFH\REHVLGH[R
'\R ' \R)]R
DQGFRUQHULQJVWLIIQHVVDVIXQFWLRQRIZKHHOORDG)]
&) ' F F )]R
VLQ DUFWDQ
)] F )]R
$OLJQLQJWRUTXHDWQRPLQDOZKHHOORDG
0]R ' ']R
&]
VLQ>
%]R & (] %]R &
DUFWDQ^
DUFWDQ
%]R
`@
ZLWKVWLIIQHVVIDFWRU
% ]R ' &&0 R &] ' ]R
SHDNIDFWRUDRUHSUHVHQWLQJKDOIWKHFRQWDFWOHQJWKDWQRPLQDOORDG
']R ' F DR '\R
DQGDOLJQLQJVWLIIQHVVDVIXQFWLRQRIZKHHOORDG)]
&0 ' W&) ' F D&)
ZKHUHDSSDUHQWO\WKHSQHXPDWLFWUDLOW
FD+DOIWKHFRQWDFWOHQJWKDLVDVVXPHG
WREHSURSRUWLRQDOZLWKWKHVTXDUHURRWRIWKHZKHHOORDG
D ' DR )] )]R
7KHUHVXOWLQJDSSUR[LPDWLRQRIWKHSQHXPDWLFWUDLOWXUQVRXWWREHTXLWHDGHTXDWH DQG PLJKW EH FRQVLGHUHG WR EH XVHG LQ WKH VHW RI 0DJLF )RUPXOD 7\UH PRGHO HTXDWLRQV( WREHGHDOWZLWKODWHURQ 7KHYDOXHVRISDUDPHWHUVXVHGIRUWKHFDOFXODWLRQVWREHFRQGXFWHGKDYHEHHQ OLVWHGLQ7DEOH 7DEOH3DUDPHWHUYDOXHVXVHGLQ6HFWLRQ
)]R 1 &\
&[
F
DR P (\
([
F F F
E
P &]
:
UH
P (]
:
F
\R
[R
F
F
F
F F
F
6(0,(03,5,&$/7<5(02'(/6
:
7KHQH[WWZRLWHPVZHZLOOGHDOZLWKLVDFKDQJHLQIULFWLRQFRHIILFLHQWIURP WR
: DQGWKHLQWURGXFWLRQRIDFDPEHUDQJOH \
\R
7KHILUVWFKDQJHFDQEHKDQGOHG
: :
E\PXOWLSO\LQJWKHFXUYHVLQUDGLDOGLUHFWLRQZLWKIDFWRU
\
\R
7RJHWKHUZLWKWKH
FKDQJHLQORDGZHILQG
\ )] ) \R )]R \R
)\ '
ZLWK
HT '
HT
&) )] \R )]R &) R \ )]
7KHIULFWLRQFRHIILFLHQWPD\EHDVVXPHGWRGHSHQGRQWKHVOLSYHORFLW\9V7R
:9 PD\EHHPSOR\HGFI
PRGHOWKHVLWXDWLRQRQZHWURDGVDGHFD\LQJIXQFWLRQ
V
HJ(T( :KHQFRQVLGHULQJWKHFRPSXWHGFKDUDFWHULVWLFVRI)LJF
QG
URZ IRU
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
&)
6+\ '
&) )] &) )]
VRWKDW
VKLIWVKRXOGDPRXQWWR
6+\
ZKLFKJLYHVULVHWR
ZLWK HT
UHSODFHGE\
6+\
)RU WKH UHSUHVHQWDWLRQ RI WKH DOLJQLQJ PRPHQW DW WKH QHZ FRQGLWLRQV WKH VLWXDWLRQLVPRUHFRPSOH[:HREVHUYHIURP)LJFWKDWIRUQRWWRRODUJHVOLS DQJOHVWKHRULJLQDOFXUYH0 ]R WHQGVWRERWKVKLIWVLGHZD\VDQGXSZDUGV,IWKH VDPHHTXLYDOHQWVOLSDQJOHDQGWKXVWKHVDPHKRUL]RQWDOVKLIW DVIRUWKH IRUFHLVHPSOR\HGWKHPRPHQWZRXOGEHFRPH]HURZKHUHDOVRWKHIRUFHFXUYH FURVVHVWKH
D[LV$PRPHQWHTXDOWR&0 6+\ZRXOGDULVHDW
WKH PRPHQW VKRXOG EHFRPH HTXDO WR &0
+RZHYHU
ZLWK D SRVLWLYH FDPEHU PRPHQW
VWLIIQHVV&0 7KLVLPSOLHVWKDWDQDGGLWLRQDOYHUWLFDOVKLIWLVUHTXLUHGHTXDOWR
69] ' &0 )] % &0 )] 6+\
7KLVDGGLWLRQDOPRPHQWFRUUHVSRQGVWRWKHVRFDOOHGUHVLGXDOWRUTXH0]UZKLFK
6(0,(03,5,&$/7<5(02'(/6
300 ]
)) \ [
[kNm]
)[
[kN]
0
]
3
)\
2
0
]U
0: γ = 0 2: γ = 10o
0
1
)\R 0
]R
0
α κ
[rad] [-]
2
αHT
0
-3 -0.3
0.3
0
)LJ,QWURGXFWLRQRI D FDPEHU DQJOH LQWKHVLGHIRUFHDQGWKHDOLJQLQJWRUTXHSXUHVOLS FKDUDFWHULVWLFVDFFRUGLQJWRWKHVLPLODULW\PHWKRG7KHORQJLWXGLQDOIRUFHSXUHVOLS
)
FKDUDFWHULVWLF
]
)
]R
1
\
\R
[
[R
LVWKHPRPHQWWKDWUHPDLQVZKHQWKHVLGHIRUFHEHFRPHVHTXDOWR]HUR)RUODUJHU YDOXHV RI WKH VOLS DQJOH WKH DGGLWLRQDO PRPHQW VKRXOG WHQG WR ]HUR DV FDQ EH REVHUYHGLQ)LJF7KLVPD\EHHDVLO\DFFRPSOLVKHGE\GLYLGLQJ E\ DWHUPOLNHF
7KH GHSHQGHQFLHV RI WKH FDPEHU VWLIIQHVVHV RQ WKH YHUWLFDO ORDG PD\ EH DVVXPHGOLQHDU:HKDYH
&) ' F )]
&0 ' F E &) UH
D
ZKHUHEGHQRWHVKDOIWKHDVVXPHGO\FRQVWDQW ZLGWKRIWKHFRQWDFWSDWFKUHWKH HIIHFWLYHUROOLQJUDGLXVDQG&) WKHORQJLWXGLQDOVOLSVWLIIQHVVZKLFKLVFRQVLGHUHG WREHOLQHDUO\LQFUHDVLQJZLWKORDG
&) ' F ) ]
E
,Q)LJWKHGLIIHUHQWVWHSVOHDGLQJWRWKHFRQGLWLRQZLWKDVPDOO FDPEHU DQJOHKDYHEHHQGHPRQVWUDWHG)LUVWWKHRULJLQDOFXUYH LVPRYHGWRWKHOHIW RYHUDGLVWDQFHHTXDOWRWKHKRUL]RQWDOVKLIW \LHOGLQJWKHLQWHUPHGLDWHFXUYH 7KHQ WKH UHVLGXDO WRUTXH LV DGGHG YHUWLFDO VKLIW SRVVLEO\ PDGH D IXQFWLRQRIVOLSDQJOHE\PXOWLSO\LQJWKHVKLIWZLWKDGHFD\LQJZHLJKWLQJIXQFWLRQ DVVXJJHVWHGDERYH7KHORDG)]DQGWKHIULFWLRQFRHIILFLHQW\KDYHEHHQNHSW HTXDOWRWKHLUUHIHUHQFHYDOXHV1DQGUHVSHFWLYHO\ 7KHORQJLWXGLQDOIRUFHDWSXUHORQJLWXQDOVOLSPD\EHPRGHOOHGLQWKHVDPH
6(0,(03,5,&$/7<5(02'(/6
ZD\ DV ZH GLG IRU WKH VLGH IRUFH (TXDWLRQV VLPLODU WR KROG IRU WKH UHIHUHQFHIXQFWLRQ$QLQIOXHQFHRIFDPEHURQWKHORQJLWXGLQDOIRUFHKRZHYHU LVDVVXPHGQRWWRRFFXU,QWKHVDPHILJXUHWKHFKDUDFWHULVWLFIRU)[YV KDV EHHQGUDZQ7KHYDOXHVRIWKHUHOHYDQWDGGLWLRQDOSDUDPHWHUVIRUWKHORQJLWXGLQDO IRUFHDQGWKHPRPHQWKDYHEHHQOLVWHGLQ7DEOH :HILQDOO\KDYHWKHIROORZLQJVLPLODULW\IRUPXODHIRUSXUHVOLSFRQGLWLRQV HLWKHUDWORQJLWXGLQDORUDWODWHUDOVOLS IRUWKHORQJLWXGDOIRUFH
)[ '
[ )] ) [R )]R [R
HT
ZLWKWKHHTXLYDOHQWORQJLWXGLQDOVOLS
&) )] [R )]R &) R [ )]
HT '
DQGIRUWKHVLGHIRUFH
)\ '
\ )] ) \R )]R \R
HT
ZLWKWKHHTXLYDOHQWVOLSDQJOHFRQWDLQLQJWKHKRUL]RQWDOVKLIW
HT '
&) )] \R )]R &) R \ )]
%
&) )] &) )]
DQGIRUWKHDOLJQLQJWRUTXH
0] '
\ )] &0 )] &) R 0 \R )]R &0 R &) )] ]R
HT %
0]U
ZLWKWKHUHVLGXDOWRUTXHFRUUHVSRQGLQJWRWKHYHUWLFDOVKLIW SURYLGHGZLWK WKHUHGXFWLRQIDFWRU
0]U '
&0 )] % W )] &) )] %F
&RPELQHG6OLS&RQGLWLRQV
:HZLOOQRZDGGUHVVWKHSUREOHPRIGHVFULELQJWKHVLWXDWLRQDWFRPELQHGVOLS 7KH DQDO\VLV RI WKH EUXVK PRGHO KDV JDLQHG FRQVLGHUDEOH LQVLJKW LQWR WKH PHFKDQLVPVWKDWSOD\DUROHDQGZHZLOOXVHWKHWKHRUHWLFDOVOLSTXDQWLWLHV
[\
RU DQGWKHPDJQLWXGH DQGZHZLOODGRSWWKHFRQFHSWVLPLODU WRWKDWRIDVVHVVLQJWKHFRPSRQHQWVRIWKHUHVXOWLQJKRUL]RQWDOIRUFHDFFRUGLQJ
6(0,(03,5,&$/7<5(02'(/6
WR DQG FRQVLGHU WKH SQHXPDWLF WUDLO W DV LQ DQG LQFOXGH H[SOLFLW FRQWULEXWLRQV RI )[ WR WKH PRPHQW DV LQGLFDWHG LQ :H KDYH IRU WKH WKHRUHWLFDOVOLSTXDQWLWLHVZLWKWKH
['
(' \
VKLIWGXHWRFDPEHULQFOXGHG D
% (
WDQ
E
%
DQG
('
[
('
%
%
ZLWK
( \
&) ) ] &) )]
,Q WKH HQVXLQJ WKHRU\ ZH ZLOO PDNH XVH RI WKH WKHRUHWLFDO VOLS TXDQWLWLHV DE ZKLFKGRHVSURGXFHWKHVORSHLQWKHFRQVWDQW FXUYHVLQWKH)\YV)[ GLDJUDPDW)[
,WVKRXOGEHQRWHGKRZHYHUWKDWH[SHULHQFHKDVVKRZQWKDW
(
WKHXVHRISUDFWLFDOVOLSTXDQWLWLHVV[DQGV\ WKDWLV
DQGWDQ
(
ZKLFKPD\EH
REWDLQHGDJDLQE\RPLWWLQJWKHGHQRPLQDWRUVLQDE PD\DOUHDG\JLYHYHU\ JRRGUHVXOWVDVZLOOEHLQGLFDWHGODWHURQLQ)LJ1RWHWKDWZKHQDSSURDFKLQJ ZKHHOORFN
[
647KLVFDOOVIRUDQDUWLILFLDOOLPLWDWLRQE\DGGLQJDVPDOOSRVLWLYH
TXDQWLW\WRWKHGHQRPLQDWRURID %HFDXVHZHDUHGHDOLQJZLWKDQLQJHQHUDOQRQLVRWURSLFW\UHZHKDYHSXUH ORQJLWXGLQDODQGODWHUDOVOLSFKDUDFWHULVWLFVWKDWDUHQRWLGHQWLFDOFI)LJ 6WLOO ZH ZLOO WDNH WKH ODWHUDO DQG ORQJLWXGLQDO FRPSRQHQWV EXW WKHQ RI WKH UHVSHFWLYHSXUHVOLSFKDUDFWHULVWLFV)\RDQG)[R,IZHZDQWWRXVHWKHWKHRUHWLFDO ZHPXVWKDYHDYDLODEOHWKHSXUHVOLSFKDUDFWHULVWLFVZLWK
VOLSTXDQWLWLHV \
[
DQG
DVDEVFLVVD7KLVFDQEHDFFRPSOLVKHGE\ILWWLQJWKHRULJLQDOGDWDDIWHUKDYLQJ
FRPSXWHGWKHYDOXHVRI WUDQVIRUPLQJ 0 ]R
\
LQWRWDQ
[
IRUHDFKYDOXHRI E\XVLQJD DQGVLPLODUIRU
DQGGHQRWHWKHUHVXOWLQJIXQFWLRQV)[R
[
)\R
\
DQG
2UVLPSOHUE\UHSODFLQJLQWKHDOUHDG\DYDLODEOHSXUHVOLSIRUFHDQG
PRPHQW IXQFWLRQV H[SUHVVHG LQ WHUPV RI WKH SUDFWLFDO VOLS TXDQWLWLHV WKH DUJXPHQW LQUDGLDQV E\WDQ
SUREDEO\DFFHSWDEOHDSSUR[LPDWLRQ DQG E\
WKHH[SUHVVLRQJLYHQEHORZGHULYHGIURP(TD
'
[
&
[
ZKLFKZRXOGEHVXFFHVVIXOLIWKH)[FKDUDFWHULVWLFKDVEHHQILWWHGIRUWKHZKROH UDQJHRISRVLWLYHGULYLQJ DQGQHJDWLYHEUDNLQJ YDOXHVRI 2WKHUZLVHLIRQO\
6(0,(03,5,&$/7<5(02'(/6
)\R
)4 ) )[R
)\
σ *
)[
σ\*
9V
wheel lock
64 )
σ * (
9V (= 9 )
σ[
)LJ &RQVWUXFWLRQRIWKHUHVXOWLQJFRPELQHGVOLSIRUFHIURPWKHSXUHVOLSFKDUDFWHULVWLFV DFFRUGLQJWR(TV
WKHEUDNLQJVLGHLVDYDLODEOH DQG
[
DQGWKHGULYLQJVLGHLVPRGHOOHGDVWKH
PLUURULPDJHRIWKHEUDNLQJVLGHRQHPLJKWEHWWHUWDNH
'
[
%_
D
[_
,I WKH WKXV REWDLQHG IXQFWLRQ RI
)
[
LV SORWWHG YHUVXV
[
WKH UHVXOWLQJ FXUYH
EHFRPHVV\PPHWULF+RZHYHULIWKHQWKHDEFVLVVDLVWUDQVIRUPHGLQWR E\XVLQJ WKHUHVXOWLQJFXUYHWXUQVRXWWREHFRPHDV\PPHWULFZLWKWKHEUDNLQJVLGH LGHQWLFDOWRWKHFKDUDFWHULVWLFZHVWDUWHGRXWZLWK7KLVDV\PPHWU\ZDVDOUHDG\ IRXQGWRRFFXUZLWKWKHEUXVKPRGHOGLVFXVVHGLQ&KDSWHU 1H[WZHVKRXOGUHDOLVHWKDWWKURXJK FDPEHUKDVEHHQDFFRXQWHGIRU DQGWKDWDVDFRQVHTXHQFH
\
LVWREHUHSODFHGE\
( \
GHILQHGE\(TE 7KH
)
UHVXOWLQJIRUFHDQGPRPHQWIXQFWLRQVDUHGHQRWHGE\
[R
[
)
\R
$V LOOXVWUDWHG LQ )LJ WKH µFRPSRQHQWV¶ QRZ EHFRPH ZLWK
( \
0
DQG
(
]R
( \
GHQRWLQJ WKH
PDJQLWXGHRIWKHWKHRUHWLFDOVOLSYHFWRU
)[ '
[
(
)
( [R
( \ (
)\ '
)\R
(
7KH ILJXUH VKRZV KRZ WKH IRUFH YHFWRU DULVHV IURP WKH LQGLYLGXDO SXUH VOLS FKDUDFWHULVWLFV$WVPDOOVOLSLQJHQHUDOWKHYHFWRUGRHVQRWUXQRSSRVLWHZLWK
9 $WZKHHOORFNKRZHYHUWKHIRUFHYHFWRUGRHV
UHVSHFWWRWKHVOLSVSHHGYHFWRU
V
6(0,(03,5,&$/7<5(02'(/6
UXQRSSRVLWHZLWKUHVSHFWWRWKHVOLSVSHHGYHFWRUEHFDXVHLWLVDVVXPHGKHUHWKDW WKHOHYHOVRIWKHLQGLYLGXDOFKDUDFWHULVWLFVDV\PSWRWHV DUHWKHVDPHIRUWKHVOLS DSSURDFKLQJ LQILQLW\ LQ WKH 0DJLF )RUPXOD ZH VKRXOG KDYH
[R
&
VLQò
[
\R
VLQò
&
\
$WODUJHVOLSWKHWKHRU\DSSDUHQWO\IDLOVWRSURSHUO\DFFRXQWIRU
WKHFRQWULEXWLRQRIWKHFDPEHUDQJOH$WZKHHOORFNRQHZRXOGH[SHFWWKHVLGH IRUFHWREHFRPH]HURDWYDQLVKLQJVOLSDQJOH7KLVZLOOQRWH[DFWO\EHWKHFDVHLQ WKH PRGHO GXH WR WKH HTXLYDOHQW WKHRUHWLFDO VLGH VOLS GHILQHG DFFRUGLQJ WR (TE ZLWK
(
JLYHQE\
)RUWKHDOLJQLQJWRUTXHZHPD\GHILQH
0 ] ' 0] % 0]U & N
F DR ) ) & F DR ) [ & F E )[ ) ]R [ \
ZLWKWKHILUVWWHUPGLUHFWO\DWWULEXWDEOHWRWKHVLGHIRUFH7KLVWHUPFDQEHZULWWHQ DV
0] ' &W N
)\
( @
%\FRQVLGHULQJ LWWXUQVRXWWKDW EHFRPHV
0] ' N
( \ (
0]R N
(
7KH ODVW WKUHH WHUPV RI (T UHVXOW IURP WKH PRPHQW H[HUWHG E\ WKH ORQJLWXGLQDOIRUFHEHFDXVHRIWKHPRPHQWDUPWKDWDULVHVWKURXJKWKHVLGHIRUFH LQGXFHGGHIOHFWLRQDSRVVLEO\LQLWLDORIIVHWRIWKHOLQHRIDFWLRQDQGWKHVLGHZD\V UROOLQJRIWKHW\UHFURVVVHFWLRQGXHWRFDPEHU)RUWKHVDNHRIVLPSOLFLW\WKHVH HIIHFWVDUHDVVXPHGQRWWREHLQIOXHQFHGE\WKHZKHHOORDG
)
:LWKWKHVLPLODULW\H[SUHVVLRQVIRUWKHSXUHVOLSIRUFHV
[\R
0N
DQGPRPHQW
]R
ZHILQDOO\REWDLQWKHIROORZLQJIRUPXODHIRUFRPELQHGVOLSFRQGLWLRQV)RUWKH ORQJLWXGDOIRUFH
)[ '
[
(
[ )] ) [R )]R [R
[ HT
ZLWKWKHHTXLYDOHQWVOLS
[ HT '
&) ) ] [R ) ]R &) R [ )]
(
DQGIRUWKHVLGHIRUFH
)\ '
( \ (
\ ) ] ) \R ) ]R \R
\ HT
6(0,(03,5,&$/7<5(02'(/6
)\ 2
8
6
[kN]
4
20
1
)] =2kN γ =5o
2
o
12
0
0
)[
α = -2o
-1 -2
0
[kN] 2
)LJD &RUQHULQJ IRUFH YHUVXV ORQJLWXGLQDO IRUFH DW D QHZ ORDG DQG FDPEHU DQJOHDV FRPSXWHGZLWKWKHVLPLODULW\PHWKRG
α = 4o 6
8 12
0]
)] =2kN γ =5o
o
20
50 o
o
2
-2
0
)[
[Nm]
-50 0
-2
[kN] 2
)LJE $OLJQLQJ WRUTXH YHUVXV ORQJLWXGLQDO IRUFH DW D QHZ ORDG DQGFDPEHUDQJOHDV FRPSXWHGZLWKWKHVLPLODULW\PHWKRG
ZLWKWKHHTXLYDOHQWVOLS
\ HT '
&) ) ] \R ) ]R &) R \ )]
(
DQGIRUWKHDOLJQLQJWRUTXH
0] '
\ )] &0 )] &) R 0 \R )]R &0 R &) )] ]R FD & R )[ )\ & F DR ) [ & F E )[ ) ]R ( \ (
N
\ HT %
0]U%
6(0,(03,5,&$/7<5(02'(/6
ZLWKWKHUHVLGXDOWRUTXH
0 ]U '
&0 ) ] % W )] &) )]
%F
\ HT
2Q EHKDOI RI H[DPSOH FDOFXODWLRQV WKH WKUHH DGGLWLRQDO QRQGLPHQVLRQDO SDUDPHWHUVFFDQGFKDYHEHHQJLYHQYDOXHVOLVWHGLQ7DEOH )RUWKHFDVHRIDORDGGLIIHUHQWIURPWKHUHIHUHQFHYDOXHDQGWKHSUHVHQFHRI DFDPEHUDQJOHWKHFRPELQHGVOLSFKDUDFWHULVWLFVIRUWKHW\UHVLGHIRUFHDQGIRU WKH DOLJQLQJ WRUTXH KDYH EHHQ DVVHVVHG E\ XVLQJ WKH DERYH HTXDWLRQV 7KH UHVXOWLQJGLDJUDPVZLWKVOLSDQJOH
XVHGDVSDUDPHWHUKDYHEHHQSUHVHQWHGLQ
WKH)LJVDDQGE'XHWRWKHXVHRIWKHWKHRUHWLFDOVOLSTXDQWLWLHVDE ZHILQGWKDWWKHFXUYHVRI)\VKRZWKHVOLJKWVORSHQHDU)[
DOVRREVHUYHGWR
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
)
\
)
]
1
test data Magic Form. + similarity method )]= 7000N
α =o 9.8
4.7
α
= 9.8
o
1
4.7
)\ )]
1.9
1.9 o
-1.9
0
0
0
0.2
0.4
)[
0.6
-κ
0.8
1 o
-1.9
)] α = "1.9o
1
)[ )]
o
4.7
-1
-0.8
-0.6
-0.4
-0.4 -0.2
0
o
9.8
)LJ
7\UH
FRPELQHG
VOLS
FKDUDFWHULVWLFVDV
REWDLQHGIURPPHDVXUHPHQWVZLWKWKH'HOIW W\UH WHVW WUDLOHU FRPSDUHG ZLWK WKH FXUYHV
0
FDOFXODWHG ZLWK WKH DLG RI WKH VLPLODULW\
0
0.2
0.4
0.6
-κ
0.8
1
PHWKRG
6(0,(03,5,&$/7<5(02'(/6
H[WUHPH FDVHV LQ SXUH VOLS VLWXDWLRQV LQ WKH OLQHDULVHG FRPELQHG VOLS VLWXDWLRQ VPDOO
DQG
DQG LQ WKH FDVH RI D ORFNHG ZKHHO
$V
LOOXVWUDWHGLQ)LJWKHPRGHOFRUUHFWO\VKRZVWKDWZKHQWKHZKHHOLVORFNHGWKH UHVXOWLQJIRUFHDFWVLQDGLUHFWLRQRSSRVLWHWRWKHVOLSVSHHGYHFWRU
9 LILQWKH V
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
DQG WDQ
KDYH EHHQ HPSOR\HG $ UHODWLYHO\ JRRG DJUHHPHQW LV
DFKLHYHGIRUWKLVSDVVHQJHUFDUW\UHLQWKHFRPELQHGVOLSVLWXDWLRQ
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
)]2UDOWHUQDWLYHO\ZHXVHRQO\WKDWSDUWRIWKHFKDUDFWHULVWLFWKDWOLHVLQ
EHWZHHQWKHWZRSHDNV7KLVHQWDLOVWKDWWKH)\YV)[FXUYHVDUHVLQJOHYDOXHGLQ WKH)[UDQJHHPSOR\HG)XUWKHUPRUHLWLVDVVXPHGWKDWWKHIULFWLRQFRHIILFLHQWLV WKHVDPHIRUORQJLWXGLQDODQGODWHUDOGLUHFWLRQVDQGLVGHQRWHGZLWK 7KH REYLRXV PDLQ HIIHFW RI WKH LQWURGXFWLRQ RI )[ RQ WKH VLGH IRUFH LV WKH ORZHULQJRIWKHPD[LPXPVLGHIRUFHWKDWFDQEHJHQHUDWHG)RUWKLVWRUHDOLVHWKH ULJKWKDQG H[SUHVVLRQV RI (TV DQG DUH WR EH PXOWLSOLHG ZLWK WKH IDFWRU
)] & )[ ' [ ) ]
ZKLOHWKHH[SUHVVLRQ LVWREHGLYLGHGE\WKHVDPHIDFWRU7KHFRUQHULQJ
6(0,(03,5,&$/7<5(02'(/6
DQG DOLJQLQJ VWLIIQHVVHV UHPDLQ XQDIIHFWHG WKURXJK WKLV RSHUDWLRQ 7KHVH VWLIIQHVVHVKRZHYHUGRGHSHQGRQWKHORQJLWXGLQDOIRUFHDVEHFRPHVREYLRXV IURPWKHFXUYHVRI)LJWKDWEHORQJWRVPDOOVLGHVOLSDQJOHV7KHIROORZLQJ IXQFWLRQVPD\VHUYHDVDFUXGHDSSUR[LPDWLRQRIWKHDFWXDOUHODWLRQVKLSV
&) ) ])[ '
&) )] & )] %
[
)] & )[
ZLWK
[
) ' & [ ) ]
Q
Q
D
'HSHQGLQJRQWKHXVHU¶VGHVLUHRQHPD\FKRRVHIRUQDYDOXHLQWKHUDQJHRIWR PRUHRUOHVVFXUYHGFKDUDFWHULVWLF&) )[ )RUWKHDOLJQLQJVWLIIQHVVZHPD\ XVHWKHIRUPXOD
&0 ) ])[ '
[
&0 )]
DQGVLPLODUIRUWKHFDPEHUVWLIIQHVV
&) ) ])[ '
[
&) )]
7KH FDPEHU PRPHQW VWLIIQHVV PD\ EH GLVUHJDUGHG 7KH ORDG GHSHQGHQFLHV &)0 )] DQG&) )] RIWKHIUHHO\UROOLQJW\UHDOUHDG\DSSHDUHGLQ(TV ,W PD\ EH DVFHUWDLQHG WKDW WKH SUHVHQW PRGHO PDNHV VXUH WKDW WKH FRUQHULQJ VWLIIQHVV YDQLVKHV ZKHQ )[ ZKHHO ORFN
6 ) ZKHHO GULYH VSLQ 64 ZKLOH DW )
]
[
)]
WKH VLGH VOLS VWLIIQHVV HTXDOV )] ZKLFK LV FRUUHFW 7KH
IXQFWLRQDO UHODWLRQVKLSV IRU )[ GR QRW SOD\ D UROH DQ\PRUH DQG WKH IULFWLRQFRHIILFLHQW\KDVEHHQUHSODFHGE\7KHLPSRVHGORQJLWXGLQDOIRUFH VKRXOGEHNHSWZLWKLQWKHERXQGDULHV)]FRV
# ) # )
[
]
7KHUHVXOWLQJHTXDWLRQVIRUWKHVLGHIRUFHDQGWKHDOLJQLQJWRUTXHZLWKWKH ORQJLWXGLQDOIRUFHVHUYLQJDVRQHRIWKHLQSXWTXDQWLWLHVQRZUHDG )RUWKHVLGHIRUFH
)\ '
)] ) [ R )]R \R
HT
ZLWKWKHHTXLYDOHQWVOLSDQJOH
HT '
&) ) ])[ R )]R )] &) R [
DQGIRUWKHDOLJQLQJWRUTXH
%
&) )])[ &) )])[
6(0,(03,5,&$/7<5(02'(/6
0] '
)] &0 )]) [ &) R 0 [ R ) ]R &) )]) [ ]R &0 R FD & R ) [ )\ & F DR )[ & F E ) [ )]R
HT
% 0]U%
ZKHUHWKHODVWWHUPVKDYHEHHQWDNHQIURP DQGWKHUHVLGXDOWRUTXHUHDGV
0 ]U '
&0 )] % W )] &) ) ] [ %F
([HUFLVHJLYHQDWWKHHQGRI6HFWLRQDGGUHVVHVWKHSUREOHPRIDVVHVVLQJ WKHVLGHIRUFHFKDUDFWHULVWLFVXVLQJWKHVLPLODULW\WHFKQLTXHZLWKWKHORQJLWXGLQDO IRUFHFRQVLGHUHGDVRQHRIWKHLQSXWTXDQWLWLHV ,Q6HFWLRQWKH0DJLF)RUPXODW\UHPRGHOZLOOEHWUHDWHGLQGHWDLO7KLV FRPSOH[ PRGHO LV FRQVLGHUDEO\ PRUH DFFXUDWH DQG ZLOO DJDLQ HPSOR\ WKH ORQJLWXGLQDOVOLS
DVLQSXWYDULDEOH
7KH0DJLF)RUPXOD7\UH0RGHO
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µ'HOIW 7\UH¶ WKH RULJLQDO GHVFULSWLRQ RI WKH DOLJQLQJ WRUTXH LV DOWHUHG WR DFFRPPRGDWHDUHODWLYHO\VLPSOHSK\VLFDOO\EDVHGFRPELQHGVOLSH[WHQVLRQ7KH SQHXPDWLF WUDLO LV LQWURGXFHG DV D EDVLV WR FDOFXODWH WKLV PRPHQW DERXW WKH YHUWLFDO D[LV FI 3DFHMND $ FRPSOHWH OLVWLQJ RI WKH PRGHO LV JLYHQ LQ 6HFWLRQ ,Q 6HFWLRQ WKH PRGHO LV IXUWKHU H[WHQGHG E\ LQWURGXFLQJ IRUPXODHIRUWKHGHVFULSWLRQRIWKHVLWXDWLRQDWODUJHFDPEHUDQGWXUQVOLS
6(0,(03,5,&$/7<5(02'(/6
0RGHO'HVFULSWLRQ
7KHJHQHUDOIRUPRIWKHIRUPXODWKDWKROGVIRUJLYHQYDOXHVRIYHUWLFDOORDG DQGFDPEHUDQJOHUHDGV
\ ' ' VLQ>& DUFWDQ ^%[ & ( %[ & DUFWDQ %[ `@
ZLWK
<; ' \[ % 69 [ ' ; % 6+
ZKHUH <
RXWSXWYDULDEOH)[)\RUSRVVLEO\0]
;
LQSXWYDULDEOHWDQ
RU
DQG %
VWLIIQHVVIDFWRU
&
VKDSHIDFWRU
'
SHDNYDOXH
(
FXUYDWXUHIDFWRU
6+
KRUL]RQWDOVKLIW
69
YHUWLFDOVKLIW
7KH 0DJLF )RUPXOD \[ W\SLFDOO\ SURGXFHV D FXUYH WKDW SDVVHV WKURXJK WKH RULJLQ [
\
UHDFKHV D PD[LPXP DQG VXEVHTXHQWO\ WHQGV WR D KRUL]RQWDO
DV\PSWRWH)RUJLYHQYDOXHVRIWKHFRHIILFLHQWV%&'DQG(WKHFXUYHVKRZV DQDQWLV\PPHWULFVKDSHZLWKUHVSHFWWRWKHRULJLQ7RDOORZWKHFXUYHWRKDYH DQRIIVHWZLWKUHVSHFWWRWKHRULJLQWZRVKLIWV6+DQG69KDYHEHHQLQWURGXFHG $QHZVHWRIFRRUGLQDWHV<; DULVHVDVVKRZQLQ)LJ7KHIRUPXODLVFDSDEOH
6+ ' 69
\
<
[P
\([) \D
arctan(%&')
' sinβ [
[ ;
π
&π
[ )LJ
β
β= &
arctan(%[)
&XUYHSURGXFHGE\WKHRULJLQDOVLQHYHUVLRQRIWKH0DJLF)RUPXOD(TV 7KH PHDQLQJRIFXUYHSDUDPHWHUVKDYHEHHQLQGLFDWHG
6(0,(03,5,&$/7<5(02'(/6
RI SURGXFLQJ FKDUDFWHULVWLFV WKDW FORVHO\ PDWFK PHDVXUHG FXUYHV IRU WKH VLGH IRUFH)\DQGLIGHVLUHGDOVRIRUWKHDOLJQLQJWRUTXH0] DQGIRUWKHIRUHDQGDIW IRUFH)[DVIXQFWLRQVRIWKHLUUHVSHFWLYHVOLSTXDQWLWLHVWKHVOLSDQJOH
DQGWKH
ORQJLWXGLQDOVOLS ZLWKWKHHIIHFWRIORDG)]DQGFDPEHUDQJOH LQFOXGHGLQWKH SDUDPHWHUV )LJXUHLOOXVWUDWHVWKHPHDQLQJRIVRPHRIWKHIDFWRUVE\PHDQVRIDW\SLFDO VLGHIRUFHFKDUDFWHULVWLF2EYLRXVO\FRHIILFLHQW'UHSUHVHQWVWKHSHDNYDOXHZLWK
$ DQGWKHSURGXFW%&'FRUUHVSRQGVWR
UHVSHFWWRWKHFHQWUDO[D[LVDQGIRU& WKHVORSHDWWKHRULJLQ[
\
7KHVKDSHIDFWRU&FRQWUROVWKHOLPLWVRIWKH
UDQJHRIWKHVLQHIXQFWLRQDSSHDULQJLQIRUPXOD DQGWKHUHE\GHWHUPLQHVWKH VKDSHRIWKHUHVXOWLQJFXUYH7KHIDFWRU%LVOHIWWRGHWHUPLQHWKHVORSHDWWKH RULJLQDQGLVFDOOHGWKHVWLIIQHVVIDFWRU7KHIDFWRU(LVLQWURGXFHGWRFRQWUROWKH FXUYDWXUHDWWKHSHDNDQGDWWKHVDPHWLPHWKHKRUL]RQWDOSRVLWLRQRIWKHSHDN )URPWKHKHLJKWVRIWKHSHDNDQGRIWKHKRUL]RQWDODV\PSWRWHWKHVKDSHIDFWRU &PD\EHFRPSXWHG
&'
& DUFVLQ
\D '
)URP%DQG&DQGWKHORFDWLRQ[PRIWKHSHDNWKHYDOXHRI(PD\EHDVVHVVHG
('
%[P & WDQ^ & ` %[P & DUFWDQ%[P
LI
& !
7KHRIIVHWV6+DQG69DSSHDUWRRFFXUZKHQSO\VWHHUDQGFRQLFLW\HIIHFWVDQG SRVVLEO\WKHUROOLQJUHVLVWDQFHFDXVHWKH)\DQG)[FXUYHVQRWWRSDVVWKURXJKWKH RULJLQ:KHHOFDPEHUPD\JLYHULVHWRDFRQVLGHUDEOHRIIVHWRIWKH)\YV FXUYHV 6XFKDVKLIWPD\EHDFFRPSDQLHGE\DVLJQLILFDQWGHYLDWLRQIURPWKHSXUHDQWL V\PPHWULFVKDSHRIWKHRULJLQDOFXUYHFI)LJFG 7RDFFRPPRGDWHVXFK DQ DV\PPHWU\ WKH FXUYDWXUH IDFWRU ( LV PDGH GHSHQGHQW RI WKH VLJQ RI WKH DEVFLVVD[
( ' (R % ( @ VJQ[
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
6(0,(03,5,&$/7<5(02'(/6
& = 0.8
\'
0.8
( = -100, -10, -2, 0, 0.7, 1
0.4 0.2
(=2
0.6
( = -100, -10, -2, 0, 0.7, 1
0.4
(=2
0.2 0
0
0
1
0.8 0.6
& = 1.0
\'
1
2
4
6
8
10
0
2
4
& = 1.3
\'
8
10
& = 1.6
\'
1
6 %& [
%& [
1
0.8
( = -100, -10, -2, 0, 0.7, 1
0.8
0.6
(=2
0.6
0.4
0.4
( = -100, -10, -2, 0, 0.7, 1
0.2
0.2
(=2
0
0
2
4
6
8
0
10
0
2
4
%& [
( = -100, -10, -2, 0, 0.7, 1 & = 2.0
\'
8
10
& = 2.4
\'
1
6 %& [
1
0.8 0.6
0
0.4
(=2
0.2 0
( = -100, -10, -2, 0, 0.7, 1 (=2
-1 0
2
4
6
8
10
0
%& [
& (
WR(T 1RWHWKDWDYDOXHRI
'
4
6
8
10
%& [
)LJ 7KHLQIOXHQFHRIWKHVKDSHIDFWRUV
YDOXHV
2
(
DQG
RQWKHDSSHDUDQFHRIWKHFXUYHDFFRUGLQJ
!GRHVQRWSURGXFHUHDOLVWLFFXUYHVZLWKSHDN
LQWKHDUJXPHQWRIWKHDUFWDQIXQFWLRQ7KHPRGLILHGIXQFWLRQUHDGV
\ ' ' VLQ>& DUFWDQ ^%[ & ( %[ & DUFWDQ %[ % + DUFWDQ %[`@
)LJXUHGHPRQVWUDWHVWKHHIIHFWRIWKHQHZFRHIILFLHQW+7RRODUJHYDOXHV PD\JLYHULVHWRDQXSZDUGFXUYDWXUHRIWKHFXUYHQHDUWKHRULJLQDVZRXOGDOVR
6(0,(03,5,&$/7<5(02'(/6
\'
& = 1.0, (=-2
\ '
& = 1.1, (=-2
1
0
-1
-1 2
+ = 0, 1, 2, 4, 8
0
0
1
& = 1.2, (=-2
1
+ = 0, 1, 2, 4, 8
+ = 0, 1, 2, 4, 8
0
\'
1
-1 0
3
1
2
3
0
1
2
% & [
%& [
3
%&[
)LJ 6KDUSQHVVRIFXUYHVQHDUWKHSHDNPD\EHLQFUHDVHGE\LQWURGXFLQJDGGLWLRQDOWHUP
+
ZLWKDGGLWLRQDOVKDUSQHVVIDFWRU
DFFRUGLQJWR(T
RFFXUDWODUJHQHJDWLYHYDOXHVRI(FIWKHORZHUGLDJUDPVRI)LJ ,QWKH HQVXLQJWH[WZHZLOOQRWXVHWKLVDGGLWLRQDOFRHIILFLHQW+ ,WPD\EHIXUWKHUPRUHRILQWHUHVWWRQRWHWKDWWKHSRVVLEO\DZNZDUGIXQFWLRQ DUFWDQ[ PD\ EH UHSODFHG E\ WKH SRVVLEO\ IDVWHU DQG DOPRVW LGHQWLFDO SVHXGR DUFWDQIXQFWLRQSVDWDQ[
[D_[_ ^E_[_D[ `ZLWKD
DQGE
7KHYDULRXVIDFWRUVDUH IXQFWLRQVRIQRUPDOORDGDQGZKHHOFDPEHUDQJOH 6HYHUDOSDUDPHWHUVDSSHDULQWKHVHIXQFWLRQV$VXLWDEOHUHJUHVVLRQWHFKQLTXHLV XVHG WR GHWHUPLQH WKHLU YDOXHV IURP PHDVXUHG GDWD DFFRUGLQJ WR D TXDGUDWLF DOJRULWKPIRUWKHEHVWILWFI2RVWHQDQG%DNNHU 2QHRIWKHLPSRUWDQW IXQFWLRQDOUHODWLRQVKLSVLVWKHYDULDWLRQRIWKHFRUQHULQJVWLIIQHVVDOPRVWH[DFWO\ JLYHQE\WKHSURGXFWRIWKHFRHIILFLHQWV%\&\DQG'\RIWKHVLGHIRUFHIXQFWLRQ %&'\
.\
M) M
%&'\ ' S
\
DWWDQ
6+ ZLWK)]DQG
) S @ @ & S
VLQ > DUFWDQ ]
)RU]HURFDPEHU WKHFRUQHULQJVWLIIQHVVDWWDLQVLWVPD[LPXPSDW)]
%&'
) \
\
S,Q
γ
S
1
α
0
S
2
)]
)LJ&RUQHULQJVWLIIQHVVYHUVXVYHUWLFDOORDGDQGWKHLQIOXHQFHRIZKHHOFDPEHU(T
6(0,(03,5,&$/7<5(02'(/6
)\
0]
α W)
\
0]U
α
6
+I W
α
6+W
)LJ 7KHDOLJQLQJWRUTXHFKDUDFWHULVWLFFRPSRVHGRIDSDUWGLUHFWO\DWWULEXWHGWRWKHVLGH IRUFHDQGDSDUWGXHWKHVRFDOOHGUHVLGXDOWRUTXHGXHWRW\UHFRQLFLW\DQGFDPEHU
)LJWKHEDVLFUHODWLRQVKLSKDVEHHQGHSLFWHG$SSDUHQWO\IRUDFDPEHUHG ZKHHOWKHFRUQHULQJVWLIIQHVVGHFUHDVHVZLWKLQFUHDVLQJ_ _1RWHWKHGLIIHUHQFHLQ FXUYDWXUH OHIW DQG ULJKW RI WKH FKDUDFWHULVWLFV DW ODUJHU YDOXHV RI WKH FDPEHU DQJOH7RDFFRPSOLVKWKLVDVSOLW(DFFRUGLQJWR(T KDVEHHQHPSOR\HG :H UHIHU WR 6HFWLRQ IRU D FRPSOHWH OLVWLQJ RI WKH IRUPXODH +HUH QRQ GLPHQVLRQDOSDUDPHWHUVKDYHEHHQLQWURGXFHG)RUH[DPSOHWKHSDUDPHWHUVLQ ZLOOEHFRPHS
)]RS.\S
)]RS.\DQGS
S.\ZLWK)]RGHQRWLQJWKH
QRPLQDOZKHHOORDG 7KHDOLJQLQJWRUTXH0]FDQQRZEHREWDLQHGE\PXOWLSO\LQJWKHVLGHIRUFH)\ ZLWK WKH SQHXPDWLF WUDLO W DQG DGGLQJ WKH XVXDOO\ VPDOO H[FHSW ZLWK FDPEHU UHVLGXDOWRUTXH0]UFI)LJ :HKDYH
0] ' & W @ )\ % 0 ]U
7KHSQHXPDWLFWUDLOGHFD\VZLWKLQFUHDVLQJVLGHVOLSDQGLVGHVFULEHGDVIROORZV
W
W
' 'W FRV > &W DUFWDQ ^ % W W & (W %W W & DUFWDQ %W
W `@
ZKHUH
W
'
WDQ
% 6 +W
6(0,(03,5,&$/7<5(02'(/6
<
+
6
\
( )
\ [ '
\
D
; [
β
[
R
&π 2
[
2 %&
'
cosβ [
β = & arctan(%[)
)LJ
&XUYHSURGXFHGE\WKHFRVLQHYHUVLRQRIWKH0DJLF)RUPXOD(T 7KHPHDQLQJ RIFXUYHSDUDPHWHUVKDYHEHHQLQGLFDWHG
7KHUHVLGXDOWRUTXHVKRZLQJDVLPLODUGHFD\
0]U ZLWK
U
'
U
' 'U FRV > DUFWDQ % U
WDQ
U @
% 6+I
,WLVVHHQWKDWERWKSDUWVRIWKHPRPHQWDUHPRGHOOHGXVLQJWKH0DJLF)RUPXOD EXWLQVWHDGRIWKHVLQHIXQFWLRQWKHFRVLQHIXQFWLRQLVHPSOR\HG,QWKDWZD\D KLOOVKDSHGFXUYHLVSURGXFHG7KHSHDNVDUHVKLIWHGVLGHZD\V 7KHUHVLGXDOWRUTXHLVDVVXPHGWRDWWDLQLWVPD[LPXP'UDWWKHVOLSDQJOH ZKHUHWKHVLGHIRUFHEHFRPHVHTXDOWR]HUR7KLVLVDFFRPSOLVKHGWKURXJKWKH KRUL]RQWDOVKLIW6+I7KHSHDNRIWKHSQHXPDWLFWUDLORFFXUVDWWDQ
6+W7KLV
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
\D '
RIWKHSRLQWZKHUHWKHFXUYHLQWHUVHFWVWKH[D[LV7KHIROORZLQJIRUPXODHKROG
&'
('
DUFFRV
%[R & WDQ^ & ` %[ R & DUFWDQ%[R
LI
& !
6(0,(03,5,&$/7<5(02'(/6
,QFDVHRIWKHSRVVLEOHSUHVHQFHRIODUJHFDPEHUDQJOHVPRWRUF\FOHV LWPD\ EHEHWWHUWRXVHLQ WKHVLGHIRUFH)\WKDWZRXOGDULVHDW
$OVRWKHVLGH
IRUFHIXQFWLRQ DQGWKHFRUQHULQJVWLIIQHVVIXQFWLRQ PD\EHPRGLILHG WREHWWHUDSSUR[LPDWHODUJHFDPEHUUHVSRQVHIRUPRWRUF\FOHW\UHVFI'H9ULHV D DQG6HFIRUDIXOOOLVWLQJRIHTXDWLRQV:HUHIHUWR6HFWLRQIRU WKH GLVFXVVLRQ RI WKH PRGHO H[WHQVLRQ IRU ODUJHU FDPEHU DQG WXUQ VOLS SDWK FXUYDWXUH DOVR DSSOLFDEOH LQ FDVH RI FRPELQHG VOLS ZLWK EUDNLQJ RU GULYLQJ IRUFHV ,QWKHSDSHURI3DFHMNDDQG%DNNHU WKHW\UH¶VUHVSRQVHWRFRPELQHG VOLSZDVPRGHOOHGE\XVLQJSK\VLFDOO\EDVHGIRUPXODH$QHZHUPRUHHIILFLHQW ZD\LVSXUHO\HPSLULF7KLVPHWKRGZDVGHYHORSHGE\0LFKHOLQDQGSXEOLVKHG E\%D\OH)RULVVLHUDQG/DIRQ ,WGHVFULEHVWKHHIIHFWRIFRPELQHGVOLSRQ WKH ODWHUDO IRUFH DQG RQ WKH ORQJLWXGLQDO IRUFH FKDUDFWHULVWLFV :HLJKWLQJ IXQFWLRQV*KDYHEHHQLQWURGXFHGZKLFKZKHQPXOWLSOLHGZLWKWKHRULJLQDOSXUH VOLSIXQFWLRQV SURGXFHWKHLQWHUDFWLRQHIIHFWVRI RQ)\DQGRI
RQ)[
7KHZHLJKWLQJIXQFWLRQVKDYHDKLOOVKDSH7KH\WDNHWKHYDOXHRQHLQWKHVSHFLDO FDVHRISXUHVOLS RU HTXDOWR]HUR :KHQIRUH[DPSOHDWDJLYHQVOLSDQJOH DIURP]HURLQFUHDVLQJEUDNHVOLSLVLQWURGXFHGWKHUHOHYDQWZHLJKWLQJIXQFWLRQ IRU)\PD\ILUVWVKRZDVOLJKWLQFUHDVHLQPDJQLWXGHEHFRPLQJODUJHUWKDQRQH EXW ZLOO VRRQ UHDFK LWV SHDN DIWHU ZKLFK D FRQWLQXRXV GHFUHDVH IROORZV 7KH FRVLQHYHUVLRQRIWKH0DJLF)RUPXODLVXVHGWRUHSUHVHQWWKHKLOOVKDSHGIXQFWLRQ
* ' ' FRV>& DUFWDQ %[ @
+HUH * LV WKH UHVXOWLQJ ZHLJKWLQJ IDFWRU DQG [ LV HLWKHU
RU WDQ
SRVVLEO\
VKLIWHG 7KHFRHIILFLHQW'UHSUHVHQWVWKHSHDNYDOXHVOLJKWO\GHYLDWLQJIURPRQH LIDKRUL]RQWDOVKLIWRIWKHKLOORFFXUV &GHWHUPLQHVWKHKHLJKWRIWKHKLOO¶VEDVH DQG % LQIOXHQFHV WKH VKDUSQHVV RI WKH KLOO &RHIILFLHQW % FRQVWLWXWHV WKH PDLQ IDFWRUUHVSRQVLEOHIRUWKHVKDSHRIWKHZHLJKWLQJIXQFWLRQV $VDQH[WHQVLRQWRWKHRULJLQDOIXQFWLRQSXEOLVKHGE\%D\OHHWDOWKHSDUW ZLWKVKDSHIDFWRU(ZLOOEHDGGHGODWHURQ7KLVH[WHQVLRQDSSHDUVWRLPSURYHWKH DSSUR[LPDWLRQLQSDUWLFXODUDWODUJHOHYHOVRIVOLSHVSHFLDOO\LQYLHZRIWKHVWULFW FRQGLWLRQ WKDW WKH ZHLJKWLQJ IXQFWLRQ * PXVW UHPDLQ SRVLWLYH IRU DOO VOLS FRQGLWLRQV )RUWKHVLGHIRUFHZHJHWWKHIROORZLQJIRUPXODH
)\ ' *\ @ )\R % 69\
ZLWKWKHZHLJKWLQJIXQFWLRQQRZH[SUHVVHGVXFKWKDWLWHTXDOVXQLW\DW
*\ '
& %\ 6 @ FRV>&\ DUFWDQ%\ 6+\ @ FRV> \ DUFWDQ
!
6(0,(03,5,&$/7<5(02'(/6
ZKHUH
6
' % 6+\
DQGIXUWKHUWKHFRHIILFLHQWV
%\ ' U%\ FRV> DUFWDQ ^ U%\ WDQ & U %\ `@
D
&\ ' U&\
E
6+\ ' U+\ % U+\ GI ]
F
69\ ' '9\
'9\ ' \ ) ] @ U9\ % U9\ GI] % U 9\
U
U
VLQ > 9\ DUFWDQ 9\
@
D
U
@ FRV > DUFWDQ 9\ WDQ @
E
ZLWKGI]WKHQRWDWLRQIRUWKHQRQGLPHQVLRQDOLQFUHPHQWRIWKHYHUWLFDOORDGZLWK UHVSHFW WR WKH DGDSWHG QRPLQDO ORDG FI QH[W VXEVHFWLRQ (T( )LJXUHGHSLFWVWKHWZRZHLJKWLQJIXQFWLRQVGLVSOD\HGERWKDVIXQFWLRQVRI DQGRI DQGWKHUHVXOWLQJIRUFHFKDUDFWHULVWLFVSDUDPHWHUVDFFRUGLQJWR7DEOH DWHQGRI6HF %HORZDQH[SODQDWLRQLVJLYHQ ,Q (T )\R GHQRWHV WKH VLGH IRUFH DW SXUH VLGH VOLS REWDLQHG IURP (T 7KHGHQRPLQDWRURIWKHZHLJKWLQJIXQFWLRQ *\ PDNHV*\ DW
7KHKRUL]RQWDOVKLIW6+\ RIWKHZHLJKWLQJIXQFWLRQDFFRPSOLVKHV WKH
VOLJKW LQFUHDVH WKDW WKH VLGH IRUFH H[SHULHQFHV DW PRGHUDWH EUDNLQJ EHIRUH WKH SHDNRI*\ LVUHDFKHGDQGWKHGHFD\RI)\FRPPHQFHV7KLVKRUL]RQWDOVKLIWPD\ EHPDGHGHSHQGHQWRQWKHYHUWLFDOORDG&\ FRQWUROVWKHOHYHORIWKHKRUL]RQWDO DV\PSWRWH,I&\
WKHZHLJKWLQJIXQFWLRQ ZLOODSSURDFK]HURZKHQ
47KLVZRXOGEHWKHFRUUHFWYDOXHIRU&
\
LI
6
LVH[SHFWHGWR EH XVHG LQWKH
HQWLUHUDQJHIURPSOXVWRPLQXVLQILQLW\,IWKLVLVQRWLQWHQGHGWKHQ&\ PD\EH FKRVHQ GLIIHUHQW IURP RQH LI *\ LV RSWLPLVHG ZLWK WKH UHVWULFWLRQ WR UHPDLQ SRVLWLYH 7KH IDFWRU %\ LQIOXHQFHV WKH VKDUSQHVV RI WKH KLOO VKDSHG ZHLJKWLQJ IXQFWLRQ$VLQGLFDWHGWKHKLOOEHFRPHVPRUHIODWZLGHU DWODUJHUVOLSDQJOHV 7KHQ % \ GHFUHDVHV DFFRUGLQJ WR D :KHQ LQ DQ H[WUHPH VLWXDWLRQ
E
DSSURDFKHV WKDWLVZKHQ9F[ ZLOOUHPDLQHTXDOWRRQHXQOHVV ZKHQDW9F[
6 %
\
ZLOOJRWR]HURDQGFRQVHTXHQWO\*\
JRHVWRLQILQLW\ZKLFKPD\HDVLO\EHWKHFDVH
6 WKHZKHHOVSHHGRIUHYROXWLRQ
DQGWKXVWKHORQJLWXGLQDOVOLS
YHORFLW\ 9V[ UHPDLQV XQHTXDO WR ]HUR 7KH TXDQWLW\ 69\ LV WKH YHUWLFDO µVKLIW¶ ZKLFKVRPHWLPHVLVUHIHUUHGWRDVWKH
LQGXFHGSO\VWHHU$WFDPEHUGXHWRWKH
DGGHGDV\PPHWU\WKHORQJLWXGLQDOIRUFHFOHDUO\SURGXFHVDWRUTXHWKDWFUHDWHV DWRUVLRQDQJOHFRPSDUDEOHZLWKDSRVVLEO\DOUHDG\SUHVHQWSO\VWHHUDQJOH7KLV VKLIWIXQFWLRQYDULHVZLWKVOLS LQGLFDWHGLQD
6(0,(03,5,&$/7<5(02'(/6
1
*[α
κ
)[ [kN]
4
1.6
κ
0.8
1.6 0.8 0.4 0.2 0.1 0.05 0
0.4 0.2 0.1
-60
0 1
*[α
0 0.05
α
60 0,2,5 10 20 40
4
α
0
-60
)[
60
[kN]
α
α
80
0, 2, 5 10 20 40 [o] 80
-1
-0.5
0
0.5
κ
1
'9\κ
200
γ
*\κ 1
69\κ 0
κ
0.5
0
0.5
κ
[-]
1
1
α
80 0
)\ [kN] 4
80
α
80
α
40 20 10 0 25
-1
-0.5
0
1 *\κ
0.5
κ
1 0,0.05 0.1 0.2 0.4
-1 4
-0.5
0
0.5
)\ [kN]
1.6
α
[ o ] 60
1
1.6
κ
0
κ
0, 0.05 0.1 0.2 0.4 0.8
0.8
-60
40 20 510 2 0
κ 0
20
α
40
[ o ] 60
)LJ 1DWXUH RI ZHLJWKLQJ IXQFWLRQV DQG WKH UHVXOWLQJ FRPELQHG VOLS ORQJLWXGLQDODQG
6
ODWHUDOIRUFHVWKHODWWHUDOVRDIIHFWHGE\WKH LQGXFHGSO\VWHHUµYHUWLFDOVKLIW¶
9\
6(0,(03,5,&$/7<5(02'(/6
$VLOOXVWUDWHGLQ)LJLWVSHDNYDOXH'9\ GHSHQGVRQWKHFDPEHUDQJOH DQGGHFD\VZLWKLQFUHDVLQJPDJQLWXGHRIWKHVOLSDQJOH
)LJXUHSUHVHQWV
PHDVXUHGGDWDWRJHWKHUZLWKWKHILWWHGFXUYHVDVSXEOLVKHGE\%D\OHHWDO 7KHFRPELQHGVOLSUHODWLRQVIRU)[DUHVLPLODUWRZKDWZHKDYHVHHQIRUWKH VLGHIRUFH+RZHYHUDYHUWLFDOVKLIWIXQFWLRQZDVQRWQHHGHGWREHLQFOXGHG,Q )LJDWKUHHGLPHQVLRQDOJUDSKLVVKRZQLQGLFDWLQJWKHYDULDWLRQRI)[DQG )\ZLWKERWK DQG 7KHLQLWLDOµ6¶VKDSHRIWKH)\YV
FXUYHVDWVPDOO GXH
WRWKHYHUWLFDOVKLIWIXQFWLRQLVFOHDUO\YLVLEOH )RU SRVVLEOH LPSURYHPHQW RI WKH JHQHUDO WHQGHQF\ RI WKH PRGHO DW ODUJHU OHYHOVRIFRPELQHGVOLSEH\RQGWKHUDQJHRIDYDLODEOHWHVWGDWDRQHPLJKWLQFOXGH DGGLWLRQDOµIDEULFDWHG¶GDWDZKLFKDUHGHULYHGIURPVLPLODULW\PHWKRGUHVXOWVDW ODUJHUYDOXHVRIWKHVOLSDQJOH$QRWKHUSRVVLELOLW\LVWKHXVDJHRIWKHFRQGLWLRQV DW ZKHHO ORFN ZKHUH RQH PLJKW DVVXPH WKDW WKH IRUFH DQG VOLS YHFWRU DUH FR OLQHDU:HWKHQKDYHIRUWKHUDWLRRIWKHFRPSRQHQWV
) 9 ' ' ) 9 [
V[
\
WDQ
V\
' &
WDQ
RU
) ') [
[R
'&
@
FRV
')
'&
[R
9 9 9 @ 9
V[
@
V
) ' &) \
[R
'&
@
VLQ
')
'&
[R
V\
V
)\
γ
3000
α=6
o
o
=5
)] = 5000N
2000
[N]
2
o o
1 -0.25
0.25
0
κ
o
-1000 -1
-2
o
o
-6
)LJ 7KHFRPELQHGVOLSVLGHIRUFHFKDUDFWHULVWLFVLQWKHSUHVHQFHRIDFDPEHUDQJOH%D\OH HWDO
6(0,(03,5,&$/7<5(02'(/6
) \
α
κ ) [ )LJ7KUHHGLPHQVLRQDOJUDSKRIFRPELQHGVOLSIRUFHFKDUDFWHULVWLFV
5HJDUGLQJWKHDOLJQLQJWRUTXHSK\VLFDOLQVLJKWLVXVHGWRPRGHOWKHVLWXDWLRQDW FRPELQHGVOLS:HZULWH
0 ' & W ]
WHT
7KHDUJXPHQWV
W
@
) %0 \
DQG
U
]U
UHT
% V )
\
@
)
[
LQFOXGLQJDVKLIW DSSHDULQJLQWKHIXQFWLRQV
IRU WKH SQHXPDWLF WUDLO DQG UHVLGXDO WRUTXH DW SXUH VLGH VOLS DUH UHSODFHG E\ HTXLYDOHQWVOLSDQJOHVDVLQGLFDWHGE\(T LQFRUSRUDWLQJWKHHIIHFWRI RQ WKHFRPSRVLWHVOLS
WHT
'
W
%
. .
[
@ VJQ
W
\
DQGVLPLODUIRU
UHT
7RDSSUR[LPDWHWKHVDPHHIIHFWRQWKHGHJUHHRIVOLGLQJLQ
WKH FRQWDFW SDWFK DV ZRXOG RFFXU ZLWK VLGH VOLS WKH ORQJLWXGLQDO VOLS
LV
PXOWLSOLHGZLWKWKHUDWLRRIWKHORQJLWXGLQDODQGODWHUDOVOLSVWLIIQHVVHV %HVLGHVDQH[WUDWHUPLVLQWURGXFHGLQ WRDFFRXQWIRUWKHIDFWWKDWD PRPHQWDUPVDULVHVIRU ) [DVDUHVXOWRIFDPEHU DQGODWHUDOW\UHGHIOHFWLRQ UHODWHG WR )\ 7KLV H[WUD WHUP PD\ JLYH ULVH WR D VLJQ FKDQJH RI WKH DOLJQLQJ
6(0,(03,5,&$/7<5(02'(/6
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
.\
DQG
\
WKHDEVFLVVD
RIWKHSQHXPDWLFWUDLOFKDUDFWHULVWLFLVFKDQJHGLQDZD\HTXDOWRWKDWRIWKHVLGH IRUFHFKDUDFWHULVWLFDQGLQDFFRUGDQFHZLWKWKHVLPLODULW\PHWKRGRI6HFWLRQ
)XOO6HWRI(TXDWLRQV
7KH 0DJLF )RUPXOD PRGHO HTXDWLRQV FRQWDLQ WKH QRQGLPHQVLRQDO PRGHO SDUDPHWHUV S T U DQG V DQG LQ DGGLWLRQ D VHW RI VFDOLQJ IDFWRUV
2WKHU
SDUDPHWHUVDQGYDULDEOHTXDQWLWLHVXVHGLQWKHHTXDWLRQVDUH J
DFFHOHUDWLRQGXHWRJUDYLW\
9F
PDJQLWXGHRIWKHYHORFLW\RIWKHZKHHOFRQWDFWFHQWUH&
9F[\ FRPSRQHQWVRIWKHYHORFLW\RIWKHZKHHOFRQWDFWFHQWUH& 9V[\FRPSRQHQWVRIVOLSYHORFLW\9VRIWKHSRLQW6 ZLWK9V\ 9U
5H
9R
UHIHUHQFHYHORFLW\
5R
XQORDGHGW\UHUDGLXV
5H
HIIHFWLYHUROOLQJUDGLXV
.9
F\
9F[9V[ IRUZDUGVSHHGRIUROOLQJ
% J5 RURWKHUVSHFLILHGYDOXH
R
UR UH
ZKHHOVSHHGRIUHYROXWLRQ ]
W\UHUDGLDOGHIOHFWLRQ!LIFRPSUHVVLRQ
)]R
QRPLQDOUDWHG ORDG
)¶]R
DGDSWHGQRPLQDOORDG
$
7KH HIIHFW RI KDYLQJ D W\UH ZLWK D GLIIHUHQW QRPLQDO ORDG PD\ EH URXJKO\ DSSUR[LPDWHGE\XVLQJWKHVFDOLQJIDFWRU
)]RN '
)]R
)
(
)]R ]R
)XUWKHUZHLQWURGXFHWKHQRUPDOL]HGFKDQJHLQYHUWLFDOORDG
GI ] '
) ] & )]RN )]RN
,QVWHDG RI WDNLQJ WKH VOLS DQJOH
(
LWVHOI LQ UDGLDQV IURP (T DV LQSXW
TXDQWLW\ RQH PD\ LQ FDVH RI YHU\ ODUJH VOLS DQJOHV DQG SRVVLEO\ EDFNZDUGV
6(0,(03,5,&$/7<5(02'(/6
UXQQLQJRIWKHZKHHOEHWWHUXVHWKHWDQJHQWRIWKHVOLSDQJOHGHILQHGDVWKHODWHUDO VOLS
( ' WDQ
@ VJQ9F[ ' &
9F\ _9F[ _
(
)RUWKHVSLQGXHWRWKHFDPEHUDQJOHZHLQWURGXFH
( ' VLQ
(
7KHORQJLWXGLQDOVOLSUDWLRLVGHILQHGDVIROORZV
'&
9V[ _9F[_
(
,I WKH IRUZDUG VSHHG 9F[ EHFRPHV RU LV HTXDO WR ]HUR RQH PLJKW DGG D VPDOO TXDQWLW\
g LQ WKH GHQRPLQDWRU RI ( ( WR DYRLG VLQJXODULW\ RU ZKHQ
WUDQVLHQWVOLSVLWXDWLRQVRFFXURQHVKRXOGXVHWKHWUDQVLHQW VOLSTXDQWLWLHVRU GHIRUPDWLRQJUDGLHQWV WDQ
NDQG NDVGHILQHGDQGXVHGLQ&KDSWHUVDQG
7RDYRLGWKHRFFXUUHQFHRIVLPLODUVLQJXODULWLHVLQWKHHQVXLQJHTXDWLRQVGXH
g
WR HJ ]HUR YHORFLW\ RU ]HUR YHUWLFDO ORDG D VPDOO DGGLWLRQDO TXDQWLW\ ZLWK VDPH VLJQ DV LWV QHLJKERXULQJ PDLQ TXDQWLW\ ZLOO EH LQWURGXFHG LQ UHOHYDQW GHQRPLQDWRUVOLNHLQWKHQH[WHTXDWLRQ )RU WKH IDFWRU FRV
DSSHDULQJ LQ WKH HTXDWLRQV IRU WKH DOLJQLQJ WRUTXH WR
SURSHUO\KDQGOHWKHFDVHRIODUJHVOLSDQJOHVDQGSRVVLEO\EDFNZDUGVUXQQLQJ9F[ ZHKDYHGHILQHG
N '
FRV ZLWK
9F[ 9FN
(
9FN ' 9F % g9
(
ZKHUHZHPD\FKRRVH
g
9
)RU WKH QRUPDOO\ HQFRXQWHUHG VLWXDWLRQV ZKHUH WXUQ VOLS PD\ EH QHJOHFWHG
64 DQGFDPEHUUHPDLQVVPDOOWKHIDFWRUV
SDWKUDGLXV 5
L
DSSHDULQJLQWKH
HTXDWLRQVDUHWREHVHWHTXDOWRXQLW\
L
'
L'
,QWKHIROORZLQJ6HFWLRQZKHUHWKHLQIOXHQFHRIVSLQWXUQVOLSDQGFDPEHU LVGHVFULEHGLQWKHH[WHQGHG0DJLF)RUPXODPRGHOWKHSURSHUH[SUHVVLRQVZLOO EHJLYHQIRUWKHVHIDFWRUVDQGDGGLWLRQDOHTXDWLRQVZLOOEHLQWURGXFHG
6(0,(03,5,&$/7<5(02'(/6
8VHU6FDOLQJ)DFWRUV 7KH IROORZLQJVHWRIVFDOLQJIDFWRUV LVDYDLODEOH7KHGHIDXOWYDOXHRIWKHVH IDFWRUVLVVHWHTXDOWRRQHH[FHSW
9
ZKLFKHTXDOV]HURLIQRWXVHG :HKDYH
SXUHVOLS )]R
QRPLQDOUDWHG ORDG
[\
SHDNIULFWLRQFRHIILFLHQW
9
ZLWKVOLSVSHHG9VGHFD\LQJIULFWLRQ
.[
EUDNHVOLSVWLIIQHVV
.\
FRUQHULQJVWLIIQHVV
&[\
VKDSHIDFWRU
([\
FXUYDWXUHIDFWRU
+[\
KRUL]RQWDOVKLIW
9[\
YHUWLFDOVKLIW
.\
FDPEHUIRUFHVWLIIQHVV
.]
FDPEHUWRUTXHVWLIIQHVV
W
SQHXPDWLFWUDLOHIIHFWLQJDOLJQLQJWRUTXHVWLIIQHVV
0U
UHVLGXDOWRUTXH
FRPELQHGVOLS [
LQIOXHQFHRQ)[
\
LQIOXHQFHRQ)\
9\
LQGXFHGµSO\VWHHU¶)\
V
0]PRPHQWDUPRI)[
RWKHU &]
UDGLDOW\UHVWLIIQHVV
0[
RYHUWXUQLQJFRXSOHVWLIIQHVV
0\
UROOLQJUHVLVWDQFHPRPHQW
7RFKDQJHIURPDUHODWLYHO\KLJKIULFWLRQVXUIDFHWRDORZIULFWLRQVXUIDFHWKH IDFWRUV
[
DQG
\
PD\EHJLYHQDYDOXHORZHUWKDQXQLW\,QDGGLWLRQWRUHIOHFW
DVOLSSHU\VXUIDFHZHW ZLWKIULFWLRQGHFD\LQJZLWKLQFUHDVLQJVOLS VSHHGRQH PD\ FKRVH IRU
9
D YDOXH ODUJHU WKDQ ]HUR HJ (TV(( 7KH
SXEOLFDWLRQV RI 'LMNV DQG RI 5HLPSHOO PD\ EH XVHIXO LQ WKLV UHVSHFW1RWHWKDWWKHVOLSVWLIIQHVVHVDUHQRWHIIHFWHGWKURXJKWKHVHFKDQJHV$ VSHFLDO GHJUHVVLYH IULFWLRQ IDFWRU
N
[\
LV LQWURGXFHG WR UHFRJQLVH WKH IDFW WKDW
: 6EXWDWDPXFKVORZHUUDWH
YHUWLFDOVKLIWVRIWKHIRUFHFXUYHVGRYDQLVKZKHQ
N
[\
$
[\
^$
[\
`VXJJHVWLRQ$
(
)RUWKHWKUHHIRUFHVDQGWKUHHPRPHQWVDFWLQJIURPURDGWRW\UHDQGGHILQHG DFFRUGLQJWRWKHGLDJUDPRI)LJWKHHTXDWLRQVILUVWWKRVHIRUWKHFRQGLWLRQ
6(0,(03,5,&$/7<5(02'(/6
RI SXUH VOLS LQFOXGLQJ FDPEHU DQG VXEVHTXHQWO\ WKRVH IRU WKH FRQGLWLRQ RI FRPELQHGVOLSUHDG
/RQJLWXGLQDO)RUFHSXUHORQJLWXGLQDOVOLS )
[R
[
''
' %6 '
S
[
'
[
'
S
(
[
'
S
&
[
'
VLQ>&
[
.
[
&[
DUFWDQ ^%
6 6
[
[
[
%
& DUFWDQ%
[
[
[
[
%6
`@
@
%S
( (
!
%S
'[
([
!
&[
]
'[
([
GI ]
GI
]
@
[
%S
([
%
GI
]
9
9 9 V
@
^
!
R
&S
([
VJQ
[
`
(
@
]
.[
.[
]
[
[
'
S
9[
'
)
]
+[
[
%S
[
.[
[
[R
]
#
.[
[
[
(
([
' ) @ S %S GI @ H[S S GI @ ' % & ' ' M) M DW ' ' & ' . & ' % g
+[
(
9[
(
[
%
&(
[
+[
@) @
[
[
(
)
(
[
+[
@S % S 9[
GI ]
9[
@
(
+[
@
GI ^_9 ]
_
F[
g % _9 9[
_ `
F[
@
@ N@
9[
(
[
/DWHUDO)RUFHSXUHVLGHVOLS )
\R
\
&
'
'
S
\
'
\
'
S
'
S
'
(
.
\
\ R
\
VLQ>&
(%6
'
\
'
@
'S
\
\
%
\
\
& DUFWDQ%
\
\
`@
%6
(
9\
( (
%S %S
'\
(\
@ &S
GI ]
@ & S
GI ^ ]
( @
'\
(\
%S
\
(\
%
9
( VJQ
9 9
\
V
`
R
@
)
]R
]
' . @ & S
\
.\
\
( @
\R
.\
\
]R
\
(
#
(
.\
)
!
(\
N VLQ> DUFWDQ ^) S )N `@ @ ' % & ' ' M) M DW ' ' ' &
.\
\ R
&(
!
\
.
\
]
(\
\
(
&\
@) @ '\
DUFWDQ ^%
+\
&\
\
\
(
(
6(0,(03,5,&$/7<5(02'(/6
% 6 6
'
\
.
& '
\
\
+\
'
S
9\
'
)
.
'
\ R
]
+\
%g
\
%S
GI
+\
]
@ ^S % S 9\
^S
.
+\
\
9\
@
+\
GI ]
%S @
(@
+\
9\
.\
(
@%
% S % S 9\
9\
GI ]
& (@
% ) S % S GI ` ' a M) M DW ' ' ' &
\ R
]
9\
9\
]
(
.\
`
@ N@
(
\
.\
\R
)
(
$OLJQLQJ7RUTXHSXUHVLGHVOLS 0
R
\R
' W
W
6
R
+W
'
T
'
'
+I
'
N '
%
'
'
W
WR
'
'
W
6
6
%6
'
'
'
T
'
U
]
WR
W
& ( % & DUFWDQ % W
W
W
W
W
@
`@ FRV
N
'
%T
+]
U
'
U
U
@
(
(
N
.
9\
(
\
(
.
%]
GI
R
)
%T
N @ T
]R
@ % T
%]
]
%]
GI
]
@ %T
%]
(% T
%]
_
( _ @
.\
!
\
(]
GI
% & \
']
%T
]
']
%T
(% T
']
(]
]
GI ]
']
GI
( @
]
@ @ VJQ 9 W
(
F[
(
@ %T %T (]
(]
( DUFWDQ% & W
W
W
#
(
(
\
(
5 ^T R
% &
( (
!
@5
(
(
]
U
I
(
GI
+]
' FRV >& DUFWDQ %
+I
% T
]
')
U
GI
+]
&]
)
W
%g
\
(]
U
W
(%
T
FRV > & DUFWDQ ^ %
U
%]
W
&
]U
+\
(
(
%T
' T %T
(
%
0
(
]UR
+W
' T % T
W
'
6
+]
.
\
&
''
(%
]UR
U
.
W
'
0
N %0
]R
]R
W
6
0
N ' &W @)
0 W
'
]R
%T
']
']
GI ]
0U
%T %T
']
']
GI ]
( .]
` FRV
N @
\
VJQ 9
%
F[
(
6(0,(03,5,&$/7<5(02'(/6
.
'
] R
'
')
.
] R
]
WR
.
'a &M0 M
\ R
5 T R
%T
]R
&' ' a M0 M
']
']
GI ]
.]
' ' ' &
DW
\
WR
.
\
\ R
DW
]R
' '
'&
(
0
(
0
/RQJLWXGLQDO)RUFHFRPELQHGVOLS )
[
*
'
*
@)
[
' FRV>&
[
' FRV>&
[ R
6
%
[
&
[
( 6
[
DUFWDQ^%
[
*
(
[R
&(
[
6
DUFWDQ ^ %
[
6
[
%
[
&DUFWDQ%
[
6
&(
+[
[
' (%6 ' U FRV> DUFWDQ U @ @ 'U ' U % U GI # 'U
%
[
6
+[
[
`@*
6
& DUFWDQ %
[ R
6
[
+[
! `@
%[
!
[
( (
&[
([
+[
([
( (
+[
%[
(
(
]
(
+[
/DWHUDO)RUFHFRPELQHGVOLS )
\
*
'
DUFWDQ^%
DUFWDQ ^%
\
&(
\
6
6
\
\
%
&DUFWDQ%
\
6
&(
+\
\
%
\
6
+\
\
`@*
6
& DUFWDQ %
\
\ R
6
+\
! `@
' %6
&
'
U
(
'
U
\
\
%\
6
+\
'
U
6
9\
'
'
9\
'
FRV> DUFWDQ ^ U
%\
(& U
`@
%\
@
\
!
(
%U
+\
(\
GI
%U
9\
) \
(
(
&\
(\
(
(
+\
U
\
'
(
9\
' FRV>&
'
%
\R
\
\ R
6
@) % 6
\
' FRV>&
\
*
*
+\
VLQ> U
]
]
GI
( (
]
9\
DUFWDQ U
@U % U 9\
#
9\
GI
]
9\
%U
9\
(
( @ FRV > DUFWDQ U
9\
( @ @
(
6(0,(03,5,&$/7<5(02'(/6
1RUPDO/RDGVHHDOVR(T )
]
'
@ )N 5 @ PD[ & ' M) M ' S S
]
]R
R
)]
]
]
]
@
]
)
&]
$
&]
N 5
]R
R
(
2YHUWXUQLQJ&RXSOHVHHDOVR6HF 0
[
'
) 5 ]
@ T & T
R
V[
V[
(%T
V[
) ) \
N @
]R
0[
(
5ROOLQJ5HVLVWDQFH0RPHQWVHHDOVR(TV
0
\
' &)
]
5
R
@ ^T
V\
DUFWDQ 9 9 U
R
%T
V\
) ) [
N `@
]R
(
0\
$OLJQLQJ7RUTXHFRPELQHGVOLS 0 0
]
'
V
]U
]
(
[
'' FRV>& DUFWDQ^%
WHT
N'
)
\
]U
(
\
'W
0
N % 0 % V@)
]
N ' & W @ )N
W
)
0
'
' WHT
UHT
5
\
' '
W
W
&( %
WHT
W
&DUFWDQ%
W
WHT
&6
0
R
W
UHT
@ ^V % V V]
W
U
% %
@
`@ FRV
WHT
N
( (
9\
]U
W
'
U
\
N % V % V
]R
[
N
.
U
) )
V]
.
' FRV > & DUFWDQ %
V]
U
V]
UHT
GI ]
@
(`@
(
V
(
@ VJQ
W
@ VJQ
U
(
(
\
.
[
N
.
\
$VHWRISDUDPHWHUYDOXHVKDYHEHHQOLVWHGLQ$SSHQGL[IRUDQH[DPSOHW\UHLQ FRQQHFWLRQZLWKWKH6:,)7W\UHPRGHOWREHGHDOWZLWKLQ&KDSWHU([DPSOHV RIFRPSXWHGFKDUDFWHULVWLFVFRPSDUHGZLWKH[SHULPHQWDOO\DVVHVVHGFXUYHVIRU ERWKSXUHVOLSDQGFRPELQHGVOLSFRQGLWLRQVDUHGLVFXVVHGLQ6HFWLRQRIWKLV FKDSWHUFI)LJV,QWKHQH[WVHFWLRQWKHHIIHFWRIKDYLQJWXUQVOLS LVPRGHOOHGDQGVRPHFDOFXODWHGFKDUDFWHULVWLFVKDYHEHHQSUHVHQWHGIRUDVHWRI K\SRWKHWLFDOPRGHOSDUDPHWHUV)LJ 6HFWLRQH[DPLQHVWKHSRVVLELOLW\
6(0,(03,5,&$/7<5(02'(/6
WRGHILQHWKHHIIHFWVRIFRQLFLW\DQGSO\VWHHUDVUHVSRQVHVWRHTXLYDOHQWFDPEHU DQGVOLSDQJOHV6HFWLRQJLYHVDSRVVLEO\EHWWHUGHVFULSWLRQWKDQ( IRU WKHRYHUWXUQLQJFRXSOHRIWUXFNFDUDQGUDFLQJW\UHVGXHWRVLGHVOLSDQGIRUWKH SDUWGXHWRFDPEHUWKDWLVLOOXVWUDWHGIRUWKHFDVHRIDPRWRUF\FOHW\UH
([WHQVLRQRIWKH0RGHOIRU7XUQ6OLS
7KH PRGHO GHVFULEHG VRIDU FRQWDLQV WKH LQSXW VOLS TXDQWLWLHV VLGH VOLS ORQJLWXGLQDO VOLS DQG ZKHHO FDPEHU ,Q WKH SUHVHQW VHFWLRQ WKH WXUQ VOLS RU LQ VWHDG\ VWDWH SDWK FXUYDWXUH LV DGGHG ZKLFK FRPSOHWHV WKH GHVFULSWLRQ RI WKH VWHDG\VWDWHIRUFHDQGPRPHQWJHQHUDWLRQSURSHUWLHVRIWKHW\UH 7XUQVOLSLVRQHRIWKHWZRFRPSRQHQWVZKLFKWRJHWKHUIRUPWKHVSLQRIWKH W\UHFI)LJ7KHWXUQVOLSLVGHILQHGKHUHDV
nW '
0
'& 9FN 5
&
DQGWKHWRWDOW\UHVSLQFI(T
n'
&
9FN
^0
& & g
VLQ `
g GHQRWHVWKHFDPEHUUHGXFWLRQ
ZLWKWKHVLQJXODULW\SURWHFWHGYHORFLW\9F¶ $JDLQ
IDFWRUIRUWKHFDPEHUWREHFRPHFRPSDUDEOHZLWKWXUQVOLS)RUUDGLDOSO\FDU
g PD\EHFRPHDVKLJKDVFDIRUVRPHWUXFNW\UHV
W\UHVWKLVUHGXFWLRQIDFWRU
HYHQVOLJKWO\RYHU ZKLOHIRUDPRWRUF\FOHW\UHWKHIDFWRULVH[SHFWHGWREH FORVH WR ]HUR OLNH ZLWK D KRPRJHQHRXV VROLG EDOO FI GLVFXVVLRQ EHORZ (T ,QWKHSUHYLRXVVHFWLRQWKHHIIHFWRIFDPEHUZDVLQWURGXFHG)RUWKHVLGHIRUFH WKLVUHVXOWHGLQDKRUL]RQWDODQGDYHUWLFDOVKLIWRIWKH)\YV
FXUYHVZKLOHIRU
WKHDOLJQLQJWRUTXHEHVLGHVWKHVPDOOHIIHFWRIWKHVKLIWHG)\FXUYHWKHUHVLGXDO WRUTXH0]UZDVDGGHGLQZKLFKDFRQWULEXWLRQRIFDPEHURFFXUV)XUWKHUPRUH FKDQJHVLQVKDSHDSSHDUHGWRRFFXUZKLFKZHUHUHSUHVHQWHGE\WKHLQWURGXFWLRQ RI LQIDFWRUVOLNH\ (\ .\ %W'WHWF7KHVHFKDQJHVPD\EHDWWULEXWHGWR FKDQJHVLQW\UHFURVVVHFWLRQDQGFRQWDFWSUHVVXUHGLVWULEXWLRQUHVXOWLQJIURPWKH ZKHHOLQFOLQDWLRQDQJOH 7KHVHVKLIWVDQGVKDSHFKDQJHVZLOOEHUHWDLQHGLQWKHPRGHOH[WHQVLRQEXW ZLOOEHH[SDQGHGWRFRYHUWKHFRPSOHWHUDQJHRIVSLQLQFRPELQDWLRQZLWKVLGH
4ZKHQSDWK
VOLSDQGDOVRORQJLWXGLQDOVOLS7KHVSLQPD\FKDQJHIURP]HURWR
FXUYDWXUH JRHV WR LQILQLW\ DQG FRQVHTXHQWO\ SDWK UDGLXV WR ]HUR ZKHQ WKH
6 :HLJKWLQJIXQFWLRQVZLOODJDLQEHLQWURGXFHGWRJUDGXDOO\UHGXFH
YHORFLW\9F
SHDNVLGHDQGORQJLWXGLQDOIRUFHVZLWKLQFUHDVLQJVSLQ$OVRFRUQHULQJVWLIIQHVV
6(0,(03,5,&$/7<5(02'(/6
α=0
α = 90o
9F
.
-ψ
.
0]
5
0]ϕR
-ψ
)\
)\ϕ 0]ϕ
)\
0 0
0.2
)\ 0]
5
0]
)\ϕR
9F
0] )\
0 0.5
D5
0
0.2
D5
0]ϕ4 0.5
)LJ7ZREDVLFVSLQIRUFHDQGPRPHQWGLDJUDPV
DQGSQHXPDWLFWUDLOZLOOEHVXEMHFWHGWRVXFKDUHGXFWLRQ 7KHWKHRUHWLFDOILQGLQJVZKLFKKDYHEHHQJDLQHGIURPWKHSK\VLFDOPRGHORI 6HFWLRQ SUHVHQWHG LQ )LJG ZLOO EH XVHG DV EDVLV IRU WKH PRGHO GHYHORSPHQW6LQFHWKHVHILQGLQJVKROGIRURQO\RQHYDOXHRIWKHYHUWLFDOORDGDQG ]HURORQJLWXGLQDOVOLSWKHLQIOXHQFHRIORDGYDULDWLRQVDQGORQJLWXGLQDOVOLSKDYH EHHQVWUXFWXUHGWHQWDWLYHO\
,Q)LJIRUWKHWZRH[WUHPHFDVHVRI
n
DQG0]YVD
W
DQG
E
WKHFRXUVHVRI)\
D5KDYHEHHQGHSLFWHG7KHSXUHVSLQVLGHIRUFHFKDUDFWHULVWLF
VKRZQLQWKHOHIWKDQGGLDJUDPDW
DOVRLQGLFDWHGLQWKHORZHUOHIWGLDJUDP
RI )LJG FDQ EH IDEULFDWHG E\ VLGHZD\V VKLIWLQJ RI WKH VLGH IRUFH FXUYH
n
EHORQJLQJWR]HURVSLQWXUQVOLS
W
DQG
XSSHURUPLGGOHOHIWGLDJUDP
ZKLOHDWWKHVDPHWLPHUHGXFLQJLWVSHDNYDOXH'\DQGLWVVORSHDWWKHFXUYHFHQWUH .\ )RUWKLVZHGHILQHWKHUHGXFWLRQIXQFWLRQVWREHVXEVWLWXWHGLQWKHSUHYLRXV (TV( 7KHSHDNVLGHIRUFHUHGXFWLRQIDFWRU
'
%
FRV > DUFWDQ^ \n
5R_nW_ % S'\n 5R_nW_ `@
ZLWKVKDUSQHVVIDFWRU
%\n ' S'\n % S'\n GI] FRV >DUFWDQ S'\n WDQ
@
DQGWKHVORSHUHGXFWLRQIDFWRU
'
S
FRV > DUFWDQ .\n
5R n W @
+HUHZLWK WKH FRQGLWLRQ LV VDWLVILHG WKDW DW
n 64 ZKHUH WKH ZKHHO LV VWHHUHG W
DERXW WKH YHUWLFDO D[LV DW D VSHHG RI WUDYHO HTXDO WR ]HUR WKDW LV YHORFLW\ RI FRQWDFWFHQWUH9F
6 DQGSDWKUDGLXV5 6OHDGLQJWR
WKHVLGHIRUFH
6(0,(03,5,&$/7<5(02'(/6
UHGXFHVWR]HURDOWKRXJKDVOLSDQJOHPD\WKHRUHWLFDOO\VWLOOH[LVW&RQVLGHULQJWKH XSSHU RU PLGGOH OHIW GLDJUDP RI )LJ G LW VHHPV WKDW WKH VLGHZD\V VKLIW VDWXUDWHV DW ODUJHUYDOXHVRIVSLQ7RPRGHOWKLVSKHQRPHQRQZKLFKDFWXDOO\ VD\VWKDWEH\RQGDFHUWDLQQHJDWLYHVOLSDQJOHQRVSLQLVODUJHHQRXJKWRPDNHWKH VLGHIRUFHYDQLVKWKHVLQHYHUVLRQRIWKHPDJLFIRUPXODLVXVHG:HKDYH
6+\n''+\nVLQ>&+\nDUFWDQ^%+\n5Rn&(+\n%+\n5Rn&DUFWDQ%+\n5Rn `@ 7KH VKDSH IDFWRU &+\n LV H[SHFWHG WR EH HTXDO RU VPDOOHU WKDQ XQLW\ 7R H[SUHVVLRQ LV DGGHGWKHVKLIWGXHWRSO\VWHHUDQGFRQLFLW\)LQDOO\ZH VXEWUDFWWKHKRUL]RQWDOGLVSODFHPHQWRIWKHSRLQWRILQWHUVHFWLRQRIWKHVLGHIRUFH YVVOLSDQJOHFXUYHZKLFKDULVHVGXHWRWKHYHUWLFDOVKLIWRIWKHFXUYHZKLFKIRU QRZLVWKRXJKWWREHVROHO\DWWULEXWHGWRWKHFDPEHUFRPSRQHQWRIVSLQ 7KLV OHDGVWRDWRWDOKRUL]RQWDOVKLIW
6+\ ' S+\ % S+\ GI] @
+\
% 6+\n & 69\ .N\
LQZKLFKWKHYHUWLFDOVKLIWGXHWR
UHDGV
69\ ' )] @ S9\ % S 9\ GI] ( @ @
.\
@ \ N
FI(T( 7KHTXDQWLW\.\¶ LVWKHVLQJXODULW\SURWHFWHGFRUQHULQJVWLIIQHVV GHILQHGE\(T( 7KURXJKWKLVPDQLSXODWLRQWKHFDPEHUVSLQVWLIIQHVVLV VROHO\ DWWULEXWHG WR WKH KRUL]RQWDO VKLIW RI WKH SRLQW RI LQWHUVHFWLRQ 6+\n FI
(TV $SSDUHQWO\WKHIDFWRUV LQ(T( QRZUHDG
' ' % 6+\n & 69\ .N\
7KHYDULRXVIDFWRUVDSSHDULQJLQ DUHGHILQHGDVIROORZV
&+\n ' S+\n ! '+\n ' S+\n % S+\n GI] @ VJQ9F[ (+\n ' S+\n # %+\n ' .\5nR ^&+\n '+\n .\ R % g. `
ZKHUHWKHVSLQIRUFHVWLIIQHVV.\5nRLVUHODWHGWRWKHFDPEHUVWLIIQHVV.\
R
&)
WKDWLVJLYHQE\(
.\5nR' .\ R &g 'M)\R M5Rn
DW
' 'n' '&)n 5 R
IRUZKLFKZHPD\GHILQH
g
' Sg n
% Sg n GI]
6(0,(03,5,&$/7<5(02'(/6
2EYLRXVO\WKLVSDUDPHWHUJRYHUQVWKHGLIIHUHQFHRIWKHUHVSRQVHWRFDPEHUZLWK UHVSHFWWRWKDWRIWXUQVOLS )RUPRGHOOLQJWKHDOLJQLQJWRUTXHZHZLOOXVHDVEHIRUHWKHSURGXFWRIWKH VLGHIRUFHDQGWKHSQHXPDWLFWUDLOLQWKHILUVWWHUPRI(TV(( ZKLOHIRU WKH VHFRQG WHUP WKH UHVLGXDO WRUTXH ZLOO EH H[SDQGHG WR UHSUHVHQW ODUJH VSLQ WRUTXH 7KHPLGGOHULJKWGLDJUDPRI)LJGVKRZVWKDWDWLQFUHDVLQJWXUQVOLSWKH UHVLGXDO RU VSLQ WRUTXH LQFUHDVHV ZKLOH WKH PRPHQW GXH WR VLGH VOLS FOHDUO\ YLVLEOHDW
n
IXQFWLRQ
PXOWLSOLHGZLWKWKHSQHXPDWLFWUDLOLQ(T(
'
W
GLPLQLVKHV7KLVGHFD\LVPRGHOOHGE\PHDQVRIWKHZHLJKWLQJ
FRV>DUFWDQT
n5 n @
'W
R
7KHVHFRQGWHUPRI(TV(( WKHUHVLGXDOWRUTXHZKLFKLQWKHSUHVHQW FRQWH[W PD\ EH EHWWHU GHVLJQDWHG DV WKH VSLQ PRPHQW LV JLYHQ E\ (TV( ,WV SHDN YDOXH 'U KDV DQ LQLWLDO YDOXH GXH WR FRQLFLW\ WKDW LV H[SHFWHGWREHWDNHQRYHUJUDGXDOO\E\DQLQFUHDVLQJWXUQVOLS,Q( WKLVLV DFFRPSOLVKHGE\WKHZHLJKWLQJIXQFWLRQ
7KHUHPDLQLQJWHUPVZLOOEH
UHSODFHG E\ WKH SHDN VSLQ WRUTXH 'Un 7KLV PHDQV WKDW WKH
¶V DSSHDULQJ LQ
( EHFRPHEHVLGHVDFFRUGLQJWR
' % 'Un
$VREVHUYHGLQWKHPLGGOHULJKWGLDJUDPRI)LJGWKHSHDNWRUTXHWKDWZLOO EHDVVXPHGWRRFFXUDWDVOLSDQJOH
6+IZKHUH)\
JURZVZLWKLQFUHDVLQJ
VSLQDQGILQDOO\VDWXUDWHV7KHPDJLFIRUPXODGHVFULEHVWKLVDVIROORZV
'Un'''UnVLQ>&'UnDUFWDQ^%'Un 5Rn& ('Un%'Un 5Rn&DUFWDQ%'Un 5Rn `@
$ LV' n7KHDV\PSWRWLFOHYHORIWKHSHDNVSLQ WRUTXHLVUHDFKHGDWn 6 4RU5 6 DQGLVGHQRWHGDV0 n4 &RQVHTXHQWO\ZLWK ,WVPD[LPXPYDOXHLI&'Un
'U
]
WKHVKDSHIDFWRU&'UnWKHPD[LPXPYDOXHLVH[SUHVVHGE\
''Un ' 0]n 4 VLQ &'Un
ZKHUHWKHPRPHQW0]n4 WKDWRFFXUVDWYDQLVKLQJZKHHOVSHHGDQGDWFRQVWDQW WXUQLQJDERXWWKHYHUWLFDOD[LVLVIRUPXODWHGDVDIXQFWLRQRIWKHQRUPDOORDG
0]n 4 ' T&Un \ 5 R)] )] )]RN @
0n
!
7KLV H[SUHVVLRQ PD\ EH FRPSDUHG ZLWK RU UHSODFHG E\ H[SUHVVLRQ IRUPXODWHGE\)UHXGHQVWHLQ 7KHVKDSHIDFWRUVLQ DUHDVVXPHGWR EHJLYHQE\FRQVWDQWSDUDPHWHUV
&'Un ' T'Un ('Un ' T'Un
!
#
6(0,(03,5,&$/7<5(02'(/6
ZKLOH
%'Un ' .] UR ^&'Un ''Un % gU &g `
ZLWK.]
UR
DVVXPHGWRGHSHQGRQWKHQRUPDOORDGDVIROORZV
.] UR' )] 5R T ']%T'] GI ]
.]
$V KDV EHHQ LQGLFDWHG ZLWK (T( ZH KDYH QRZ IRU WKH FDPEHU PRPHQW VWLIIQHVV
.] R ' .] UR & 'WR .\ R
' &0
DQGFRQVHTXHQWO\IRUWKHPRPHQWVWLIIQHVVDJDLQVWVSLQ
.]5nR ' .] R &g
' &0n 5R
1RZWKDWWKHIRUPXODHIRUWKHSHDNRIWKHVSLQWRUTXH'UnKDVEHHQGHYHORSHG ZH PXVWFRQVLGHUWKHUHPDLQLQJFRXUVHRI0]UZLWKWKHVOLSDQJOH
)RU WKLV
(T( LVHPSOR\HG$FFRUGLQJWRWKHPLGGOHULJKWGLDJUDPRI)LJGWKH FXUYHV EHFRPH IODWWHU DV WKH VSLQ LQFUHDVHV /RJLFDOO\ IRU
n 6 4 WKH PRPHQW
VKRXOGEHFRPHLQGHSHQGHQWRIWKHVOLSDQJOH7KHVKDUSQHVVLVFRQWUROOHGE\WKH IDFWRU%UZKLFKZHOHWJUDGXDOO\GHFUHDVHWR]HURZLWKLQFUHDVLQJVSLQ)RUWKLV ZHLQWURGXFHLQ(T( WKHZHLJKWLQJIXQFWLRQ
'
n 5 n @
FRV>DUFWDQT
%U
R
7KHIDFWRU&UFRQWUROVWKHDV\PSWRWLFOHYHOZKLFKFRUUHVSRQGVWRWKHWRUTXHDW
E
,QWKDWVLWXDWLRQWKHWXUQVOLS
n ' & 0 _9 _ ' 5 ZKLFKFRUUHVSRQGVWR F\
WKHGHILQLWLRQJLYHQE\(T 7KHPRPHQW0]nWKDWLVJHQHUDWHGZKHQWKH ZKHHO PRYHV DW
E
LQFUHDVHV ZLWK LQFUHDVLQJ FXUYDWXUH 5 XS WR LWV
PD[LPXPYDOXHWKDWLVDWWDLQHGDW5
DQGHTXDOV0]n4:HXVHWKHIRUPXOD
0]n ' 0]n 4 @ @ DUFWDQ T&Un 5R_n _ @ *\
ZLWKSDUDPHWHUT&Un7KLVTXDQWLW\PD\EHGLIILFXOWWRDVVHVVH[SHULPHQWDOO\WKH YDOXHLVH[SHFWHGWREHDUHDVRQDEOHHVWLPDWH,QDQ\FDVHWKHDUJXPHQWRI
E
VKRXOGUHPDLQ 7KLVPRPHQWDW LVPXOWLSOLHGZLWKWKHZHLJKWLQJ IXQFWLRQ *\ ( WR DFFRXQW IRU WKH DWWHQXDWLRQ WKURXJK WKH DFWLRQ RI ORQJLWXGLQDOVOLS :HREWDLQIRUWKHIDFWRU
' @ DUFFRV>0]n
&ULQ(T( XVLQJ( ZLWK_
'Un_ % gU @
_
)LQDOO\ZHPXVWWDNHFDUHRIWKHZHLJKWLQJIXQFWLRQ
_
U
6 4
ZKLFKLVLQWURGXFHGLQWKH
H[SUHVVLRQV ( IRU )[ WR OHW WKH ORQJLWXGLQDO IRUFH GLPLQLVK ZLWK LQFUHDVLQJVSLQ:HGHILQH
'
% 5Rn @
FRV >DUFWDQ [n
6(0,(03,5,&$/7<5(02'(/6
ZLWK
%[n ' S'[n % S'[n GI] FRV >DUFWDQ S'[n
:LWK WKH IDFWRUV
@
GHWHUPLQHG DQG VXEVWLWXWHG LQ WKH HTXDWLRQV
( WKHGHVFULSWLRQRIWKHVWHDG\VWDWHIRUFHDQGPRPHQWJHQHUDWLRQKDV EHHQFRPSOHWHG7RVKRZWKDWWKHIRUPXODHSURGXFHTXDOLWDWLYHFRUUHFWUHVXOWV DFROOHFWLRQRIFRPSXWHGFXUYHVKDYHEHHQSUHVHQWHGLQWKHGLDJUDPVRI)LJ ,Q7DEOHWKHYDOXHVRIWKHPRGHOSDUDPHWHUVKDYHEHHQOLVWHG7KHGLDJUDPV VKRZWKDWWKHIRUPXODHDUHSHUIHFWO\FDSDEOHWRDWOHDVWTXDOLWDWLYHO\DSSUR[LPDWH WKHFXUYHVWKDWKDYHEHHQFRPSXWHGZLWKWKHEUXVKVLPXODWLRQPRGHORI&KDSWHU FI)LJ ,WKDVQRWEHHQDWWHPSWHGWRILQGDEHVWILWIRUWKHSDUDPHWHUV
7DEOH
0DJLF)RUPXODK\SRWKHWLFDOSDUDPHWHUYDOXHV3DUDPHWHUVWKDWJRYHUQWKHLQIOXHQFH RIORQJLWXGLQDOVOLSDQGFKDQJHVLQYHUWLFDOORDGKDYHEHHQJLYHQWKHWHQWDWLYHYDOXH ]HUR
)]
1)]R
15R
P5H
5RD
P
S&[
S'[
S'[
S([
S([
S([
S([
S.[
S.[
S.[
S+[
S+[
S9[
S9[
S&\
S'\
S'\
S'\
S(\
S(\
S(\
S(\
S.\
S.\
S.\
S+\
S+\
S+\
S9\
S9\
S9\
S9\
T%]
T%]
T%]
T%]
T%]
T%]
T%]
T&]
T']
T']
T']
T']
T']
T']
T']
T']
T(]
T(]
T(]
T(]
T(]
T+]
T+]
T+]
T+]
U%[
U%[
U&[
U+[
U%\
U%\
U%\
U&\
U+\
U9\
U9\
U9\
U9\
U9\
U9\
VV]
VV]
VV]
VV]
S.\n
S'[n
S'[n
S'\n
S+\n
S+\n
S+\n
S+\n
Sg n
Sg n
T'Wn
T%Un
T'Un
T'Un
T&Un
T&Un
S'\n
S'\n
6(0,(03,5,&$/7<5(02'(/6
α = -4,
4
)\
3
-2, 0, 2, 4, 6, 12, 20
α = -4,
o
γ=ϕ=0
0]
W
o
12
20
2
[kN] 1
-1
α = -4o
-2
-3
-2
-1
α = -4,
4
)\
0
)[
α = 20
100
1
2
3
α = -4o
-3
[kN]
-2, 0, 2, 4, 6, 12, 20
4
-2
-1
α = -4,
o
γ=8
o
0]
ϕW = 0 4
2
0
α = -4o
-1 -2
-3
-2
-1
1
2
3
[kN] o
γ = 8o
200
ϕW = 0
100
α = -4o
4
0 [Nm] -100
0
0
)[
-2, 0, 2, 4, 6, 12, 20
20
[kN] 1
0
20
-200
)[
0
1
2
3
-3
[kN]
200
8 4 0
2
150
0
α = -4o
1
-2
-1
)[
o
3
[kN]
W
-200
3
)\
γ=ϕ=0
0 [Nm] -100
0
o
-2, 0, 2, 4, 6, 12, 20
200
4
0
1
2
3
[kN]
α = -4o 0 4
100
]
0
o
8 50
0 0 [Nm] -50
-1 -2 0
0.2
0.4
-100 0
0.6
Dϕ =-D/5
Dϕ =-D/5
0.2
W
D5= -0.15, -0.1, -0.05, 0, 0.05, 0.1, 0.15, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7
[kN]
200
0]
2 1 0
0.7
150 100 50 0
0
0.7
0.6
D5= -0.15, -0.1, -0.05, 0, 0.05, 0.1, 0.15, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7
3
)\
0.4
W
0
-50
-1
[Nm] -100
-2
D5=-0.15
-150
-3 -20 -15 -10
-5
α
0 5 [deg]
10
15
20
D5=-0.15
-200 -20 -15 -10
-5
α
0 5 [deg]
10
15
20
)LJ 0DJLF)RUPXODUHVXOWVIRUVWHDG\VWDWHUHVSRQVHRIIRUFHV)[\DQGPRPHQW0]WRVOLS
n
DQJOH VOLSUDWLR FDPEHUDQJOH DQGSDWKFXUYDWXUH
W
5
6(0,(03,5,&$/7<5(02'(/6
3O\6WHHUDQG&RQLFLW\
,QWKHIRUPXODHIRUWKHVLGHIRUFH)\DQGWKHDOLJQLQJWRUTXH0]WKHYHUWLFDOVKLIW 69\WKHKRUL]RQWDOVKLIW6+\DQGWKHUHVLGXDOWRUTXHSHDNYDOXH'UFRQWDLQWHUPV ZKLFKSURGXFHWKHLQLWLDOYDOXHVRIVLGHIRUFHDQGDOLJQLQJWRUTXHWKDWRFFXUDW VWUDLJKWDKHDGUXQQLQJDW
7KHVHLQLWLDOYDOXHVDUHNQRZQWREHWKHUHVXOW
RIFRQLFLW\DQGSO\VWHHUZKLFKDUHFRQQHFWHGZLWKQRQV\PPHWU\RIWKHW\UH FRQVWUXFWLRQ7KHVHWZRSRVVLEOHVRXUFHVUHVXOWLQPDUNHGO\GLIIHUHQWEHKDYLRXU RIWKHW\UHZKHQDWJHRPHWULFDOO\]HURVLGHVOLSWKHW\UHLVUROOHGIRUZDUGVRU EDFNZDUGV,IZHZRXOGKDYHDW\UHWKDWH[KLELWVSO\VWHHUEXWQRFRQLFLW\WKH JHQHUDWHG VLGH IRUFH ZLOO SRLQW WR WKH RSSRVLWH GLUHFWLRQ ZKHQ WKH ZKHHO LV FKDQJHGIURPIRUZDUGWREDFNZDUGUROOLQJ7KLVZRXOGDOVREHWKHFDVHZKHQRQ DWHVWULJWKHZKHHOPRYHVDWDVPDOOVWHHUDQJOH
DQGWKHURDGVXUIDFHPRWLRQ
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
7KHGHIRUPDWLRQVRIWKHW\UHUROOLQJRQDIULFWLRQVXUIDFHUHVHPEOHWKRVHWKDW ZRXOG RFFXU ZLWK D W\UH IUHH RI FRQLFLW\ DQG SO\VWHHU WKDW UROOV DW D VPDOO
)\ EDFNZDUGV
)\
)\
IRUZDUGV
)FRQ
ψ α
SO\
pure conicity
)LJ
)SO\
)SO\ )SO\ ψ pure ply-steer
)FRQ
ψ
α
SO\
combination
6LGHIRUFHYVVWHHUDQJOHFKDUDFWHULVWLFVDWIRUZDUGDQGEDFNZDUGUROOLQJVKRZLQJWKH FDVHVRISXUHFRQLFLW\SXUHSO\VWHHUDQGDFRPELQDWLRQRIERWKVLWXDWLRQV%HORZ WKHDVVRFLDWHGIRRWSULQWVRQDIULFWLRQOHVVVXUIDFHKDYHEHHQGHSLFWHG
6(0,(03,5,&$/7<5(02'(/6
FDPEHUDQGVLGHVOLSDQJOHUHVSHFWLYHO\,WLVWKHUHIRUHWHPSWLQJWRDVVXPHWKDW LQWKHVHFRPSDUDEOHFDVHVWKHPRPHQW0]DQGWKHIRUFH)\UHVSRQGDWWKHVDPH UDWHZKLFKZRXOGPHDQWKDWWKHDVVRFLDWHGSQHXPDWLFWUDLOVDUHHTXDOIRUSO\ VWHHU DQG VLGH VOLS DQG IRU FRQLFLW\ DQG FDPEHU 7KH DVVXPSWLRQ LV SDUWO\ VXSSRUWHG E\ DQ H[WHQVLYH H[SHULPHQWDO LQYHVWLJDWLRQ RI /HH ZKLFK DVVHVVHGDVWURQJOLQHDUFRUUHODWLRQEHWZHHQFRQLFLW\DQGWKHGLIIHUHQFHEHWZHHQ VOLSDQJOHVZKHUH)\DQG0]EHFRPHHTXDOWR]HUR,Q)LJWKLVGLIIHUHQFHLV GHVLJQDWHGDV
R
/HHIRXQGWKDWDW
R
FRQLFLW\LVDOPRVW]HUR7KLVPHDQVWKDW
IRU YDQLVKLQJ FRQLFLW\ WKH UHPDLQLQJ SO\VWHHU SURGXFHV D PRPHQW WKDW LV DSSUR[LPDWHO\ HTXDO WR PLQXV WKH SQHXPDWLF WUDLO IRU VLGH VOLS WLPHV WKH VLGH IRUFH:LWKWKHLQWURGXFWLRQRIWKHVPDOO HTXLYDOHQWVOLSDQJOH
0SO\ . & &0
SO\ ZLWK
&0 ' W&)
SO\
ZHKDYH
DQG
)SO\ . &)
SO\
$OVRDFFRUGLQJWR/HHWKHUHVLGXDOWRUTXHDW]HURVLGHIRUFH0]ULQ)LJLV VWURQJO\FRUUHODWHGZLWK
0]U . &0
R
DQGWKXVZLWKFRQLFLW\:HILQGIURPWKHGLDJUDP
R
,I ZH PD\ IXUWKHU DVVXPH WKDW IRU WKH FRQLFLW\ IRUFH DQG PRPHQW D VLPLODU FRUUHVSRQGHQFHZLWKFDPEHUUHVSRQVHH[LVWVZHZRXOGKDYHDIWHULQWURGXFLQJDQ HTXLYDOHQWFDPEHUDQJOH
0FRQ . &0
FRQ
FRQ ZLWK
&0 ' W &)
ZKHUHW ! UHSUHVHQWVWKHGLVWDQFHRIWKHSRLQWRIDSSOLFDWLRQRIWKHUHVXOWLQJ FDPEHUWKUXVWLQIURQWRIWKHFRQWDFWFHQWUHWKDWLVQHJDWLYHWUDLO 7KHFRQLFLW\ IRUFHEHFRPHV
0]
)\
no camber:
γ=0
&)α
0]U δo
&0α
α
)LJ7KHVLWXDWLRQQHDU]HURVOLSDQJOHIRUFHDQGPRPHQWGXHWRFRQLFLW\DQGSO\VWHHU
6(0,(03,5,&$/7<5(02'(/6
)FRQ . &)
FRQ
:LWKWKHVHDVVXPSWLRQVLWLVSRVVLEOHWRHVWLPDWHWKHFRQWULEXWLRQVRISO\VWHHU DQGFRQLFLW\LQWKHLQLWLDOYDOXHVRIVLGHIRUFHDQGDOLJQLQJWRUTXHDW
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
&0 &) % &0 &)
'
DQG FRQVHTXHQWO\ ZH FDQ ILQG WKH FDPEHU DQJOH
IURP WKH GLVWDQFH
,Q D
VLPLODUIDVKLRQWKHFRQLFLW\ZLOOEHDVVHVVHGLQWHUPVRIWKHHTXLYDOHQWFDPEHU DQJOH DQG DIWHU WKDW WKH SDUW DWWULEXWHG WR SO\VWHHU FDQ EH GHWHUPLQHG DQG H[SUHVVHG LQ WHUPV RI WKH HTXLYDOHQW VOLS DQJOH )LJXUH LOOXVWUDWHV WKH FRQYHUVLRQWRHTXLYDOHQWDQJOHV:HILQGVLPLODUWRWKHLQYHUVHRI
FRQ
'
R
&0 &) % &0 &)
no conicity no ply-steer
)\
0]
γ
0]Uγ
&)α γ=0
&0γ γ &)γ γ ∆αγ
δγ
&0α
α
α
I
)LJ &KDUDFWHULVWLFVRIDW\UHZLWKRXWSO\VWHHURUFRQLFLW\QHDUWKHRULJLQDW DQGZLWKRXWFDPEHU
ZLWK
6(0,(03,5,&$/7<5(02'(/6
0]
no camber:
γ =0
)\
0]U
conicity compensated
γ = -γFRQ
)\
0 ]
α
α
SO\
∆αo δo )LJ
α
I
&KDUDFWHULVWLFVRIDW\UHZLWKSO\VWHHUDQGFRQLFLW\VKRZLQJFRQYHUVLRQWRHTXLYDOHQW FDPEHUDQGVOLSDQJOH
7KHHTXLYDOHQWVOLSDQJOHFDQQRZEHREWDLQHGIURP
'
SO\
R
&
&) &)
9
VJQ F[
FRQ
7KHIRXUVOLSDQGFDPEHUVWLIIQHVVHVDUHDYDLODEOHIURP(TV( 7KHVLJQRIWKHIRUZDUGYHORFLW\KDVEHHQSURSHUO\LQWURGXFHG 7KHQH[WWKLQJZHKDYHWRGRLVH[SUHVVLQJ
R
DQG
R
LQWHUPVRIWKHVKLIWV
DQG WKH LQLWLDO UHVLGXDO WRUTXH DV GHILQHG LQ 6HFWLRQ :H REWDLQ ZLWK (
R
' 6+I
DQGIURP DQG)LJ
R'
)\ 0 ] % ' 0]U &0 &) &0
ZLWK&0 DFFRUGLQJWR(T( DQGWKHLQLWLDOUHVLGXDOWRUTXHIURP(
0]U ' 'UR ' )] 5RT']%T '] GI] @
0U
)LQDOO\WKHLQLWLDOVLGHIRUFHDQGWRUTXHDUHWREHUHPRYHGIURPWKHHTXDWLRQVE\ SXWWLQJWKHSDUDPHWHUVS+\S+\S9\S9\T ']DQGT']RUWKHVFDOLQJIDFWRUV +\
9\
DQG
0U
HTXDOWR]HURDQGE\UHSODFLQJLQ(TV( WKHRULJLQDOVLGH
VOLSLQSXWYDULDEOH
(
HII
'
WDQ
%
WDQ
SO\ VJQ
@VJQ9[E\LWVHIIHFWLYHYDOXH
9[
DQG LQ (TV( WKH RULJLQDO FDPEHU
VLQ E\ WKH HIIHFWLYH WRWDO
6(0,(03,5,&$/7<5(02'(/6
γeff
)\
0]
γeff = 0 (γ = -γcon )
camber ply-steer conicity
α eff
)LJ7KHILQDOGLDJUDPZLWKHIIHFWLYHVLGHVOLSDQGFDPEHUYDULDEOHV
FDPEHU
(
HII
'
%
VLQ
FRQ
ZKHUH LW VKRXOG EH UHDOLVHG WKDW ERWK
SO\
DQG
FRQ
DUH VPDOO TXDQWLWLHV 7KH
UHVXOWLQJGLDJUDPRI)LJVKRZVFKDUDFWHULVWLFVWKDWSDVVWKURXJKWKHRULJLQ ZKHQWKHHIIHFWLYHFDPEHUDQJOHLVHTXDOWR]HUR)RUWKHW\UHUXQQLQJ RQ WKH RWKHU VLGH RI WKH YHKLFOH WKH VDPH PRGHO FDQ EH XVHG EXW ZLWK
FRQ
LQ
FKDQJHGLQVLJQ 7KHWKHRU\DERYHLVEDVHGRQFRQVLGHUDWLRQVQHDUWKHRULJLQRIWKHVLGHIRUFH DQGDOLJQLQJWRUTXHYVVOLSDQJOHGLDJUDPV7KHLQWURGXFWLRQRIHIIHFWLYHVOLSDQG FDPEHUDQJOHVPD\KRZHYHUJLYHULVHWRVOLJKWFKDQJHVLQSHDNOHYHOVRIWKHVLGH IRUFHDQGSUREDEO\DOVRRIWKHDOLJQLQJWRUTXH)RUWKHIRUPHUWKHVLWXDWLRQPD\ EH UHSDLUHG E\ PRYLQJ WKH IRUFH FKDUDFWHULVWLF LQ D GLUHFWLRQ SDUDOOHO WR WKH WDQJHQWDW)\
6
9\
'
UHVXOWLQJLQDQDGGLWLRQDOYHUWLFDOVKLIW
) @ S
]
9\
% S9\ GI ] & S9\ % S9\ GI]
HII
DQGDQDVVRFLDWHGDGGLWLRQDOKRUL]RQWDOVKLIW
6+\ ' & 69\ .N\
7\UH3XOO ,ILGHQWLFDOW\UHVZRXOGEHILWWHGRQWKHIURQWD[OHRIDQDXWRPRELOHEXWZLWKWKH FRQLFLW\ IRUFHV SRLQWLQJ LQ WKH VDPH GLUHFWLRQ DQG WKH YHKLFOH PRYHV DORQJ D VWUDLJKW OLQH WKDW LV VLGH IRUFHV DUH HTXDO WR ]HUR D VWHHULQJ WRUTXH PXVW EH DSSOLHG WKDW RSSRVHV WKH UHVLGXDO WRUTXHV 0]U JHQHUDWHG E\ WKH IURQW W\UHV )LJ 7KLVLVDFWXDOO\RQO\DSSUR[LPDWHO\WUXHEHFDXVHZHPD\QHJOHFWWKH WUDLOV FRQQHFWHG ZLWK SO\VWHHU DQG FRQLFLW\ ZLWK UHVSHFW WR WKH YHKLFOH ZKHHO
6(0,(03,5,&$/7<5(02'(/6
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