7\UHDQG9HKLFOH '\QDPLFV +DQV%3DFHMND
3URIHVVRU(PHULWXV 'HOIW8QLYHUVLW\RI7HFKQRORJ\ &RQVXOWDQW712$XWRPRWLYH...
191 downloads
910 Views
15MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
7\UHDQG9HKLFOH '\QDPLFV +DQV%3DFHMND
3URIHVVRU(PHULWXV 'HOIW8QLYHUVLW\RI7HFKQRORJ\ &RQVXOWDQW712$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
WKDWLV DGULYLQJIRUFH,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 GULYLQJUDQJHRIVOLS LQWKHGHQRPLQDWRURI
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[LPXPYDOXH UDGLXVRIµIULFWLRQFLUFOH¶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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
)LJ9HKLFOHPRGHOVKRZLQJWKUHHGHJUHHVRIIUHHGRP ODWHUDO\DZDQGUROO
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
ZKLFKUHVXOW
IURPVXVSHQVLRQVSULQJVDQGDQWLUROOEDUV 7KHIRUHDQGDIWSRVLWLRQRIWKHFHQWUHRIJUDYLW\RIWKHERG\LVGHILQHGE\D DQGE LWVKHLJKWIROORZVIURPWKHGLVWDQFHK¶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
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 / DQGULJKW 5 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 WRUTXHUHVXOWVIURPWKHVLGHIRUFH RIFRXUVHDOVRIURPWKHKHUHQRWFRQVLGHUHG 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 \
,QDGGLWLRQWKHVLGHIRUFH EXWDOVRWKHIRUHDQGDIWIRUFH 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
SRVLWLYH OHDQLQJ
RXWZDUGV )RUWKHOHIWDQGULJKWZKHHOVZHKDYHWKHLQLWLDODQJOHV
'& ' &
L/R
L/R
LR
L5R
LR
L5R
' '
LR
LR
$GGLQJDOOUHOHYDQWFRQWULEXWLRQV WR WRJHWKHU\LHOGVWKHWRWDOVWHHU DQJOHIRUHDFKRIWKHZKHHOV 7KHHIIHFWLYHFRUQHULQJVWLIIQHVVRIDQD[OH&HIILLVQRZGHILQHGDVWKHUDWLR RIWKHD[OHVLGHIRUFHDQGWKHYLUWXDOVOLSDQJOH7KLVDQJOHLVGHILQHGDVWKHDQJOH EHWZHHQWKHGLUHFWLRQRIPRWLRQRIWKHFHQWUHRIWKHD[OHL DFWXDOO\DWURDGOHYHO ZKHQWKHYHKLFOHYHORFLW\ZRXOGEHYHU\ORZDQGDSSURDFKHV]HUR WKHQDOVR)\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 DOLJQLQJWRUTXHV OHIWDQGULJKW GLVWRUWLRQVZLOODULVHUHVXOWLQJLQDQLQFUHPHQWDO VWHHUDQJOH
F
RIWKHIURQWZKHHOV
ZLOOEHQHJDWLYHLQ)LJIRUWKHFDVHRI
F
MXVWVWHHUFRPSOLDQFH 6LQFHORDGWUDQVIHULVQRWFRQVLGHUHGLQWKLVH[DPSOHWKH VLWXDWLRQDWWKHOHIWDQGULJKWZKHHOVDUHLGHQWLFDO LQLWLDOWRHDQGFDPEHUDQJOHV EHLQJGLVUHJDUGHG 7KHIURQWW\UHVOLSDQJOHLVGHQRWHGZLWK DQJOHRIWKHD[OHLVGHQRWHGZLWK
D
'
ZKHUHERWK
&
7KHµYLUWXDO¶VOLS
DQGHTXDOV FI)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[OHFRQVLGHUHG DWVWHDG\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[OH XSWROLIWRIIRI
WKH LQQHU W\UH DIWHU ZKLFK WKH RWKHU D[OH PD\ WDNH RYHU WR DFFRPPRGDWH WKH LQFUHDVHGWRWDOORDGWUDQVIHU 'UDZWKHLQGLYLGXDOFXUYHVRI)\/DQG)\5 IRUWKHOHIWDQGULJKWW\UH DVDIXQFWLRQ RI
ZKLFKDSSHDUWRDULVHXQGHUWKHORDGWUDQVIHUFRQGLWLRQFRQVLGHUHGKHUH
)LQDOO\ SORW WKHVH IRUFHV DV D IXQFWLRQ RI WKH YHUWLFDO ORDG )] UDQJLQJ IURP 1 1RWHWKHYDULDWLRQRIWKHODWHUDOIRUFHRIDQLQGLYLGXDO OHIWRUULJKW W\UH LQWKLVVDPHUDQJHRIYHUWLFDOORDGZKLFKPD\EHFRYHUHGLQDOHIWDQGLQDULJKW KDQGWXUQDWLQFUHDVLQJVSHHGRIWUDYHOXQWLO DQGSRVVLEO\EH\RQG WKHPRPHQWWKDW RQHRIWKHZKHHOV WKHLQQHUZKHHO 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)LJ 7KH 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
)LJ 6LPSOHFDUPRGHOZLWKVLGHIRUFHFKDUDFWHULVWLFVIRUIURQWDQGUHDU GULYHQ D[OH
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
ZKLFKDOORZVWKHXVHRIDWZRZKHHOYHKLFOHPRGHO7KHVWHHUDQGVOLSDQJOHVZLOO EHUHVWULFWHGWRUHODWLYHO\VPDOOYDOXHV7KHQWKHYDULDWLRQRIWKHJHRPHWU\PD\ EHUHJDUGHGWRUHPDLQOLQHDUWKDWLV FRV
.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)LJ XVHGLQWKHVXEVHTXHQWVHFWLRQFDQEHHDVLO\ DVVHVVHG ,Q 6XEVHFWLRQ WKH HTXDWLRQV IRU WKH FDU ZLWK WUDLOHU ZLOO EH HVWDEOLVKHG7KHSRVVLEOHLQVWDELOLW\RIWKHPRWLRQZLOOEHVWXGLHG :HZLOOHPSOR\/DJUDQJH¶VHTXDWLRQVWRGHULYHWKHHTXDWLRQVRIPRWLRQ)RU DV\VWHPZLWKQGHJUHHVRIIUHHGRPQ JHQHUDOLVHG 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
VHWVRIYDULDEOHVDUH WKHGRWVUHIHUULQJWRGLIIHUHQWLDWLRQZLWKUHVSHFWWRWLPH
' ;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\8LVEXLOWXSLQWKHVXVSHQVLRQVSULQJV LQFOXGLQJWKHUDGLDO 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
[]
'
)
)
[
[
&)
%) %) \
& PKN X0 & UY '
D)
% PKN % PKN Y0 % 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(T E DUHWREHUHJDUGHG)RUDOLQHDUPRGHOWKHORDGWUDQVIHUFDQEHQHJOHFWHGLILQLWLDO OHIWULJKWRSSRVLWH ZKHHODQJOHVDUHGLVUHJDUGHG:HKDYHDWVWHDG\VWDWH HIIHFW 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 % PKN Y0 % 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
)URPWKHHTXDWLRQV EDQGF 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
P Y0
% 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\DZLQWKHGLUHFWLRQRIWKHIRUFH LIEHKLQGWKHQDJDLQVWWKH IRUFH7KHSRLQWLVRILQWHUHVWZKHQGLVFXVVLQJWKHVWHHULQJFKDUDFWHULVWLFVDQG VWDELOLW\
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
/LQHDU6WHDG\6WDWH&RUQHULQJ6ROXWLRQV
:HDUHLQWHUHVWHGLQWKHSDWKFXUYDWXUH 5 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
7KHVWHHUDQJOHYHUVXVODWHUDODFFHOHUDWLRQDWFRQVWDQWSDWKFXUYDWXUH OHIWJUDSK 7KH GLIIHUHQFHLQVOLSDQJOHYHUVXVODWHUDODFFHOHUDWLRQDQGWKHUHTXLUHGVWHHUDQJOHDWD JLYHQSDWKFXUYDWXUH ULJKWJUDSK 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
'
)
]
&
&
)
]
&
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
)
\
'
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
!
$VLVVKRZQ
E\WKHH[SUHVVLRQV WKHVLJQVRI DQGVDUHGLIIHUHQW&RQVHTXHQWO\DVRQH PLJKWH[SHFWZKHQWKHFHQWULIXJDOIRUFHLVFRQVLGHUHGDVWKHH[WHUQDOIRUFHD YHKLFOHDFWVRYHUVWHHUHGZKHQWKHQHXWUDOVWHHUSRLQWOLHVLQIURQWRIWKHFHQWUHRI JUDYLW\DQGXQGHUVWHHUHGZKHQ6OLHVEHKLQGWKHFJ$VZHZLOOVHHODWHURQWKH DFWXDO QRQOLQHDU YHKLFOH PD\ FKDQJH LWV VWHHULQJ FKDUDFWHU ZKHQ WKH ODWHUDO DFFHOHUDWLRQ LQFUHDVHV ,W DSSHDUV WKHQ WKDW WKH GLIIHUHQFH LQ VOLS DQJOH LV QR ORQJHUGLUHFWO\UHODWHGWRWKHXQGHUVWHHUJUDGLHQW &RQVLGHUDWLRQRI(T UHYHDOVWKDWLQWKHOHIWKDQGJUDSKRI)LJWKH GLIIHUHQFH LQ VOLS DQJOH FDQ EH PHDVXUHG DORQJ WKH RUGLQDWH VWDUWLQJ IURP WKH YDOXHO5,WLVRILQWHUHVWWRFRQYHUWWKHGLDJUDPLQWRWKHJUDSK VKRZQRQWKH ULJKWKDQGVLGHRI)LJZLWKRUGLQDWHHTXDOWRWKHGLIIHUHQFHLQVOLSDQJOH,Q WKDWZD\WKHGLDJUDPEHFRPHVPRUHIOH[LEOHEHFDXVHWKHYDOXHRIWKHFXUYDWXUH 5 PD\ EH VHOHFWHG DIWHUZDUGV 7KH KRUL]RQWDO GRWWHG OLQH LV WKHQ VKLIWHG YHUWLFDOO\DFFRUGLQJWRWKHYDOXHRIWKHUHODWLYH FXUYDWXUHO5FRQVLGHUHG7KH GLVWDQFHWRWKHKDQGOLQJOLQHUHSUHVHQWVWKHPDJQLWXGHRIWKHVWHHUDQJOH )LJXUHGHSLFWVWKHUHVXOWLQJVWHDG\VWDWHFRUQHULQJPRWLRQ7KHYHKLFOH VLGHVOLSDQJOH KDVEHHQLQGLFDWHG,WLVRILQWHUHVWWRQRWHWKDWDWORZVSHHGWKLV DQJOHLVQHJDWLYHIRUULJKWKDQGWXUQV%H\RQGDFHUWDLQYDOXHRIVSHHGWKHW\UH VOLSDQJOHVKDYHEHFRPHVXIILFLHQWO\ODUJHDQGWKHYHKLFOHVOLSDQJOHFKDQJHVLQWR SRVLWLYHYDOXHV,Q([HUFLVHWKHVOLSDQJOH
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
&
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
)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
9 9
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\ZHOOZLWKWKH UHVSRQVHWLPHVIRXQGLQYHKLFOHPRGHOVLPXODWLRQ VWXGLHV$UHPDUNDEOHUHVXOWLVWKDWIRUDQXQGHUVWHHUHGDXWRPRELOHWKHUHVSRQVH WLPHLVVPDOOHUWKDQIRUDQRYHUVWHHUHGFDU
)RUFHG/LQHDU9LEUDWLRQV
7KHFRQYHUVLRQRIWKHHTXDWLRQVRIPRWLRQ LQWRWKHVWDQGDUGVWDWHVSDFH UHSUHVHQWDWLRQ LV XVHIXO ZKHQ WKH OLQHDU V\VWHP SURSHUWLHV DUH WKH VXEMHFW RI LQYHVWLJDWLRQ7KHV\VWHPDWKDQGLVRIWKHVHFRQGRUGHUDQGKHQFHSRVVHVVHVWZR VWDWHYDULDEOHVIRUZKLFKZHFKRRVH YDQGU7KHV\VWHPLVVXEMHFWHGWRDVLQJOH LQSXWVLJQDO WKHVWHHUDQJOH 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 YHKLFOH WKHODWHUDODFFHOHUDWLRQD\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
''
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
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
UDG a 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\VWHP EHFDXVHRIWKHM WHUPLQWKHQXPHUDWRUWKH\DZUDWHDPSOLWXGHUHVSRQVHWHQGVWRDGHFD\DWD G%
SHURFWDYHUDWH LISORWWHGLQORJORJVFDOH 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\UHVSRQVHIXQFWLRQV YHKLFOHSDUDPHWHUVDFFRUGLQJWR7DEOH
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
GHJUHHV7KHSKDVHLQFUHDVHDWORZIUHTXHQFLHVDQGKLJKHUVSHHGVLVGXHWRWKH SUHVHQFHRIWKHVSHHG9LQWKDWVDPHWHUP$WVSHHGVEH\RQGDSSUR[LPDWHO\WKH FKDUDFWHULVWLFVSHHGWKHFRUUHVSRQGLQJ ODVW WHUPLQWKHGHQRPLQDWRUKDVOHVV LQIOXHQFHRQWKHLQLWLDOVORSHRIWKHSKDVHFKDUDFWHULVWLF7KHODWHUDODFFHOHUDWLRQ UHVSRQVH VKRZQLQWKHFHQWUHJUDSKRI)LJJLYHVDILQLWHDPSOLWXGHDW IUHTXHQFLHVWHQGLQJWRLQILQLW\EHFDXVHRIWKHSUHVHQFHRI
LQWKHQXPHUDWRU
)RUWKHVDPHUHDVRQWKHSKDVHODJJRHVEDFNWR]HURDWODUJHIUHTXHQFLHV7KH VLGHVOLSSKDVHUHVSRQVHWHQGVWRGHJUHHV DWODUJHUVSHHGV 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)LJ URWDWHG DQWLFORFNZLVH 7KH GLDJUDP PD\ EH FRPSOHWHG E\ DWWDFKLQJ WKH JUDSK WKDW VKRZV IRU D VHULHV RI VSHHGV9WKHUHODWLRQVKLSEHWZHHQODWHUDODFFHOHUDWLRQ LQJXQLWV 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 EHHQLQGLFDWHG ZKHHOEDVHO 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
VSHHG 9
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\WKHPDLQEUDQFKRIKDQGOLQJFXUYHLVGUDZQ IURQW UHDU
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
)LJ 7\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\UDWLRQN N
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\RFFXU ODUJHVWHHUDQJOHDVLQGLFDWHGLQPLGGOHGLDJUDP 7KHORZHUGLDJUDPVKRZVWKHFDVHZLWKFRPSOHWHXQGHUVWHHUIHDWXULQJDVWDEOHPDLQ EUDQFK
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
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
RIWKHQRUPDOLVHGD[OHFKDUDFWHULVWLFVVWDUWVWRVKRZ
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
([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[OHFKDUDFWHULVWLFV DEDQGF VKRZQLQWKH JUDSKRI)LJ7KHGLPHQVLRQVRIWKHYHKLFOHPRGHODUH D
E
òO
P)RU
WKHW\UHVZHPD\HPSOR\D[OHFKDUDFWHULVWLFVGHVFULEHGE\WKH0DJLF)RUPXOD
)\
'VLQ>&DUFWDQ^%
( %
:HGHILQH WKHSHDNVLGHIRUFH' VRWKDW%
DUFWDQ %
`@
:) DQGWKHFRUQHULQJVWLIIQHVV& ]
)
%&'
F) )]
F) & )RUWKHVL[W\UHD[OHFRQILJXUDWLRQVWKHSDUDPHWHUYDOXHVKDYH
EHHQJLYHQLQWKHWDEOHEHORZ
D[OH
FDVH
F)
&
(
DE
F
D
E
F
IURQW
UHDU
'HWHUPLQHIRUHDFKRIWKHWKUHHFRPELQDWLRQV WZRGU\RQHZHW
7KHKDQGOLQJFXUYH FI)LJ
7KHFRPSOHWHKDQGOLQJGLDJUDP FI)LJ
7KHSRUWLRQRIWKHFXUYHVZKHUHWKHYHKLFOHVKRZVDQRYHUVWHHUQDWXUH
7KHVWDELOLW\ERXQGDU\ DVVRFLDWHGZLWKWKHVHRYHUVWHHUUDQJHV LQWKH D\JYHUVXV
,QGLFDWHLQWKHGLDJUDP RULQDVHSDUDWHJUDSK
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<
PRYHPHQW DORQJ D FXUYHG WUDFN 7KH VOLS DQJOHV RI WKH WZR UHDU D[OHV DUH GLIIHUHQWVRWKDWDFRXQWHUDFWLQJWRUTXHDULVHV7KLVWRUTXHJHWVODUJHUZKHQWKH WXUQLQJUDGLXVEHFRPHVVPDOOHU7KLVPD\IRULQVWDQFHRFFXUDWDJLYHQOHYHORI ODWHUDODFFHOHUDWLRQ:KHQDWWKLVOHYHOWKHVSHHGEHFRPHVORZHUWKHFXUYDWXUH PXVW EHFRPH ODUJHU DQG WKH RSSRVLQJ WRUTXH ZLOO LQFUHDVH ZKLFK HQWDLOV DQ LQFUHDVHGIURQWVWHHUDQJOHWRJHQHUDWHDODUJHUVLGHIRUFHQHHGHGWREDODQFHWKH YHKLFOH 7KLV LQFUHDVHG VWHHU DQJOH JRHV RQ WRS RI WKH VWHHU DQJOH ZKLFK ZDV DOUHDG\ODUJHUEHFDXVHRIWKHLQFUHDVHGO5+HUHOLVWKHDYHUDJHZKHHOEDVH &RQVHTXHQWO\ LQ WKH KDQGOLQJ GLDJUDP WKH SRLQWV RQ WKH KDQGOLQJ FXUYH EHORQJLQJ WR WKH ORZHU VSHHG OLH PRUH WR WKH OHIW )RU D GHWDLOHG VWXG\ RQ WKLV VSHFLDOVXEMHFWZHUHIHUWR:LQNOHU
$VVHVVPHQWRIWKH,QIOXHQFHRIWKH3QHXPDWLF7UDLORQWKH+DQGOLQJ&XUYH
6R IDU WKH GLUHFW LQIOXHQFH RI WKH SQHXPDWLF WUDLOV KDYH QRW EHHQ WDNHQ LQWR DFFRXQW$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 7KHUHVXOWLVDILUVWRUGHUQRQOLQHDUHTXDWLRQ XVLQJN
,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 WKHWUDMHFWRULHV VROXWLRQFXUYHV 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 YHUWLFDOLQWHUFHSWV G
'
JO 9
)
\
G
'&
JO
)
\
)
G
ß
DE
% %
]
)
% &
LQWHUFHSWVXQGHU G
\
]
9
)
ß4
)
KRUL]RQWDOLQWHUFHSWV G
'
)
]
\
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$OVRVKRZQ WKHWKUHH 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 s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
RQ WKH
VWDELOLW\PDUJLQ GLVWDQFHEHWZHHQVWDEOHSRLQWDQGVHSDUDWUL[ 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\PDUJLQ V\VWHPRI)LJ WRS
δ
δ
=0
= 0.04
δ
= 0.15
0.1
0.1
)LJ,QIOXHQFHRIVWHHULQJRQWKHVWDELOLW\PDUJLQ V\VWHPRI)LJ ERWWRP
7<5(&+$5$&7(5,67,&6$1'9(+,&/(+$1'/,1*$1'67$%,/,7<
IRU WKHWZRYHKLFOHVFRQVLGHUHGLQ)LJ7KHV\VWHPRI)LJLVFOHDUO\ PXFKPRUHVHQVLWLYH$QLQFUHDVHLQ EXWDOVRDQLQFUHDVHLQVSHHG9 UHGXFHV
WKHVWDELOLW\PDUJLQXQWLOLWLVWRWDOO\YDQLVKHGDVVRRQDVWKHWZRVLQJXODUSRLQWV PHUJH DOVRWKHFRUUHVSRQGLQJSRLQWV,DQG,,RQWKHKDQGOLQJFXUYHRI)LJ DQGWKHGRPDLQEUHDNVRSHQ$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
VLWXDWLRQDSSHDUVWREHVWDEOHDJDLQ IRFXV DVKDVEHHQVWDWHGEHIRUH)RUWKH XQGHUVWHHUHGYHKLFOHRI)LJVWDELOLW\LVSUDFWLFDOO\DOZD\VHQVXUHG )RUDIXUWKHUDSSUHFLDWLRQRIWKHSKDVHGLDJUDPLWLVRILQWHUHVWWRGHWHUPLQH WKHQHZLQLWLDOVWDWH URYR DIWHUWKHDFWLRQRIDODWHUDOLPSXOVHWRWKHYHKLFOH FI )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 DQGGLUHFWLRQRIWKHLPSXOVH6 N
,P
DE (YLGHQWO\DQLPSXOVHDFWLQJDWWKH
UHDU LQRXWZDUGGLUHFWLRQ 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)]LR L
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[ ZKLFKPLJKWEH PLQXVWKHIRUZDUGFRPSRQHQWRI 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 LQ7DEOH DERYH(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 VHWVRIFXUYHVKDYHEHHQSORWWHG RQHVHWIRUFRQVWDQWYDOXHVRIWKHYHKLFOHVLGH 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
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
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
)LJXUH JLYHVDWRSYLHZRIWKHV\VWHPZLWKWKUHHGHJUHHVRIIUHHGRP7KH
nDQGWKHYHKLFOH
DOWHUQDWLYHVHWRIWKUHHYDULDEOHVYUDQGWKHDUWLFXODWLRQDQJOH
YHORFLW\9 DSDUDPHWHU ZKLFKDUHQRWFRQQHFWHGWRWKHLQHUWLDOD[HVV\VWHP ; < KDV EHHQ LQGLFDWHG DV ZHOO DQG ZLOO EH HPSOR\HG ODWHU RQ 7KH NLQHWLF HQHUJ\IRUWKLVV\VWHPEHFRPHVLIZHQHJOHFWDOOWKHWHUPVRIWKHVHFRQGRUGHU RIPDJQLWXGH SURGXFWVRIYDULDEOHV
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
< )LJ 6LQJOHWUDFNPRGHORIFDUWUDLOHUFRPELQDWLRQ
)\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 LQDGGLWLRQ K
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$7,216
LQSXWYHFWRU
ρ κ Ω
RXWSXWYHFWRU
UDGLDOGHIOHFWLRQ
)
]
ORQJLWXGLQDOVOLS
)
[
VSHHGRIUHYROXWLRQ
0
α ODWHUDOVOLSDQJOH ϕ VSLQWXUQVOLS γ FDPEHUDQJOH
)
\
\
QRUPDOORDG ORQJLWXGLQDOIRUFH UROOLQJUHVLVWPRPHQW FRUQHULQJ VLGH IRUFH
0
]
VHOI DOLJQLQJWRUTXH
0
[
RYHUWXUQLQJFRXSOH
)LJ,QSXWRXWSXWTXDQWLWLHV URDGVXUIDFHFRQVLGHUHGIODW
EHWUHDWHGDVLQSXWYDULDEOHVLQWRWKHV\VWHP)LJXUHSUHVHQWVWKHµYHFWRUV¶RI LQSXWDQGRXWSXWFRPSRQHQWV,QWKLVGLDJUDPWKHW\UHLVDVVXPHGWREHXQLIRUP DQGWRPRYHRYHUDIODWURDGVXUIDFH7KHLQSXWYHFWRUVWHPVIURPPRWLRQVRIWKH ZKHHOUHODWLYHWRWKHURDG$SUHFLVHGHILQLWLRQRIWKHVHLQSXWTXDQWLWLHVLVJLYHQ LQWKHQH[WVHFWLRQ )RUVPDOOGHYLDWLRQVIURPWKHVWUDLJKWDKHDGPRWLRQDOLQHDUGHVFULSWLRQRIWKH EHKDYLRXUPD\EHJLYHQ7KHQLWLVDGYDQWDJHRXVWRUHFRJQL]HWKHIDFWWKDWWKH UHVSRQVHV WR WKH V\PPHWULF DQG DQWLV\PPHWULF PRWLRQV RI WKH DVVXPHGO\ V\PPHWULFZKHHOW\UHV\VWHPFDQEHFRQVLGHUHGDVXQFRXSOHG)LJXUHVKRZV WKHVHSDUDWHIXQFWLRQEORFNVZLWKWKHLQSXWDQGRXWSXWTXDQWLWLHV+HUHZHKDYH DOVRFRQVLGHUHGWKHSRVVLELOLW\RILQSXWIURPYDULDWLRQVLQURDGVXUIDFHJHRPHWU\ DQG IURP W\UH QRQXQLIRUPLWLHV UHVXOWLQJ LQ HJ RXWRIURXQGQHVV VWLIIQHVV YDULDWLRQVDQGµEXLOWLQ¶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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$7,216
VXUIDFHFRQGLWLRQV JURRYHV 7KHWLPHUDWHRIFKDQJHRIWKLVUHODWLYHSRVLWLRQLV QHHGHGQRWRQO\IRUSRVVLEOHK\VWHUHVLVHIIHFWVEXWPDLQO\IRUWKHGHWHUPLQDWLRQ RIWKHVRFDOOHGµVOLS¶RIWKHZKHHOZLWKUHVSHFWWRWKHJURXQG ,IWKHURDGVXUIDFHQHDUWKHFRQWDFWSDWFKFDQEHDSSUR[LPDWHGE\DIODWSODQH WKDW LV ZKHQ WKH VPDOOHVW FRQVLGHUHG ZDYHOHQJWK RI WKH GHFRPSRVHG VXUIDFH YHUWLFDOSURILOHLVODUJHZLWKUHVSHFWWRWKHFRQWDFWOHQJWKDQGLWVDPSOLWXGHVPDOO WKHGLVWDQFHRIWKHZKHHOFHQWUHWRWKHURDGSODQHDQGWKHDQJOHEHWZHHQZKHHO SODQHDQGWKHQRUPDOWRWKHURDGVXUIDFHZLOOVXIILFHLQDGGLWLRQWRWKHVHYHUDO VOLSTXDQWLWLHVDQGWKHUXQQLQJVSHHGRIWKHZKHHO )RUWKHGHILQLWLRQRIWKHYDULRXVPRWLRQDQGSRVLWLRQLQSXWTXDQWLWLHVOLVWHGLQ )LJLWLVKHOSIXOWRFRQVLGHU)LJ$QXPEHURISODQHVKDYHEHHQGUDZQ 7KHURDGSODQHDQGWKHZKHHOFHQWUHSODQH ZLWKOLQHRILQWHUVHFWLRQDORQJWKH XQLWYHFWRUO DQGWZRSODQHVQRUPDOWRWKHURDGSODQHRQHRIZKLFKFRQWDLQVWKH YHFWRUODQGWKHRWKHUWKHXQLWYHFWRUVZKLFKLVGHILQHGDORQJWKHZKHHOVSLQD[LV )URPWKHILJXUHIROORZVWKHGHILQLWLRQRIWKHFRQWDFWFHQWUH&DOVRGHVLJQDWHGDV WKHSRLQWRILQWHUVHFWLRQ RIWKHWKUHHSODQHV 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\WKHXQLWYHFWRUQ SRVLWLYHXSZDUGV )LJXUH DOVR VKRZV WZR V\VWHPV RI D[HV EHVLGHV WKH LQHUWLDO WULDG )LUVWO\ ZH KDYH LQWURGXFHGWKHURDGFRQWDFWD[HVV\VWHP &[\] RIZKLFKWKH[D[LVSRLQWV IRUZDUGVDORQJWKHOLQHRILQWHUVHFWLRQ O WKH]D[LVSRLQWVGRZQZDUGVQRUPDO WRWKHURDGSODQH Q DQGWKH\D[LVSRLQWVWRWKHULJKWDORQJWKHWUDQVYHUVHXQLW YHFWRUW6HFRQGO\WKHZKHHOD[OHV\VWHPRID[HV $
KDVEHHQGHILQHG
ZLWKWKH D[LVSDUDOOHOWRWKH[D[LVWKH D[LVDORQJWKHZKHHOVSLQGOHD[LV V DQGWKH D[LVDORQJWKHUDGLXV U 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 FRQVLGHUDEOH PRWRUF\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\ORZIULFWLRQ LF\ 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 $FFRUGLQJWRWKLVGHILQLWLRQZHZLOOKDYHWKHVLWXDWLRQWKDWZKHQDZKHHOUROOV IUHHO\ WKDWLV DW 0 D ORQJLWXGLQDOVOLS
DWFRQVWDQWVSHHGRYHUDIODWHYHQURDGVXUIDFHWKH
LVHTXDOWR]HUR7KLVQRWZLWKVWDQGLQJWKHIDFWWKDWDWIUHH
UROOLQJ VRPH IRUH DQG DIW GHIRUPDWLRQV ZLOO RFFXU EHFDXVH RI WKH SUHVHQFH RI K\VWHUHVLVLQWKHW\UHWKDWJHQHUDWHVDUROOLQJUHVLVWDQFHPRPHQW0\7KURXJKWKLV DUROOLQJUHVLVWDQFHIRUFH)U
0\UDULVHVZKLFKQHFHVVDULO\LVDFFRPSDQLHGE\
WDQJHQWLDOGHIRUPDWLRQV:HPD\DJUHHWKDWDWWKHLQVWDQWRIREVHUYDWLRQSRLQW 6 WKDWOLHVRQWKHVOLSFLUFOHDQGLVDWWDFKHGWRWKHZKHHOULPKDVUHDFKHG LWV ORZHVWSRVLWLRQWKDWLV RQWKHOLQHDORQJWKHUDGLXVYHFWRUU$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
7KHORQJLWXGLQDOVOLS VRPHWLPHVFDOOHGWKHVOLSUDWLR LVGHQRWHGE\ DQGPD\ EHWHQWDWLYHO\GHILQHGDVWKHUDWLRRIORQJLWXGLQDOVOLSYHORFLW\9V[RISRLQW6DQG WKHIRUZDUGVSHHGRIWKHZKHHOFHQWUH9[
'&
9V[ 9[
RUZLWK
'&
9[& UH 9[
7KLVDJDLQKROGVIRUDZKHHORQDIODWURDGDQGZLWK
0
$PRUHJHQHUDODQG
SUHFLVHGHILQLWLRQRI ZLOOEHJLYHQODWHURQ7KHVLJQRIWKHORQJLWXGLQDOVOLS KDVEHHQFKRVHQVXFKWKDWDWGULYLQJZKHQ)[! LVSRVLWLYHDQGDWEUDNLQJ ZKHQ)[ LVQHJDWLYH:KHQWKHZKHHOLVORFNHG
ZHREYLRXVO\KDYH
,QWKHOLWHUDWXUHWKHV\PEROV RU6 LVPRUHFRPPRQO\XVHGWRGHQRWHWKH VOLSUDWLR 7KHDQJXODUVSHHGRIUROOLQJ
U
PRUHSUHFLVHO\GHILQHGIRUWKHFDVHRIPRYLQJ
RYHUXQGXODWHGURDGVXUIDFHVLVWKHWLPHUDWHRIFKDQJHRIWKHDQJOHEHWZHHQWKH UDGLXVFRQQHFWLQJ6DQG$ 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 DGGLWLRQ 9U
9[1RWHWKDWDWZKHHOORFN
U
DQGZLWK
0
LQ
WKHDQJXODUVSHHGRIUROOLQJ
LVQRWHTXDOWR]HURZKHQWKHZKHHOPRYHVRYHUDURDGZLWKDFXUYHGYHUWLFDO
U
SURILOH WKHQQRWDOZD\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 QRUPDOWRWKHURDGDERXWWKHQRUPDOWRWKHURDG7KHQ FI)LJ
' & %0
VLQ
7KLVHTXDWLRQIRUPVDFRUUHFWEDVLVIRUDJHQHUDOGHILQLWLRQRI OHYHOURDGVXUIDFHV,WVFRPSXWDWLRQLVVWUDLJKWIRUZDUGLI
DOVRRQQRQ
LVDYDLODEOHIURP
ZKHHOG\QDPLFVFDOFXODWLRQV
7KHORQJLWXGLQDOUXQQLQJVSHHG9F[ LVGHILQHGDVWKHORQJLWXGLQDOFRPSRQHQW
RIWKHYHORFLW\RISURSDJDWLRQRIWKHLPDJLQDU\SRLQW& RQUDGLXVYHFWRUU LQ WKHGLUHFWLRQRIWKH[D[LV YHFWRUO ,QFDVHWKHZKHHOLVPRYHGLQVXFKDZD\
WKDWWKHVDPHSRLQWUHPDLQVLQFRQWDFWZLWKWKHURDGZHZRXOGKDYH9F[
9 V[
7KLVFRUUHVSRQGVWRZKHHOORFNZKHQWKHURDGLVIODWDQGWKHYHKLFOHSLWFKUDWH LV]HUR)RUDIUHHO\UROOLQJW\UHWKHORQJLWXGLQDOUXQQLQJVSHHGHTXDOVWKHOLQHDU
VSHHGRIUROOLQJ 9F[ 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
RILQWHUVHFWLRQ XQLWYHFWRUO DERXWWKH]D[LVQRUPDOWRWKHURDG FI)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
\
] )LJ5RWDWLRQDOVOLSUHVXOWLQJIURPSDWKFXUYDWXUHDQGZKHHOFDPEHU VOLSDQJOH
%$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
SRLQW4ZLWKUHVSHFWWRWKHLQHUWLDOIUDPH 2 [ \ ] LVJLYHQE\YHFWRUT7KH QRUPDOWRWKHURDGSODQHLVGHILQHGE\XQLWYHFWRUQ7KHORFDWLRQRIDUHIHUHQFH SRLQW%RIWKHYHKLFOHLVGHILQHGE\YHFWRUEDQGWKHORFDWLRQRIWKHZKHHOFHQWUH $E\ED FI)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 EHFRPHV FI)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 DQGUZLWKUHVSHFWWRDQLQHUWLDOWULDG FKRRVH
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]HUR 9V[
'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 LQORQJLWXGLQDOGLUHFWLRQ 9V[
$JDLQZHDVVXPHUH U)RUWKHVDNHRIVLPSOLI\LQJ
WKHFRPSOH[SUREOHPLWLVDVVXPHGWKDWWKHZKHHOFHQWUHKHLJKW+LVDJLYHQFRQVWDQW
δ
%
X ε
+ IL[HG
a
$ s
X r
& )LJ 6LGHYLHZRIIURQWSDUWRIYHKLFOH PRWRUF\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)LJ DV\VWHPRID[HV 2 [ \ ] 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 EHWWHU WKHSRLQWRILQWHUVHFWLRQWUDYHOVZLWKDQDVVXPHGO\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[LV WKDWLVWKH\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
WKHHIIHFWRIWKHWLPHUDWHRIFKDQJHRIWKHFDPEHUDQJOH ZKHHO
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 RIFRQWDFW UR \
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\UH OHIW DQGDFDUW\UH ULJKW LQFDPEHUHGSRVLWLRQWRXFKLQJWKH DVVXPHGO\ IULFWLRQOHVV URDG VXUIDFH 7KH IRUPHU ZLWKRXW DQG WKH ODWWHU ZLWK WRUVLRQDQGEHQGLQJRIWKHFDUFDVVDQGEHOW7RS UHDUYLHZ ERWWRP WRSYLHZ
%$6,&7<5(02'(//,1*&216,'(5$7,216
)RUKRPRJHQHRXVUROOLQJERGLHV HJDUDLOZD\ZKHHORUDUXEEHUEDOO ZLWK FRXQWHUVXUIDFHVDOUHDG\SDUDOOHOEHIRUHWRXFKLQJWRUVLRQDERXWWKHORQJLWXGLQDO D[LV DQG ODWHUDO EHQGLQJ GR QRW RFFXU SO\VWHHU DQG FRQLFLW\ DUH DEVHQW DQG KRUL]RQWDOFRPSUHVVLRQPD\EHQHJOHFWHG UR \
¶VYDQLVK 0RUHRYHUZHWKHQKDG
UR \VLQ VRWKDWIRUWKLVVSHFLDOFDVH UHGXFHVWR
[0R ' [0 ' & ^UR & \ VLQ \0R ' \0 ' &[ VLQ @
`@
)RUDW\UHWKHWHUPVZLWK DUHDSSURSULDWH&LUFXPIHUHQWLDOFRPSUHVVLRQ
[
GHFUHDVHVWKHHIIHFWLYHUROOLQJUDGLXVUHWKDWLVGHILQHGDWIUHHUROOLQJ$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
HIIHFWRIWKHWHUPZLWKVLQ LQWKHVHFRQGHTXDWLRQ )RUDPRWRUF\FOHW\UH ZLWK D FURVV VHFWLRQ DSSUR[LPDWHO\ IRUPLQJ D VHFWRU RI D FLUFOH WRXFKLQJ DW FDPEHULVDFFRPSOLVKHGSUDFWLFDOO\ZLWKRXWWRUVLRQDERXWDORQJLWXGLQDOD[LVDQG WKH DVVRFLDWHG ODWHUDO EHQGLQJ RI WKH W\UH QHDU WKH FRQWDFW ]RQH )LJXUH LOOXVWUDWHVWKHH[SHFWHGGHIRUPDWLRQVDQGWKHUHVXOWLQJPXFKVPDOOHUFXUYDWXUH RI WKH FRQWDFW OLQH RQ D IULFWLRQOHVV VXUIDFH IRU WKH FDU W\UH UHODWLYH WR WKH FXUYDWXUHH[KLELWHGE\WKHPRWRUF\FOHW\UH6LQFHRQDVXUIDFHZLWKIULFWLRQWKH UROOLQJW\UHZLOOEHGHIRUPHGWRDFTXLUHDVWUDLJKWFRQWDFWOLQHWKLVREVHUYDWLRQ PD\H[SODLQWKHUHODWLYHO\ORZFDPEHUVWLIIQHVVRIWKHFDUDQGWUXFNW\UH $VPHQWLRQHGDERYHFLUFXPIHUHQWLDOFRPSUHVVLRQRIEHOWDQGWUHDGUHVXOWLQJ IURP WKH QRUPDO ORDGLQJ SURFHVV VRPHZKDW FRXQWHUDFWHG E\ WKH SUHVHQFH RI K\VWHUHVLV DOVR UHSUHVHQWHG E\ WKH IDFWRU
[
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 ORFDWLRQLQWKHFRQWDFWSDWFKDQGRIWKHWLPH X
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:HKDYH ZKHQGLVUHJDUGLQJWKH
%$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 TXDQWLWLHV LHGHIOHFWLRQVDQGLQSXW VOLS 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\VWHHU FI(TV
/HWXVQRZFRQVLGHUWKHVSHFLDOFDVHRIDIUHHO\UROOLQJW\UHVXEMHFWHGWRRQO\
VPDOOODWHUDOVOLSDQGVSLQ
_ _ © _ _ © UH © _5_ZLWK5 WKHLQVWDQWDQHRXV
UDGLXVRISDWKFXUYDWXUHDQGWKXV_
n _©UH DQGQHJOHFWHIIHFWVRILQLWLDOQRQ
W
SDUDOOHOOLW\RIWRXFKLQJVXUIDFHVDVZHOODVµEXLOWLQ¶ORDGDQGFDPEHULQGXFHG GHIRUPDWLRQHIIHFWV DOO ¶V
9V[ '
9V\ ' & 9F
:HWKHQKDYH DSSUR[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\UHRQDXQLWDUHD ZLWKFRPSRQHQWV
%$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
GHIRUPDWLRQJUDGLHQWVLQWKHDUHDRIDGKHVLRQ 9J 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\UHPRGHODWFRPELQHGORQJLWXGLQDO EUDNH 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
]
LV WRR GLIILFXOW WR GHDO ZLWK DQDO\WLFDOO\ $OVR WKH LQWURGXFWLRQ RI FDUFDVV FRPSOLDQFHZLOOEHGHPRQVWUDWHG)RUIXUWKHUVWXG\ZHUHIHUWRWKHRULJLQDOZRUN RI:LOOXPHLW 3DFHMND D 6KDUS *LSVHU *XR DQG0DVWLQX DQGWKHVWDWHRIWKHDUWSDSHURI3DFHMNDDQG6KDUS $OWKRXJKLWLVSRVVLEOHWRGHYHORSDPRGHOIRUQRQVWHDG\VWDWHFRQGLWLRQVE\ SXUHO\HPSLULFDOPHDQVPRVWUHODWLYHO\VLPSOHDQGPRUHFRPSOH[WUDQVLHQWDQG G\QDPLFW\UHPRGHOVDUHEDVHGRQWKHSK\VLFDOQDWXUHRIWKHW\UH,WLVRILQWHUHVW WRQRWHWKDWIRUDSURSHUGHVFULSWLRQRIW\UHEHKDYLRXUDWWLPHYDU\LQJFRQGLWLRQV DQHVVHQWLDOSURSHUW\PXVWEHUHSUHVHQWHGLQWKHSK\VLFDOPRGHOVEHORQJLQJWR ERWKULJKWKDQGFDWHJRULHVRI)LJ7KDWLVWKHODWHUDODQGVRPHWLPHVDOVRWKH IRUH DQG DIW FRPSOLDQFH RI WKH FDUFDVV /HVV FRPSOH[ QRQVWHDG\VWDWH W\UH PRGHOVIHDWXUHRQO\FDUFDVVFRPSOLDQFHZLWKRXWWKHLQFOXVLRQRIHODVWLFWUHDG HOHPHQWV,QVWHDG\VWDWHPRGHOVWKHLQWURGXFWLRQRIVXFKDIOH[LELOLW\LVRIWHQQRW UHTXLUHG2QO\WRSURSHUO\UHSUHVHQWWKHVHOIDOLJQLQJWRUTXHLQFDVHRIDEUDNHG RU GULYHQ ZKHHO FDUFDVV ODWHUDO FRPSOLDQFH LV QHHGHG 7\UH LQHUWLD EHFRPHV LPSRUWDQWDWKLJKHUVSHHGVDQGIUHTXHQFLHVRIWKHZKHHOPRWLRQ7KHSUREOHPRI HVWDEOLVKLQJQRQVWHDG\VWDWHW\UHPRGHOVLVDGGUHVVHGLQ&KDSWHUVDQG LQVXFFHVVLYHOHYHOVRIFRPSOH[LW\WRPHHWFRQGLWLRQVRILQFUHDVLQJGLIILFXOW\ &RQGLWLRQVEHFRPHPRUHGHPDQGLQJZKHQIRUH[DPSOH WKHZKHHOPRWLRQ JLYHVULVHWRODUJHUYDOXHVRIVOLSZKLFKQRORQJHUSHUPLWVDQDSSUR[LPDWHOLQHDU GHVFULSWLRQ RI WKH IRUFH DQG PRPHQW JHQHUDWLQJ SURSHUWLHV FRPELQHG VOLS
%$6,&7<5(02'(//,1*&216,'(5$7,216
RFFXUVSRVVLEO\LQFOXGLQJZKHHOFDPEHUDQGWXUQVOLS ODUJHFDPEHURFFXUV ZKLFK PD\ QHFHVVLWDWH WKH FRQVLGHUDWLRQ RI WKH GLPHQVLRQV RI WKH W\UH FURVV VHFWLRQ WKH IULFWLRQ FRHIILFLHQW FDQ QRW EH DSSUR[LPDWHG DV D FRQVWDQW TXDQWLW\EXWPD\YDU\ZLWKVOLGLQJYHORFLW\DQGVSHHGRIWUDYHODVRFFXUVRQZHW RULF\VXUIDFHV WKHZDYHOHQJWKRIWKHSDWKRIFRQWDFWSRLQWVDWQRQVWHDG\ VWDWH FRQGLWLRQV FDQ QR ORQJHU EH FRQVLGHUHG ODUJH ZKLFK PD\ UHTXLUH WKH LQWURGXFWLRQRI WKH ODWHUDODQGORQJLWXGLQDOFRPSOLDQFHRIWKHFDUFDVV WKH ZDYHOHQJWKEHFRPHVUHODWLYHO\VKRUWZKLFKPD\QHFHVVLWDWHWKHFRQVLGHUDWLRQRI DILQLWHFRQWDFWOHQJWK UHWDUGDWLRQHIIHFW DQGSRVVLEO\WKHFRQWDFWZLGWK DWWXUQ VOLSDQGFDPEHU WKHVSHHGRIWUDYHOLVODUJHVRWKDWW\UHLQHUWLDEHFRPHVRI LPSRUWDQFH LQ SDUWLFXODU LWV J\URVFRSLF HIIHFW WKH IUHTXHQF\ RI WKH ZKHHO PRWLRQKDVUHDFKHGDOHYHOWKDWUHTXLUHVWKHLQFOXVLRQRIWKHILUVWRUHYHQKLJKHU PRGHVRIYLEUDWLRQRIWKHEHOW WKHYHUWLFDOSURILOHRIWKHURDGVXUIDFHFRQWDLQV YHU\VKRUWZDYHOHQJWKVZLWKDSSUHFLDEOHDPSOLWXGHVDVZRXOGRFFXULQWKHFDVH RI UROOLQJ RYHU D VKRUW REVWDFOH RU FOHDW WKHQ DPRQJVW RWKHU WKLQJV WKH W\UH HQYHORSLQJ SURSHUWLHV VKRXOG EH DFFRXQWHG IRU PRWLRQV EHFRPH VHYHUH ODUJH VOLS DQG KLJK VSHHG ZKLFK PD\ QHFHVVLWDWH PRGHOOLQJ WKH HIIHFW RI WKH ZDUPLQJ XS RI WKH W\UH LQYROYLQJ SRVVLEO\ WKH LQWURGXFWLRQ RI WKH W\UH WHPSHUDWXUHDVDPRGHOSDUDPHWHU$OOLWHPVPHQWLRQHG H[FHSWWKHODVWRQH ZLOO EHDFFRXQWHGIRULQWKHUHPDLQGHURIWKLVERRN
&KDSWHU 7+(25<2)67($'<67$7(6/,3)25&( $1'020(17*(1(5$7,21
,QWURGXFWLRQ
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µEUXVK¶PRGHORI)URPPDQGRI-XOLHQ FI +DGHNHO IRUUHIHUHQFHV ZKRGLGQRWFRQVLGHUFDUFDVVFRPSOLDQFH)LDOD DQG )UHXGHQVWHLQ GHYHORSHG WKHRULHV LQ ZKLFK WKH FDUFDVV GHIOHFWLRQ LV DSSUR[LPDWHG E\ D V\PPHWULF SDUDEROD %|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
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
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¶V PRGHO V\PPHWULF SDUDEROLF FDUFDVV GHIOHFWLRQ G PRGHO RI )URPP EUXVKPRGHOZLWKULJLGFDUFDVV 7KHWUHDGHOHPHQWVWLIIQHVVLVWKHVDPHIRUHDFK PRGHO7KHSDUDPHWHUVLQWKHFDVHVDEDQGFKDYHEHHQFKRVHQLQVXFKDZD\
)\
d
2000
b
a,c
[N] 1000
0
0]
0
5
10
α
[deg]
0
[Nm]
d
b
c
-20
a
)LJ &RPSDULVRQRI FDOFXODWHG FKDUDFWHULVWLFVIRUIRXUGLIIHUHQWW\UHPRGHOVZLWKWUHDG HOHPHQWVDQGQRQV\PPHWULFDOSUHVVXUHGLVWULEXWLRQDWDJLYHQZKHHOYHUWLFDOORDG D VWULQJE EHDPF )LDODG EUXVKLHZLWKULJLGFDUFDVV IURP)UDQN
7+(25<2)67($'<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[SUHVVLRQIRUWKHVLGHIRUFHYHUVXVVOLSDQJOH LISDUDEROLFSUHVVXUHGLVWULEXWLRQ 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
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
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\GHFUHDVLQJ QRWFRQVLGHUHGE\)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 %HUJPDQ 3DFHMND 6DNDL DOVRQWKGHJUHH 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$7,21
wheel slip speed
9
speed of travel 9 slip angle
9V\
α
α
Ω
SXUHVLGH VOLS
)\
)[
&
SXUHEUDNH
)[
&
)\
)]
9V[
VOLS
9V
9
α FRPELQHG
) W
VOLS
9
A
maximum deflection
)\ 9V\
)\
9
B
C sliding
)\
D
B
adhesion
α
A
C
α
0 full sliding
)\
0] W)\
D
SXUHVLGH
)\
)LJ
VOLS
7KHEUXVKW\UHPRGHO 7RSOHIW YLHZRIGULYHQDQGVLGHVOLSSLQJW\UH 7RSULJKW WKHW\UHDWGLIIHUHQWVOLSFRQGLWLRQV %RWWRPOHIW WKHW\UHDWSXUHVLGHVOLSIURPVPDOOWRODUJHVOLSDQJOH %RWWRPULJKW WKHUHVXOWLQJVLGHIRUFHDQGDOLJQLQJWRUTXHFKDUDFWHULVWLFV
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
GHIOHFWLRQVDUHGHYHORSHGDQGFRUUHVSRQGLQJIRUFHVDQGPRPHQWDULVH7KHWUHDG HOHPHQWVPRYHIURPWKHOHDGLQJHGJH DWWKHULJKWKDQGVLGHRIWKHSLFWXUHV WR WKH WUDLOLQJ HGJH 7KH WLS RI WKH HOHPHQW ZLOO DV ORQJ DV WKH DYDLODEOH IULFWLRQ DOORZVDGKHUHWRWKHJURXQG WKDWLVLWZLOOQRWVOLGHRYHUWKHURDGVXUIDFH $W WKH VDPH WLPH WKH EDVH SRLQW RI WKH HOHPHQW UHPDLQV LQ WKH ZKHHO SODQH DQG PRYHVEDFNZDUGVZLWKWKHOLQHDUVSHHGRIUROOLQJ9U WKDWLVHTXDOWR
UH ZLWK
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
7+(25<2)67($'<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
\
$VWKHGLVWULEXWLRQRIWKHGHIOHFWLRQVRIWKHHOHPHQWVKDVQRZEHHQHVWDEOLVKHG WKH WRWDO IRUFH )\ DQG WKH PRPHQW 0] FDQ EH DVVHVVHG E\ LQWHJUDWLRQ RYHU WKH FRQWDFWOHQJWK OLNHLQ(T EXWQRZVHSDUDWHIRUWKHVOLGLQJUDQJHD [ [W DQGWKHDGKHVLRQUDQJH[W [ D )RUFRQYHQLHQFHZHLQWURGXFHWKHQRWDWLRQIRU WKHVOLS
\
'
WDQ
7KHUHVXOWLQJIRUPXODIRUWKHIRUFHUHDGV LI
* *#
VO
)\ ' )] & VJQ ' )] \ \ & \ \ % DQGLI
* *$
VO
\ \
EXW ò
)\ ' )] VJQ DQGIRUWKHPRPHQW LI
* *#
D
VO
0 ] ' & )] D & VJQ ' & )] D \ \ & \ \ % ZLWKSHDNYDOXH )]DW
0] '
\
\ \
&
\ ,I
\ \
* *$
VO
EXW ò D
7KHSQHXPDWLFWUDLOWZKLFKLQGLFDWHVWKHGLVWDQFHEHKLQGWKHFRQWDFWFHQWUH& ZKHUHWKHUHVXOWDQWVLGHIRUFH)\LVDFWLQJEHFRPHV LI
* *#
VO
W' &
0] & \ \ % \ \ & ' D )\ & \ \ % \ \
\ \
DQGLI
* *$
W'
VO
EXW ò
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
&KDUDFWHULVWLFVRIWKHVLPSOHEUXVKPRGHO VLGHIRUFHDOLJQLQJWRUTXHDQGSQHXPDWLF WUDLOYVVOLSDQJOH
7KHVHUHODWLRQVKLSVKDYHEHHQVKRZQJUDSKLFDOO\LQ)LJ$WYDQLVKLQJVOLS DQJOHH[SUHVVLRQ UHGXFHVWR
W ' WR ' &
0] )\
' D
6
7KLVYDOXHLVVPDOOHUWKDQQRUPDOO\HQFRXQWHUHGLQSUDFWLFH7KHLQWURGXFWLRQRI DQHODVWLFFDUFDVVZLOOLPSURYHWKLVTXDQWLWDWLYHDVSHFW7KHQWKHPRUHUHDOLVWLF
. DPD\EHDFKLHYHG FI3DFHMND IRUDVWUHWFKHGVWULQJPRGHO
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 $OVRLQ WKLVUHVSHFWWKHLQWURGXFWLRQRIDQHODVWLFFDUFDVV LQSDUWLFXODUZKHQLWVODWHUDO
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
)\ µ)] 8
1
)
\
[N] 6
[deg]
8
4000
0.01
5
7
6
4
4
5
3
3000
2
3
2
2000 1000
-0.1
0]
0
µ
D )]
α
1
1 0
0.03 W[m]
2
1 0
0.02 4
3
0
0.005
5000
α
5
θ \ = 5, µ 1 D 2 = 13)] x10-5
) [kN] ]
20
[deg]
40
60
0] [Nm]
)LJ 7KHVRFDOOHG*RXJKSORWIRUWKHEUXVKPRGHOQRQGLPHQVLRQDODQGZLWKGLPHQVLRQ XVLQJDQDVVXPHGORDGYVFRQWDFWOHQJWKUHODWLRQVKLS
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
)\
)\
)[kN] ]
5
[N]
27.5 14 20000 25
10
15 12
)[kN] ]
[N]
20
25
10
20000
30
8
20
80 2
α [deg]
W
[mm]
0
20 10
15 10000
5
15
12
25 30
8
50
4
10000 10
10 14
20
40
6
15
5 25
6
10
4 W
5
2
40 50
[mm]
α [deg]
0 0
)LJ
250
[Nm]
500 0 ]
0
200
[Nm]
400 0 ]
/HIW *RXJKSORWRIWUXFNW\UH PHDVXUHGRQDGU\URDGVXUIDFH )UHXGHQVWHLQ
F
5LJKW EUXVKPRGHOZLWKWUHDGHOHPHQWVWLIIQHVV
S
)
GHFD\LQJOLQHDUO\ZLWK
]
7+(25<2)67($'<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 HTXDOWRWKHHIIHFWLYHUROOLQJUDGLXVUH GHILQHGDWIUHHUROOLQJ EHORZWKHZKHHO FHQWUH$WIUHHUROOLQJE\GHILQLWLRQWKHVOLSSRLQW6KDVDYHORFLW\HTXDOWR]HUR
Ω
9U U H 9[
D
UH
[
U 6
9V[ X
∆W 9V[ ∆W 9[ ∆W 9U
6
9V[
D[ X
)LJ6LGHYLHZRIWKHEUXVKW\UHPRGHODWEUDNLQJ QRVOLGLQJFRQVLGHUHG
7+(25<2)67($'<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 ZKHHOFHQWUH 9F[
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[
$QHOHPHQWWKHWLSRIZKLFKDGKHUHVWRWKHJURXQGDQGWKHEDVHSRLQWLVPRYHG WRZDUGVWKHUHDURYHUDGLVWDQFHD[IURPWKHOHDGLQJHGJH IRUZKLFKDWLPHVSDQ W
D[ 9ULVQHHGHG KDVGHYHORSHGDGHIOHFWLRQLQORQJLWXGLQDOGLUHFWLRQ
X ' & 9V[ D & [ 9U
7KHVDPHH[SUHVVLRQPD\EHREWDLQHGE\LQWHJUDWLRQRIWKHIXQGDPHQWDOHTXDWLRQ DQGQRWLQJWKDW9J[ FRQGLWLRQX
DW[
D
]
MXMW
¶V
DQGILQDOO\XVLQJWKHERXQGDU\
7+(25<2)67($'<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[ %
,QWHUPVRIWKHDOWHUQDWLYHGHILQLWLRQRIORQJLWXGLQDOVOLS WKHµWKHRUHWLFDO¶VOLSWR EHXVHGLQWKHVXEVHTXHQWVHFWLRQDQGGHILQHGDV
['
&
9V[ ' 9U %
QRWHZHUHVWULFWRXUVHOYHVWRQRQQHJDWLYHVSHHGVRIUROOLQJ 9U
$ 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\RFFXU VD\&) DERXWODUJHUWKDQ&) ZKLFKLVGXHWRWKHODWHUDO WRUVLRQDO FRPSOLDQFH RI WKH FDUFDVVRIWKHDFWXDO W\UH 6WLOO LW LV H[SHFWHG WKDW TXDOLWDWLYH VLPLODULW\ RI ERWK SXUH VOLS FKDUDFW HULVWLFVUHPDLQV
,QWHUDFWLRQEHWZHHQ/DWHUDODQG/RQJLWXGLQDO6OLS FRPELQHGVOLS
)RUWKHDQDO\VLVRIWKHLQIOXHQFHRIORQJLWXGLQDOVOLS RUORQJLWXGLQDOIRUFH RQWKH ODWHUDO IRUFH DQG PRPHQW JHQHUDWLRQ SURSHUWLHV ZH VKDOO IRU WKH VDNH RI PDWKHPDWLFDOVLPSOLFLW\UHVWULFWRXUVHOYHVWRWKHFDVHRIHTXDOORQJLWXGLQDODQG ODWHUDOVWLIIQHVVHVRIWKHWUHDGHOHPHQWV LVRWURSLFPRGHO LH
FS' FS[' F S\
DQGHTXDODQGFRQVWDQWIULFWLRQFRHIILFLHQWV
' [' \
$JDLQDSDUDEROLFSUHVVXUHGLVWULEXWLRQLVFRQVLGHUHG )LJXUH GHSLFWVWKHGHIRUPDWLRQVZKLFKDULVHZKHQWKHW\UHPRGHOZKLFK UXQVDWDJLYHQVOLSDQJOH LVGULYHQRUEUDNHG'XHWRWKHHTXDOVWLIIQHVVLQDOO
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
9U
9V
D
D
9
λ
2
D
9
[
σ
α
[
XY
]
δ
2
)
α
[
)
σ[
0
’inclined’ parabola
D
)
\
)
GULYLQJ
\
9V
9U
2
9
α
[
)
[
α
σ
]
0
-2
)
9
λ
D
δ
σ[
D
)
EUDNLQJ
\
)
\
V[
9 6
[
)
] )LJ
)]
[
T]
9HFWRUGLDJUDPDQGGHIRUPDWLRQRIWKHEUXVKPRGHOUXQQLQJDWDJLYHQVOLSDQJOHIRU WKHFDVHVRIGULYLQJDQGEUDNLQJ
KRUL]RQWDO GLUHFWLRQV DQG WKH LVRWURSLF IULFWLRQ SURSHUWLHV WKH GHIOHFWLRQV DUH GLUHFWHGRSSRVLWHWRWKHVOLSVSHHGYHFWRU
9V
DOVRLQWKHVOLGLQJUHJLRQ,QWKLV
ODWWHUUHJLRQWKHWLSVRIWKHHOHPHQWVVOLGHRYHUWKHURDGZLWKVOLGLQJVSHHG
T
9J
GLUHFWHGRSSRVLWHWRWKHORFDOIULFWLRQIRUFH SHUXQLWFRQWDFWOHQJWK 7KHZKROH GHIRUPDWLRQKLVWRU\RIDWUHDGHOHPHQWZKLOHUXQQLQJWKURXJKWKHFRQWDFWDUHDLV DRQHGLPHQVLRQDOSURFHVVDORQJWKHGLUHFWLRQRI
9V
7KHYHORFLW\RISURJUHVVLRQRIDEDVHSRLQWWKURXJKWKHFRQWDFWOHQJWKLVWKH UROOLQJVSHHG9U DJDLQDVVXPHGQRQQHJDWLYH 7KHGHIOHFWLRQUDWHRIDQHOHPHQW LQWKHDGKHVLRQUHJLRQLVHTXDOWRWKHVOLSVSHHG9V7KHWLPHZKLFKHODSVHVIURP WKHSRLQWRIHQWUDQFHWRWKHSRLQWDGLVWDQFH[LQIURQWRIWKHFRQWDFWFHQWUHHTXDOV
W ' D&[ 9U
7+(25<2)67($'<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]RQWDOFRQWDFWIRUFHDFWLQJRQWKHWLSVRIWKHHOHPHQWV SHUXQLW 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 @)\
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
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)LJ RQHDW 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$7,21
WR )] LV GLUHFWHG RSSRVLWH WR WKH VSHHG YHFWRU RI WKH ZKHHO FRLQFLGHVZLWKWKHVOLSVSHHGYHFWRU
9
9 ZKLFK WKHQ
V
([SHULPHQWDOHYLGHQFH HJ)LJ VXSSRUWVWKHQDWXUHRIWKHWKHRUHWLFDO FXUYHVIRUWKHIRUFHVDVVKRZQLQ)LJ2IWHQWKHVKDSHDSSHDUVWREHPRUH DV\PPHWULF WKDQ SUHGLFWHG E\ WKH VLPSOH EUXVK PRGHO 7KLV PD\ EH GXH WR D VOLJKWLQFUHDVHLQWKHFRQWDFWOHQJWKZKLOHEUDNLQJ PDNLQJWKHW\UHVWLIIHU DQG E\WKHEUDNHIRUFHLQGXFHGVOLSDQJOHRIWKHFRQWDFWSDWFKDWVLGHVOLS7KLVLV DFFRPSOLVKHGWKURXJKWKHWRUVLRQRIWKHFDUFDVVLQGXFHGE\WKHPRPHQWDERXW WKHYHUWLFDOD[LVWKDWLVH[HUWHGE\WKHEUDNLQJIRUFHZKLFKKDVVKLIWHGLWVOLQHRI DFWLRQ GXH WR WKH ODWHUDO GHIOHFWLRQ FRQQHFWHG ZLWK WKH VLGH IRUFH 7KH PRUH DGYDQFHGPRGHOWREHGHYHORSHGIXUWKHURQLQWKLVFKDSWHUZLOOWDNHWKHODWWHU HIIHFWLQWRDFFRXQW 7KHPRPHQWFXUYHVSUHVHQWHGLQWKHORZHUGLDJUDPRI)LJVKRZDPRUH RUOHVVV\PPHWULFDOEHOOVKDSH$VH[SHFWHGWKHDOLJQLQJWRUTXHEHFRPHVHTXDO WR ]HUR ZKHQ WRWDO VOLGLQJ RFFXUV
$
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 $QRWKHU LPSRUWDQW DGYDQWDJH RI SXWWLQJ VLQ
LQVWHDG RI WDQ
DORQJ WKH
DEVFLVVDLVWKDW DIWHUKDYLQJFRPSOHWHGWKHGLDJUDPIRUQHJDWLYHYDOXHVRIVLQ UHVXOWLQJLQDQRGGO\V\PPHWULFJUDSK WKHFRPSOHWHUDQJHRI
LVFRYHUHG WKH
9PD\VZLQJDURXQGRYHUWKHZKROHUDQJHRIE$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$7,21
)\
radius 1.1)]
12o o 8 o
6 4
o
)[
)]
1000N
0
-1000N
EUDNLQJ
GULYLQJ
-0.08
-0.04
κ= -0.02
0.02
0.04
0.06
0.08
α =2o
)]
) ) FKDUDFWHULVWLFVIRUW\UHPHDVXUHGRQGU\LQWHUQDOGUXPZLWKGLDPHWHU
)LJ
\
[
RIP IURP+HQNHU
K)\
0]
GULYLQJ
EUDNLQJ
α =8o α =+4 oo α =-4
)[
2000N
α =8o 0
2000N
α =4 o
-100Nm
)LJ
&RPELQHG VOLS VLGH IRUFH DQG PRPHQW YHUVXV ORQJLWXGLQDO IRUFHFKDUDFWHULVWLFV PHDVXUHGDWWZRYDOXHVRIWKHVOLSDQJOHIRUDW\UHRQDGU\IODWVXUIDFH IURP 1RUGHHQDQG&RUWHVH
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
YR YF
0
]
&
XF
α
9 [
[
)
α
0
]
\
\
) )LJ
)
([WHQGHGW\UHEUXVKPRGHOVKRZLQJRIIVHWDQGGHIOHFWLRQRIFDUFDVVOLQH VWUDLJKW DQGSDUDOOHOWRZKHHOSODQH
ZKHUHWKHFRPSOLDQFHFRHIILFLHQWFKDVEHHQLQWURGXFHGWKDWLVGHILQHGE\
F
'
g\ &
&
F\
g[
&
F[
+HUH &F[ DQG &F\ GHQRWH WKH ORQJLWXGLQDO DQG ODWHUDO FDUFDVV VWLIIQHVVHV UHVSHFWLYHO\ DQG
g
[
DQG
g
\
WKH HIIHFWLYH IUDFWLRQV RI WKH DFWXDO GLVSODFHPHQWV
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
PPKDVEHHQGHSLFWHG
+HUH FDUFDVV FRPSOLDQFH KDV EHHQ GLVUHJDUGHG DQG RQO\ WKH ODVW WHUP RI KDVEHHQDGGHG:HVHHWKDWDPRPHQWLVJHQHUDWHGDOUHDG\DW]HURVOLS
7+(25<2)67($'<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
LQDVXLWDEOHUDQJH HJIURP 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
&DPEHUDQG7XUQLQJ 6SLQ
)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 ZKHHOSODQHDQGWKHQRUPDOWRWKHURDG FI)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 $WWKHULJKW DWRSYLHZRIWKHSHULSKHUDOOLQHRIWKHQRQUROOLQJZKHHOFRQVLGHUHGDV EHLQJDSDUWRIDQLPDJLQDU\EDOOSUHVVHGDJDLQVWDIODWVXUIDFH
7+(25<2)67($'<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 $IWHULQWURGXFLQJFS[
DQGFS\ DUHD LQ [ DQG \ GLUHFWLRQ UHVSHFWLYHO\ DQG DVVXPLQJ VPDOO VSLQ DQG KHQFH YDQLVKLQJVOLGLQJZHFDQFDOFXODWHWKHODWHUDOIRUFHDQGWKHPRPHQWDERXWWKH YHUWLFDOD[LVE\LQWHJUDWLRQRYHUWKHFRQWDFWDUHD:HREWDLQ
)\ ' FNS\ D E n ' &)n n
0] ' FNS[ D E
n'
&0n n
RULQWHUPVRISDWKFXUYDWXUH LI
)\ ' &&)n 0 ] ' &&0n
5
5
&
&
/ RUFRQVWDQW DQGFDPEHUDQJOH
&g
' &)n & % &)
&g
& % &
UH
UH
5
' &0n
5
VPDOO
0
7+(25<2)67($'<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 ODWHUDOGHIOHFWLRQLVDOVR DSSUR[LPDWHO\ SDUDEROLFWKDWQRVOLGLQJZLOORFFXUXS WR D FHUWDLQ FULWLFDO YDOXH RI VSLQ
n
VO
ZKHUH WKH DGKHVLRQ OLPLW LV UHDFKHG
nDQG0
WKURXJKRXWWKHFRQWDFWOHQJWK8SWRWKLVSRLQW)\YDULHVOLQHDUO\ZLWK
]
UHPDLQVHTXDOWR]HUR $FFRUGLQJWR(T ZLWK
g DVVXPHGWRWDNH D YDOXHWKDWLVPLQLPDOO\
HTXDO WR ]HUR VSLQ GXH WR FDPEHU WKHRUHWLFDOO\ FDQQRW H[FHHG WKH YDOXH UH &RQVHTXHQWO\DWODUJHUYDOXHVRIVSLQWXUQVOLSPXVWEHLQYROYHG%H\RQGWKH FULWLFDOYDOXH
n
VO
WKHVLWXDWLRQEHFRPHVTXLWHFRPSOH[7KHGLVFXVVLRQPD\EH
VLPSOLILHGE\FRQVLGHULQJDWXUQWDEOHRQWRSRIZKLFKWKHZKHHOUROOVZLWKLWV VSLQ D[LV IL[HG 7KH FRQGLWLRQ RI DGKHVLRQ LV VDWLVILHG ZKHQ WKH GHIOHFWLRQV UHPDLQ ZLWKLQ WKH ERXQGDULHV JLYHQ E\ WKH SDUDEROD¶V Y PD[ DV LQGLFDWHG LQ )LJD ,Q WKH VDPH ILJXUH WKH FRUUHVSRQGLQJ VLWXDWLRQ DW FDPEHU KDV EHHQ LQGLFDWHGDWWKHVDPHYDOXHRIVSLQZLWKFXUYDWXUH5 RIWKHGHIOHFWHGSHULSKHUDO OLQH \
R
(T RQD
VXUIDFHHTXDOWRWKHSDWKFXUYDWXUH5
6OLGLQJ RFFXUV DV VRRQ DV WKH SRLQWV RI WKH WDEOH VXUIDFH FDQ FURVVWKH
DGKHVLRQERXQGDU\YPD[7KLVRFFXUVVLPXOWDQHRXVO\IRUDOOWKHSRLQWVLQWKH IURQWKDOIRIWKHFRQWDFWOLQHZKHQWKHSDWKFXUYDWXUHH[FHHGVWKHFXUYDWXUHRI WKH DGKHVLRQ ERXQGDU\ 2QFH WKH FRQWDFW SRLQW DUULYHV LQ WKH UHDU KDOI RI WKH FRQWDFW]RQH DW[
WKHSRLQWFDQPDLQWDLQDGKHVLRQEHFDXVHQRZWKHSRLQW
RQ WKH WDEOH PRYHV WRZDUGV WKH LQVLGH RI WKH DGKHVLRQ ERXQGDULHV 7KH SRLQW [V FI)LJEF
IROORZVDFLUFOHXQWLOWKHRSSRVLWHERXQGDU\LVUHDFKHGDW[
ZKHUHWKHGHIRUPDWLRQYLVRSSRVLWHLQVLJQDQGUHDFKHVLWVPD[LPXPYDOXHYPD[
n
DIWHUZKLFKVOLGLQJRFFXUVDJDLQ:LWKLQFUHDVLQJWXUQVOLS
5 WKLVODWWHU
VOLGLQJ]RQHJURZV$WWKHVDPHWLPHWKHVLGHIRUFH)\GHFUHDVHVDQGWKHWRUTXH
n!n
0]WKDWDULVHVIRU
VO
LQFUHDVHVXQWLOWKHVLWXDWLRQLVUHDFKHGZKHUH5DQG)\
DSSURDFK]HURDQG0]DWWDLQVLWVPD[LPXPYDOXH W\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 FRQWDFWOLQHLQWKHDGKHVLRQUDQJHLVWKHQFRQYHUWHGWRDVWUDLJKW KRUL]RQWDO OLQH 7KHJUDSKLVREWDLQHGIURPWKHWXUQVOLSJUDSKE\VXEWUDFWLQJIURPDOOWKHFXUYHV WKHGHIOHFWLRQYn4WKDWZRXOGRFFXULIIXOODGKHVLRQFDQEHPDLQWDLQHG HJDW
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$7,21
WXUQVOLSJUDSK
ϕ
5
wheel plane
9F
contact line
[ Yϕ
Y
Y PD[
\ FDPEHUJUDSK
YPD[
Y
peripheral line on frictionless surface
\γο
contact line
\
5γ
[
9F
D
’large’WXUQVOLS
ϕ
YPD[
wheel plane
5
[
contact line
YPD[
4 (as if µ 64)
Y
Y ϕ 0]
(contact line if
)\
µ 64) E
\
’large’WXUQVOLS wheel plane
(|5| >>D)
ϕ
YPD[
[
contact line
YPD[
\
upward shifted Yϕ 4
γ
4 (as if µ 64)
Y ϕ Y
0]
\ PD[/
HTXLYDOHQWJUDSK
Y
\γο
contact line
(
\ PD[5
\
V
[V
)\
=0
[ V
\
\γο Y PD[
ϕ4 )
Y
γο
PD[
\ Y
[
F
)LJ 7KHW\UHEUXVKPRGHOZLWK]HURZLGWKUROOLQJZKLOHWXUQLQJRUDWDFDPEHUDQJOH D DWIXOODGKHVLRQ7XUQLQJDWODUJHVSLQVKRZLQJVOLGLQJDWWKHIURQWKDOIDQGDWWKH UHDUHQG EF ZLWKSDUDEROLFDSSUR[LPDWLRQRIWKHFLUFXODUSDWKLQ F
7+(25<2)67($'<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$7,21
*
*
LI WDQ
_
\
#
_
0 ] ' & )] D
*
*
DQGLI WDQ
0] '
_
\
\VO
\ \
_!
\
(
&
\ \
(
%
\ \
(
&
\ \
\VO
D
:HPD\LQWURGXFHDSQHXPDWLFWUDLOW WKDWPXOWLSOLHGZLWKWKHIRUFH)\ GXHWR WKHVOLSDQJOH ZLWKWKHODVWWHUPRI RPLWWHG SURGXFHVWKHPRPHQW0]
*
LI WDQ
*
_
\
#
_
\VO
\
( 0 & % ( W '& ' D ( ) & % ]
\
\
\
\
\
(
&
(
\
DQGHOVH W
\
\
\
\
\
7KHJUDSKRI)LJFODULILHVWKHFRQILJXUDWLRQRIWKHYDULRXVFXUYHVDQG WKHLU PXWXDO UHODWLRQVKLS )RU GLIIHUHQW YDOXHV RI WKH FDPEHU DQJOH
WKH
FKDUDFWHULVWLFV IRU WKH IRUFH DQG WKH PRPHQW YHUVXV WKH VOLS DQJOH KDYH EHHQ FDOFXODWHG ZLWK WKH DERYH HTXDWLRQV DQG SUHVHQWHG LQ )LJ 7KH FRUUHVSRQGLQJ *RXJK SORW KDV EHHQ GHSLFWHG LQ )LJ 7KH UHODWLRQVKLS EHWZHHQ DQGWKHVSLQ
nIROORZVIURP(T
7KH FXUYHV HVWDEOLVKHG VKRZ JRRG TXDOLWDWLYH DJUHHPHQW ZLWKPHDVXUHG
0
)\ Wα , ] D Dµ)] µ)]
θ\ = 5 (1-ε γ ) D = 1 UH 3
α
0.2
W 0
]
1
W )
α \α
)\ )
\α
)\γ
0
1/θ \
1/θ\*-0.2
1/θ\*+ -1
tanα
γ = 10
)LJ%DVLFFRQILJXUDWLRQRIWKHFKDUDFWHULVWLFVYVVLGHVOLSDWFDPEHUDQJOH
O
GHJ
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
)\ µ)]
θ\ = 5
(1-εγ ) D = 1 UH 3
20
10
γ
0
[deg]
1 -10
0] -20
-20
0
-10
α
20 10
)\
0 -10 -20
-1 -0.2
0
γ
20 [deg]
[deg]
]
Dµ)] )LJ7KHFDOFXODWHGVLGHIRUFHDQGPRPHQWFKDUDFWHULVWLFVDWYDULRXVFDPEHUDQJOHV
FKDUDFWHULVWLFV6RPHGHWDLOVLQWKHLUIHDWXUHVPD\EHGLIIHUHQWZLWKUHVSHFWWR H[SHULPHQWDO HYLGHQFH ,Q WKH QH[W VHFWLRQ ZKHUH WKH VLPXODWLRQ PRGHO LV LQWURGXFHGWKHHIIHFWRIYDULRXVRWKHUSDUDPHWHUVOLNHWKHZLGWKRIWKHFRQWDFW SDWFKDQGWKHSRVVLEO\FDPEHUGHSHQGHQWDYHUDJHIULFWLRQFRHIILFLHQWRQWKHSHDN VLGHIRUFHZLOOEHGLVFXVVHG 7KHQH[WLWHPWREHDGGUHVVHGLVWKHUHVSRQVHWRODUJHVSLQLQWKHSUHVHQFHRI VLGH VOLS )LJXUHV DE UHIHU WR WKLV VLWXDWLRQ /DUJH VSLQ ZLWK WZR VOLGLQJ UDQJHVRFFXUVZKHQWKHIROORZLQJWZRFRQGLWLRQVDUHIXOILOOHG
θ\ = 5
(1-εγ ) D UH
)\
µ)]
1 =3
1
α 20
-1
10
-2 0 -3
0
-4
-10
-5 -6
-20
3 2 1
0
[deg]
6 5 4
γ
[deg]
-1 )LJ 7KHFRUUHVSRQGLQJ*RXJKSORW
0.2 0]
Dµ)]
7+(25<2)67($'<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
DQGLQVHFRQGVOLGLQJUDQJH D [ [V
\ ' &$ D & [ D
n
VJQ
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
’large’WXUQVOLS ZLWKVOLSDQJOH
wheel plane
α
contact line
α
YPD[
upward shifted Y ϕ4
γ
ϕ
YPD[
Y
HTXLYDOHQWJUDSK
\ PD[/
Y
\γο
contact line
α ϕ
range for to be ’large’
α
\ V
\ V [V
\
(
ϕ4 )
Y
\ PD[5
α
[ V
’large’VSLQ
γ
9F
Yϕ4
\
[
[ 9F
D
α <0
HTXLYDOHQWJUDSK
\ PD[/
α ϕ
range for to be ’large’
\γο
α
[ \ PD[5
α
\
9F
\PD[/ α ϕ
9F
range for to be ’large’
α
α \ PD[5
\
’small’ VSLQ
γ
[ V
HTXLYDOHQWJUDSK
α
α >0
E
α <0
α ϕ
[
\ PD[/ \γο
range for to be ’large’ \ PD[5
[
α
\
9F
\PD[/ α ϕ
α
range for to be ’large’
9F
[
α
\ PD[5 \
α >0 F
)LJ
7KHPRGHOUXQQLQJDWODUJHVSLQ WXUQLQJDQGHTXLYDOHQWFDPEHU DWDUHODWLYHO\VPDOO DE RUODUJHSRVLWLYHRUQHJDWLYHVOLSDQJOH F
7+(25<2)67($'<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 DUHGHILQHGDV OLNHLQ
\ ' &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 `
$IWHULQWURGXFLQJWKHWKHRUHWLFDOVOLSTXDQWLWLHVIRUWKHWZRDWWDFKHGZKHHOV
[/5
'&
9
V[/5
9
\
'&
U
9 9
V\
0
'&
9
U
U
ZHILQGIRUWKHJUDGLHQWVRIWKHGHIOHFWLRQVLQWKHDGKHVLRQ]RQH ZKHUH9J
LIVPDOOVSLQLVFRQVLGHUHG VOLGLQJRQO\DWWKHUHDU
MX
/5
M[ MY M[
'&
V[/5
' & &[ \
% & g
U
VLQ
H
ZKLFK\LHOGVDIWHULQWHJUDWLRQIRUWKHGHIOHFWLRQVLQWKHDGKHVLRQ]RQH [ [W
X ' D& [ /5
Y ' D&[
[/5
% ò D &[
\
% & g
U
7KHWUDQVLWLRQSRLQWIURPDGKHVLRQWRVOLGLQJDW[
VLQ
H
[WFDQEHDVVHVVHGZLWKWKH
DLGRIWKHFRQGLWLRQ
F H ' T S
/5
]
7+(25<2)67($'<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
YHU\ SRZHUIXO DQG FDQ EH XVHG HLWKHU XQGHU VWHDG\VWDWH RU WLPH YDU\LQJ FRQGLWLRQV,QWKHODWWHUQRQVWHDG\VLWXDWLRQZHPD\GLYLGHWKHFRQWDFWOHQJWK LQWR D QXPEHU RI ]RQHV RI HTXDO OHQJWK LQ HDFK RI ZKLFK D WUHDG HOHPHQW LV IROORZHG,QFDVHRIWXUQLQJRUFDPEHUWKHFRQWDFWSDWFKVKRXOGEHGLYLGHGLQWR VHYHUDOSDUDOOHOURZVRIHOHPHQWV:KLOHPRYLQJ WKURXJKWKH]RQHVWKHIRUFHV DFWLQJ RQ WKH HOHPHQWV DUH FDOFXODWHG DQG LQWHJUDWHG $IWHU KDYLQJ PRYHG FRPSOHWHO\WKURXJKD]RQHWKHLQWHJUDWLRQSURGXFHVWKH]RQHIRUFHV7KHVHIRUFHV DFWRQWKHEHOWDQGWKHFRUUHVSRQGLQJGLVWRUWLRQLVFDOFXODWHG:LWKWKHXSGDWHG EHOW GHIOHFWLRQ WKH QH[W SDVVDJH WKURXJK WKH ]RQHV LV SHUIRUPHG DQG WKH FDOFXODWLRQLVUHSHDWHG +HUHZHZLOOUHVWULFWWKHGLVFXVVLRQWRVWHDG\VWDWHVOLSFRQGLWLRQVDQGWDNH D VLQJOH ]RQH ZLWK OHQJWK HTXDO WR WKH FRQWDFW OHQJWK ,Q 6HFWLRQ DQ LQWURGXFWRU\GLVFXVVLRQKDVEHHQJLYHQDQGUHIHUHQFHKDVEHHQPDGHWRDQXPEHU RI VRXUFHV LQ WKH OLWHUDWXUH 7KH FRPSOHWH OLVWLQJ RI WKH VLPXODWLRQ SURJUDP 7UHDG6LPZULWWHQLQ0DWODEFRGHLVJLYHQLQ$SSHQGL[)RUGHWDLOVZHPD\ UHIHUWRWKLVSURJUDP )LJXUHGHSLFWVWKHPRGHOZLWKGHIOHFWHGEHOWDQGWKHWUHDGHOHPHQWWKDW KDV PRYHGIURPWKHOHDGLQJHGJHWRDFHUWDLQSRVLWLRQLQWKHFRQWDFW]RQH,Q
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[WHUQDOIRUFH WKURXJKWKHODWHUDOFRPSOLDQFH 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
)LQLWHWUHDGZLGWK WZRURZVRIWUHDGHOHPHQWVIOH[LEOHFDUFDVV ZLWK DQGZLWKRXWFDPEHU )LJF
&RPELQHGODWHUDODQGWXUQVOLS WZRURZVIOH[LEOHFDUFDVV )LJG
3QHXPDWLFWUDLODWSXUHVLGHVOLS IOH[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\
$OVRWKHIRUHDQGDIWIRUFH) [
WHQGVWRGHFUHDVHDIWHUKDYLQJUHDFKHGLWVSHDNYDOXH7KHLQZDUGHQGLQJVRIWKH FXUYHVDWFRQVWDQWYDOXHVRIVOLSDQJOHVKRZQLQWKHERWWRPGLDJUDPDUHW\SLFDO HVSHFLDOO\IRUDW\UHUXQQLQJRQZHWURDGVXUIDFHV7KHSHDNYDOXHVWKHPVHOYHV ZLOOGHFUHDVHZKHQWKHVSHHGRIWUDYHOLVLQFUHDVHG QRWVKRZQ ZKLOHWKHLQLWLDO
7+(25<2)67($'<67$7(6/,3)25&($1'020(17*(1(5$7,21
VORSHV VOLSVWLIIQHVVHV 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 3XUHWXUQVOLSFKDUDFWHULVWLFVDFFRUGLQJWRWKHPRGHO OHIWULJLGFDUFDVVZLWKRQHRU WZRURZVRIHOHPHQWV DQGUHVXOWVIURPH[SHULPHQWVZLWKDELDVSO\WUXFNW\UH ULJKW
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
$SSOLFDWLRQ %UDNLQJXSWR:KHHO/RFN
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
P X0 & YU ' )[ % )[ P Y0 % XU ' )\ % )\ PN U0 ' D)\ & E)\
ZLWK
WDQ WDQ
' & Y % DU X ' & Y & EU X
DQG
)\ ' ) \ )\ ' )\
)] )]
)RUDSURSHUVLPXODWLRQRIWKHPRWLRQWKHVOLSUDWLR¶V ZKHHOVSHHGVRIUHYROXWLRQ
VKRXOGUHVXOWIURPWKH
L
ZKLFKZRXOGUHTXLUHDGGLWLRQDOGHJUHHVRIIUHHGRP L
)RU WKLV RFFDVLRQ ZH ZLOO HPSOR\ DQ DOWHUQDWLYH DSSURDFK WKDW LQYROYHV WKH LQWURGXFWLRQRIIXQFWLRQVIRUWKHVLGHIRUFHLQZKLFKLWVGLUHFWGHSHQGHQFHRQWKH EUDNLQJHIIRUW RUEUDNHSUHVVXUH LVLQFOXGHG7KLVLVSRVVLEOHZKHQµGU\¶IULFWLRQ LV DVVXPHG WR RFFXU EHWZHHQ W\UH DQG URDG DQG WKH FXUYHV RI WKH ULJKWPRVW
7+(25<2)67($'<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 )LJWKHFKDUDFWHULVWLFVKDYHEHHQUHSURGXFHG RQHFXUYHIRUWKHIUHHO\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 OLQHLVUHDFKHGZKHUHDOVRWKLVZKHHOJHWVORFNHG QRWHWKDWVLQ
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
$VDUHVXOWWKH(TV UHGXFHWR
P X0 ' & % & )]
P Y0 %
X PN U0 %
_ _
& % & Y % PX %
X
_ _
X
_ _
D & % E & U %
X
_ _
D& & E&
U'
D& & E& Y '
)XUWKHUPRUHZHKDYHLQWURGXFHGWKHDEVROXWHYDOXHRIWKHIRUZDUGVSHHGXLQWKH GHQRPLQDWRUVWRDOORZIRUWKHFRQVLGHUDWLRQRIQHJDWLYHYDOXHVRIXZKLOHDWWKH UHPDLQV XQFKDQJHG FI WKH FRUUHVSRQGLQJ HTXDWLRQV
VDPH WLPH WKH VLJQ RI
IRUWKHQRQGULYHQRUEUDNHGYHKLFOH 1HJDWLYHYDOXHVRIXFRUUHVSRQGWR WKHFDVHRIORFNHGIURQWZKHHOV (OLPLQDWLRQRIWKHODWHUDOYHORFLW\YIURPHTXDWLRQV \LHOGV
P N X U % P &N % % D % N & % E & _X_U0% % O & & & PX_X_ D& & E& U '
:KHQFRPSDUHGZLWK(T FRQVLGHULQJ LWLVQRWHGWKDWLQWKHVHFRQG
FRHIILFLHQW RI WKH WHUP N & KDV GLVDSSHDUHG DQG WKDW N % KDV EHHQ DGGHG7KLVLVGXHWRWKHGLIIHUHQWLDWLRQRIXLQWKHHOLPLQDWLRQSURFHVVRIY:H KDGRULJLQDOO\LQWKHVHFRQGFRHIILFLHQWRI WKHWHUP
PN X0 VJQ X ' &N D PJ % % ' &N & % % O
ZKLFKH[SODLQVWKHFKDQJHVREVHUYHG,QRWKHUUHVSHFWV(T LVVLPLODUWR WKH KRPRJHQHRXV YHUVLRQ RI (T +RZHYHU DQ LPSRUWDQW GLIIHUHQFH DSSHDUVLQWKHFRHIILFLHQWVZKLFKDUHQRZGHSHQGHQWRQWKHWLPHEHFDXVHRIWKH SUHVHQFHRIWKHOLQHDUO\ ZLWKWLPHGHFUHDVLQJVSHHGX:HKDYHWKHDGGLWLRQDO HTXDWLRQIRUX E\LQWHJUDWLRQRIWKHILUVWRI(TV
7+(25<2)67($'<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'GHQRWLQJWKHFRQVWDQWVRILQWHJUDWLRQ JRYHUQHGE\WKHLQLWLDOYDOXHVRI WKHVWDWHYDULDEOHVYDQGU DQG
UHSUHVHQWLQJWKHHLJHQYDOXHVWKDWLV WKHURRWV
RIWKHFKDUDFWHULVWLFHTXDWLRQRIWKHVXEVWLWXWLYHGLIIHUHQWLDOHTXDWLRQ
$ % % % & & XR_XR_ ' '
&RPSDULVRQZLWKWKHFRHIILFLHQWVRI(T UHYHDOVWKHH[SUHVVLRQVIRUWKH TXDQWLWLHV$%&DQG'6LQFHDSSDUHQWO\WKHILUVWWKUHHDUHDOZD\VSRVLWLYH DQGLQWKHFDVHFRQVLGHUHG & LV PXFKVPDOOHUWKDQ&ZKLFKPDNHV'DOVRD SRVLWLYH TXDQWLW\ ZH H[SHFW WKDW WKH VXEVWLWXWLYH V\VWHP FDQ RQO\ EHFRPH XQVWDEOHEHFDXVHRIDSRVVLEO\QHJDWLYHWKLUGFRHIILFLHQW7KHQRQHRIWKHURRWV OHWVVD\
EHFRPHVSRVLWLYH ZKLOHUHPDLQLQJUHDO 2EYLRXVO\WKLVFDQRQO\
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
XFULW '
JO )] & &
)URPWKLVH[SUHVVLRQLWFDQEHVHHQWKDWZKHQDULJLGIURQWW\UHLVFRQVLGHUHG VR
6 4 ZHREWDLQWKHVLPSOHIRUP X
WKDW&
FULW
% JO 7KHLQIOXHQFHRIDQHODVWLF
IURQWW\UHLVQRWYHU\JUHDWDVWKHIURQWFRUQHULQJVWLIIQHVVPD\UDQJHIURPWR
7+(25<2)67($'<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 FDVHRIIURQWZKHHOVORFNHG X ZKHUHREYLRXVO\WKHURRWVUHPDLQQHJDWLYH DQGEHFRPHFRPSOH[DWKLJKHUVSHHGVRIWUDYHO UHDOSDUW GRWWHGFXUYH $VD SUDFWLFDOLQGLFDWLRQIRUWKHGHJUHHRILQVWDELOLW\ZHPD\HPSOR\WKHKHLJKWRIWKH KRUL]RQWDODV\PSWRWHWRZKLFK
/
X64
WHQGVZKHQX
6 4 )RUWKDWZHILQG
J @ DE @ & & O N )]
'
ZKLFKVKRZVWKDWLQVWDELOLW\GHFUHDVHVZKHQWKHIURQWW\UHFRUQHULQJVWLIIQHVVLV ORZHU DQG EHFRPHV PRUH LQ EDODQFH ZLWK WKH KHUH YHU\ ORZ UHDU µFRUQHULQJ VWLIIQHVV¶ ,WDOVREHFRPHVFOHDUWKDWZKHQWKHIURQWZKHHOVDUHORFNHGDV )] WKHGHJUHHRILQVWDELOLW\UHGXFHVWR]HURUHSUHVHQWLQJLQGLIIHUHQW
ZHOO &
VWDELOLW\ 7KHFRPSOHWHVROXWLRQRI(T FDQEHIRXQGE\DSSO\LQJWKH/RPPHO WUDQVIRUPDWLRQ RI WKH %HVVHO GLIIHUHQWLDO HTXDWLRQ DQG XVLQJ WKH PRGLILHG
λ1,2 front wheels locked
rear wheels locked
λ1 -2
0
Xcrit
2
λ1,X 4
X µJO
λ1 λ2
λ2
)LJ 5HDO SDUW RI WKH HLJHQYDOXHV RI WKH VXEVWLWXWLYH OLQHDUV\VWHPZLWKFRQVWDQW FRHIILFLHQWV
7+(25<2)67($'<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 $EUDPRZLW] DQG 6WHJXQ FI (T IRUWKHWUDQVIRUPHGHTXDWLRQDQGSIRUWKHVROXWLRQ ,QWKHSDSHURI.RLWHUDQG3DFHMND WKHH[DFWVROXWLRQKDVEHHQJLYHQ WRJHWKHUZLWKH[DPSOHVRIWKHQXPHULFDOO\FRPSXWHGVROXWLRQIRUWKHFRPSOHWH QRQOLQHDUV\VWHP)RUWKHVSHFLDOFDVHZLWKSDUDPHWHUVD
E
N%
&)]
WKHVROXWLRQRIWKHOLQHDUHTXDWLRQ LVSUHVHQWHGLQ)LJ7KHH[DFW
VROXWLRQ LV FRPSDUHG ZLWK WKH DSSUR[LPDWH RQH RI WKH VXEVWLWXWLYH HTXDWLRQ7KHVROXWLRQLQWKHIRUPRIDVWDEOHDQGDQXQVWDEOHEUDQFKKDVEHHQ GUDZQ LQ WKH X U SODQH FRQVWLWXWLQJ WKH SURMHFWLRQ RI WKH FRPSOHWH GLPHQVLRQDOWUDMHFWRU\LQWKH XYU VSDFH$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
LQFOXGLQJWKHWKLUGWHUP ZLWKW 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([SRQHQWLDODUFWDQJHQWSDUDEROLF XS WRLWVPD[LPXP DQGK\SHUEROLFWDQJHQWIXQFWLRQV GLIIHUHQFHRIWZR 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\UHUXQVDWLWVUDWHG QRPLQDO ORDG )]R DWFDPEHUHTXDOWR]HUR WR]HUR
DWHLWKHUIUHHUROOLQJ
RUDWVLGHVOLSHTXDO
DQGRQDJLYHQURDGVXUIDFH R $VLPLODUVKDSHPHDQVWKDWWKH
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¶&¶'¶DQG(¶IRUWKHQRQGLPHQVLRQDO
)] = 1.0
)\ = )\
0
200 lb. 600 lb. 1000 lb. 1400 lb. 1800 lb.
µ )] Magic Formula fit
-1.0 -10
= 0.714, & \ = 1.40, ' 1.000, ( = -0.20 \ \ % \
-5
0
α= )LJ
&)α()] ) µ )]
5
10
tanα
5HVXOWRI5DGW¶VQRQGLPHQVLRQDOLVDWLRQRIW\UHFKDUDFWHULVWLFVVKRZLQJWKDWWKHW\UH VLGHIRUFHFKDUDFWHULVWLFVPHDVXUHGDWGLIIHUHQWORDGVUHGXFHWRYLUWXDOO\WKHVDPH FXUYHZKHQWKHIRUFHDQGWKHVOLSDQJOHDUHQRUPDOL]HGDVLQGLFDWHG
6(0,(03,5,&$/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 QRUPDOLVLQJWKHPRPHQWDQGWKHVOLSDQJOHDVLQGLFDWHG WUHSUHVHQWVWKHSQHXPDWLF 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,&$/7<5(02'(/6
IRUFHWDNHVSODFH SHDNOHYHO DQG DFKDQJHLQVORSHDWYDQLVKLQJVLGHVOLS 7KHILUVWPRGLILFDWLRQFDQEHFUHDWHGE\PXOWLSO\LQJWKHFKDUDFWHULVWLFERWK
LQ YHUWLFDO DQG KRUL]RQWDO GLUHFWLRQ ZLWK WKH UDWLR )] )]R 7KH KRUL]RQWDO PXOWLSOLFDWLRQLVQHHGHGWRQRWGLVWXUEWKHRULJLQDOVORSH:HREWDLQIRUWKHQHZ IXQFWLRQ
)\ '
)] ) ) ]R \R
HT
ZLWKWKHHTXLYDOHQWVOLSDQJOH
)]R )]
HT '
%HFDXVHREYLRXVO\WKHGHULYDWLYHRI)\RZLWKUHVSHFWWRLWVDUJXPHQW LVHTXDOWRWKHRULJLQDOFRUQHULQJVWLIIQHVV&)
M)\ M
M
HT
M
'
)
G \R G HT
HT
R
WKHVDPHYDOXHIRUWKHVORSHDW
)] G)\R ) ]R G HT
'
DW
ZHILQGIRUWKHGHULYDWLYHRI
)\ZLWKUHVSHFWWR
HT
' &) R
ZKLFKSURYHVWKDWWKHVORSHDWWKHRULJLQRIWKHFKDUDFWHULVWLFLVQRWDIIHFWHGE\ WKH VXFFHVVLYH PXOWLSOLFDWLRQV 7KH VHFRQG VWHS LQ WKH PDQLSXODWLRQ RI WKH
&α & α
)\
) R
3 [kN]
1
)
)
)
2
0
[kN/rad]
]
&α
1 2
)
2
1
)
)
1
0
\
15
0
0
2
]
-1
1
[kNm]
0
0
0.3 [rad]
α
&α
10
0
[kNm/rad]
]R
&α
]
0 R
-3
)LJ
&α
0
0
)
8VLQJWKHVLPLODULW\PHWKRGWRDGDSW
0 FXUYHVWRDQHZOHYHORIORDG
DQG
]
0 2
-2
300
)
&α
\R
0 1
2
&α
30
]R
\
2
4 [kN]
)
]
)LJ &RUQHULQJ DQGDOLJQLQJVWLII QHVVYHUVXVZKHHOORDG
6(0,(03,5,&$/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
$OLJQLQJWRUTXHDWQRPLQDOZKHHOORDG
0]R ' ']R
&]
VLQ>
%]R & (] %]R &
DUFWDQ^
DUFWDQ
%]R
`@
ZLWKVWLIIQHVVIDFWRU
% ]R ' &&0 R &] ' ]R
SHDNIDFWRU DRUHSUHVHQWLQJKDOIWKHFRQWDFWOHQJWKDWQRPLQDOORDG
']R ' F DR '\R
DQGDOLJQLQJVWLIIQHVVDVIXQFWLRQRIZKHHOORDG)]
&0 ' W&) ' F D&)
ZKHUHDSSDUHQWO\WKHSQHXPDWLFWUDLOW
FD+DOIWKHFRQWDFWOHQJWKDLVDVVXPHG
WREHSURSRUWLRQDOZLWKWKHVTXDUHURRWRIWKHZKHHOORDG
D ' DR )] )]R
7KHUHVXOWLQJDSSUR[LPDWLRQRIWKHSQHXPDWLFWUDLOWXUQVRXWWREHTXLWHDGHTXDWH DQG PLJKW EH FRQVLGHUHG WR EH XVHG LQ WKH VHW RI 0DJLF )RUPXOD 7\UH PRGHO HTXDWLRQV ( WREHGHDOWZLWKODWHURQ 7KHYDOXHVRISDUDPHWHUVXVHGIRUWKHFDOFXODWLRQVWREHFRQGXFWHGKDYHEHHQ OLVWHGLQ7DEOH 7DEOH3DUDPHWHUYDOXHVXVHGLQ6HFWLRQ
)]R 1 &\
&[
F
DR P (\
([
F F F
E
P &]
:
UH
P (]
:
F
\R
[R
F
F
F
F F
F
6(0,(03,5,&$/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
ZKHUHEGHQRWHVKDOIWKH DVVXPHGO\FRQVWDQW ZLGWKRIWKHFRQWDFWSDWFKUHWKH HIIHFWLYHUROOLQJUDGLXVDQG&) WKHORQJLWXGLQDOVOLSVWLIIQHVVZKLFKLVFRQVLGHUHG WREHOLQHDUO\LQFUHDVLQJZLWKORDG
&) ' F ) ]
E
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
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,&$/7<5(02'(/6
WR DQG FRQVLGHU WKH SQHXPDWLF WUDLO W DV LQ DQG LQFOXGH H[SOLFLW FRQWULEXWLRQV RI )[ WR WKH PRPHQW DV LQGLFDWHG LQ :H KDYH IRU WKH WKHRUHWLFDOVOLSTXDQWLWLHVZLWKWKH
['
(' \
VKLIWGXHWRFDPEHULQFOXGHG D
% (
WDQ
E
%
DQG
('
[
('
%
%
ZLWK
( \
&) ) ] &) )]
,Q WKH HQVXLQJ WKHRU\ ZH ZLOO PDNH XVH RI WKH WKHRUHWLFDO VOLS TXDQWLWLHV DE ZKLFKGRHVSURGXFHWKHVORSHLQWKHFRQVWDQW FXUYHVLQWKH)\YV)[ GLDJUDPDW)[
,WVKRXOGEHQRWHGKRZHYHUWKDWH[SHULHQFHKDVVKRZQWKDW
(
WKHXVHRISUDFWLFDOVOLSTXDQWLWLHVV[DQGV\ WKDWLV
DQGWDQ
(
ZKLFKPD\EH
REWDLQHGDJDLQE\RPLWWLQJWKHGHQRPLQDWRUVLQ DE PD\DOUHDG\JLYHYHU\ JRRGUHVXOWVDVZLOOEHLQGLFDWHGODWHURQLQ)LJ1RWHWKDWZKHQDSSURDFKLQJ ZKHHOORFN
[
647KLVFDOOVIRUDQDUWLILFLDOOLPLWDWLRQE\DGGLQJDVPDOOSRVLWLYH
TXDQWLW\WRWKHGHQRPLQDWRURI D %HFDXVHZHDUHGHDOLQJZLWKDQLQJHQHUDOQRQLVRWURSLFW\UHZHKDYHSXUH ORQJLWXGLQDODQGODWHUDOVOLSFKDUDFWHULVWLFVWKDWDUHQRWLGHQWLFDO FI)LJ 6WLOO ZH ZLOO WDNH WKH ODWHUDO DQG ORQJLWXGLQDO FRPSRQHQWV EXW WKHQ RI WKH UHVSHFWLYHSXUHVOLSFKDUDFWHULVWLFV)\RDQG)[R,IZHZDQWWRXVHWKHWKHRUHWLFDO ZHPXVWKDYHDYDLODEOHWKHSXUHVOLSFKDUDFWHULVWLFVZLWK
VOLSTXDQWLWLHV \
[
DQG
DVDEVFLVVD7KLVFDQEHDFFRPSOLVKHGE\ILWWLQJWKHRULJLQDOGDWDDIWHUKDYLQJ
FRPSXWHGWKHYDOXHVRI WUDQVIRUPLQJ 0 ]R
\
LQWRWDQ
[
IRUHDFKYDOXHRI E\XVLQJ D DQGVLPLODUIRU
DQGGHQRWHWKHUHVXOWLQJIXQFWLRQV )[R
[
)\R
\
DQG
2UVLPSOHUE\UHSODFLQJLQWKHDOUHDG\DYDLODEOHSXUHVOLSIRUFHDQG
PRPHQW IXQFWLRQV H[SUHVVHG LQ WHUPV RI WKH SUDFWLFDO VOLS TXDQWLWLHV WKH DUJXPHQW LQUDGLDQV E\WDQ
SUREDEO\DFFHSWDEOHDSSUR[LPDWLRQ DQG E\
WKHH[SUHVVLRQJLYHQEHORZ GHULYHGIURP(T D
'
[
&
[
ZKLFKZRXOGEHVXFFHVVIXOLIWKH)[FKDUDFWHULVWLFKDVEHHQILWWHGIRUWKHZKROH UDQJHRISRVLWLYH GULYLQJ DQGQHJDWLYH EUDNLQJ YDOXHVRI 2WKHUZLVHLIRQO\
6(0,(03,5,&$/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\(T E 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 WKHOHYHOVRIWKHLQGLYLGXDOFKDUDFWHULVWLFV DV\PSWRWHV DUHWKHVDPHIRUWKHVOLS DSSURDFKLQJ LQILQLW\ LQ WKH 0DJLF )RUPXOD ZH VKRXOG KDYH
[R
&
VLQ ò
[
\R
VLQ ò
&
\
$WODUJHVOLSWKHWKHRU\DSSDUHQWO\IDLOVWRSURSHUO\DFFRXQWIRU
WKHFRQWULEXWLRQRIWKHFDPEHUDQJOH$WZKHHOORFNRQHZRXOGH[SHFWWKHVLGH IRUFHWREHFRPH]HURDWYDQLVKLQJVOLSDQJOH7KLVZLOOQRWH[DFWO\EHWKHFDVHLQ WKH PRGHO GXH WR WKH HTXLYDOHQW WKHRUHWLFDO VLGH VOLS GHILQHG DFFRUGLQJ WR (T E 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 SDUDPHWHUVF FDQGFKDYHEHHQJLYHQYDOXHVOLVWHGLQ7DEOH )RUWKHFDVHRIDORDGGLIIHUHQWIURPWKHUHIHUHQFHYDOXHDQGWKHSUHVHQFHRI DFDPEHUDQJOHWKHFRPELQHGVOLSFKDUDFWHULVWLFVIRUWKHW\UHVLGHIRUFHDQGIRU WKH DOLJQLQJ WRUTXH KDYH EHHQ DVVHVVHG E\ XVLQJ WKH DERYH HTXDWLRQV 7KH UHVXOWLQJGLDJUDPVZLWKVOLSDQJOH
XVHGDVSDUDPHWHUKDYHEHHQSUHVHQWHGLQ
WKH)LJVDDQGE'XHWRWKHXVHRIWKHWKHRUHWLFDOVOLSTXDQWLWLHV DE 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)LJ 7KHIRUPXODLVFDSDEOH
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)[DVIXQFWLRQVRIWKHLUUHVSHFWLYHVOLSTXDQWLWLHV WKHVOLSDQJOH
DQGWKH
ORQJLWXGLQDOVOLS ZLWKWKHHIIHFWRIORDG)]DQGFDPEHUDQJOH LQFOXGHGLQWKH SDUDPHWHUV )LJXUH LOOXVWUDWHVWKHPHDQLQJRIVRPHRIWKHIDFWRUVE\PHDQVRIDW\SLFDO VLGHIRUFHFKDUDFWHULVWLF2EYLRXVO\FRHIILFLHQW'UHSUHVHQWVWKHSHDNYDOXH ZLWK
$ 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\PPHWULFVKDSHRIWKHRULJLQDOFXUYH FI)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$VXLWDEOHUHJUHVVLRQWHFKQLTXHLV XVHG WR GHWHUPLQH WKHLU YDOXHV IURP PHDVXUHG GDWD DFFRUGLQJ WR D TXDGUDWLF DOJRULWKPIRUWKHEHVWILW FI2RVWHQDQG%DNNHU 2QHRIWKHLPSRUWDQW IXQFWLRQDOUHODWLRQVKLSVLVWKHYDULDWLRQRIWKHFRUQHULQJVWLIIQHVV DOPRVWH[DFWO\ JLYHQE\WKHSURGXFWRIWKHFRHIILFLHQWV%\&\DQG'\RIWKHVLGHIRUFHIXQFWLRQ %&'\
.\
M) M
%&'\ ' S
\
DWWDQ
6+ ZLWK)]DQG
) S @ @ & S
VLQ > DUFWDQ ]
)RU]HURFDPEHU WKHFRUQHULQJVWLIIQHVVDWWDLQVLWVPD[LPXPSDW)]
%&'
) \
\
S,Q
γ
S
1
α
0
S
2
)]
)LJ&RUQHULQJVWLIIQHVVYHUVXVYHUWLFDOORDGDQGWKHLQIOXHQFHRIZKHHOFDPEHU(T
6(0,(03,5,&$/7<5(02'(/6
)\
0]
α W)
\
0]U
α
6
+I W
α
6+W
)LJ 7KHDOLJQLQJWRUTXHFKDUDFWHULVWLFFRPSRVHGRIDSDUWGLUHFWO\DWWULEXWHGWRWKHVLGH IRUFHDQGDSDUWGXHWKHVRFDOOHGUHVLGXDOWRUTXH GXHWRW\UHFRQLFLW\DQGFDPEHU
)LJWKHEDVLFUHODWLRQVKLSKDVEHHQGHSLFWHG$SSDUHQWO\IRUDFDPEHUHG ZKHHOWKHFRUQHULQJVWLIIQHVVGHFUHDVHVZLWKLQFUHDVLQJ_ _1RWHWKHGLIIHUHQFHLQ FXUYDWXUH OHIW DQG ULJKW RI WKH FKDUDFWHULVWLFV DW ODUJHU YDOXHV RI WKH FDPEHU DQJOH7RDFFRPSOLVKWKLVDVSOLW(DFFRUGLQJWR(T KDVEHHQHPSOR\HG :H UHIHU WR 6HFWLRQ IRU D FRPSOHWH OLVWLQJ RI WKH IRUPXODH +HUH QRQ GLPHQVLRQDOSDUDPHWHUVKDYHEHHQLQWURGXFHG)RUH[DPSOHWKHSDUDPHWHUVLQ ZLOOEHFRPH S
)]RS.\S
)]RS.\DQGS
S.\ZLWK)]RGHQRWLQJWKH
QRPLQDOZKHHOORDG 7KHDOLJQLQJWRUTXH0]FDQQRZEHREWDLQHGE\PXOWLSO\LQJWKHVLGHIRUFH)\ ZLWK WKH SQHXPDWLF WUDLO W DQG DGGLQJ WKH XVXDOO\ VPDOO H[FHSW ZLWK FDPEHU UHVLGXDOWRUTXH0]U FI)LJ :HKDYH
0] ' & W @ )\ % 0 ]U
7KHSQHXPDWLFWUDLOGHFD\VZLWKLQFUHDVLQJVLGHVOLSDQGLVGHVFULEHGDVIROORZV
W
W
' 'W FRV > &W DUFWDQ ^ % W W & (W %W W & DUFWDQ %W
W `@
ZKHUH
W
'
WDQ
% 6 +W
6(0,(03,5,&$/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
IRUPXODWLRQKDVSURYHQWRJLYHYHU\JRRGDJUHHPHQWZLWKPHDVXUHGFXUYHV7KH DGYDQWDJHZLWKUHVSHFWWRWKHHDUOLHUYHUVLRQVZKHUHIRUPXOD LVXVHGIRU WKHDOLJQLQJWRUTXHDVZHOOLVWKDWZHKDYHQRZGLUHFWO\DVVHVVHGWKHIXQFWLRQIRU WKHSQHXPDWLFWUDLOZKLFKLVQHHGHGWRKDQGOHWKHFRPELQHGVOLSVLWXDWLRQ ,Q)LJWKHEDVLFSURSHUWLHVRIWKHFRVLQHEDVHGFXUYHKDYHEHHQLQGLFDWHG VXEVFULSWVRIIDFWRUVKDYHEHHQGHOHWHGDJDLQ $JDLQ'LVWKHSHDNYDOXH&LV DVKDSHIDFWRUGHWHUPLQLQJWKHOHYHO\DRIWKHKRUL]RQWDODV\PSWRWHDQGQRZ% LQIOXHQFHV WKH FXUYDWXUH DW WKH SHDN LOOXVWUDWHG ZLWK WKH LQVHUWHG SDUDEROD )DFWRU(PRGLILHVWKHVKDSHDWODUJHUYDOXHVRIVOLSDQGJRYHUQVWKHORFDWLRQ[R
\D '
RIWKHSRLQWZKHUHWKHFXUYHLQWHUVHFWVWKH[D[LV7KHIROORZLQJIRUPXODHKROG
&'
('
DUFFRV
%[R & WDQ^ & ` %[ R & DUFWDQ %[R
LI
& !
6(0,(03,5,&$/7<5(02'(/6
,QFDVHRIWKHSRVVLEOHSUHVHQFHRIODUJHFDPEHUDQJOHV PRWRUF\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\ILUVWVKRZDVOLJKWLQFUHDVHLQPDJQLWXGH EHFRPLQJODUJHUWKDQRQH 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'UHSUHVHQWVWKHSHDNYDOXH VOLJKWO\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 DQGWKHUHVXOWLQJIRUFHFKDUDFWHULVWLFV SDUDPHWHUVDFFRUGLQJWR7DEOH 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$VLQGLFDWHGWKHKLOOEHFRPHVPRUHIODW ZLGHU DWODUJHUVOLSDQJOHV 7KHQ % \ GHFUHDVHV DFFRUGLQJ WR D :KHQ LQ DQ H[WUHPH VLWXDWLRQ
E
DSSURDFKHV WKDWLVZKHQ9F[ ZLOOUHPDLQHTXDOWRRQHXQOHVV ZKHQDW9F[
6 %
\
ZLOOJRWR]HURDQGFRQVHTXHQWO\*\
JRHVWRLQILQLW\ZKLFKPD\HDVLO\EHWKHFDVH
6 WKHZKHHOVSHHGRIUHYROXWLRQ
DQGWKXVWKHORQJLWXGLQDOVOLS
YHORFLW\ 9V[ UHPDLQV XQHTXDO WR ]HUR 7KH TXDQWLW\ 69\ LV WKH YHUWLFDO µVKLIW¶ ZKLFKVRPHWLPHVLVUHIHUUHGWRDVWKH
LQGXFHGSO\VWHHU$WFDPEHUGXHWRWKH
DGGHGDV\PPHWU\WKHORQJLWXGLQDOIRUFHFOHDUO\SURGXFHVDWRUTXHWKDWFUHDWHV DWRUVLRQDQJOHFRPSDUDEOHZLWKDSRVVLEO\DOUHDG\SUHVHQWSO\VWHHUDQJOH7KLV VKLIWIXQFWLRQYDULHVZLWKVOLS LQGLFDWHGLQ D
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
$VLOOXVWUDWHGLQ)LJLWVSHDNYDOXH'9\ GHSHQGVRQWKHFDPEHUDQJOH DQGGHFD\VZLWKLQFUHDVLQJPDJQLWXGHRIWKHVOLSDQJOH
)LJXUHSUHVHQWV
PHDVXUHGGDWDWRJHWKHUZLWKWKHILWWHGFXUYHVDVSXEOLVKHGE\%D\OHHWDO 7KHFRPELQHGVOLSUHODWLRQVIRU)[DUHVLPLODUWRZKDWZHKDYHVHHQIRUWKH VLGHIRUFH+RZHYHUDYHUWLFDOVKLIWIXQFWLRQZDVQRWQHHGHGWREHLQFOXGHG,Q )LJDWKUHHGLPHQVLRQDOJUDSKLVVKRZQLQGLFDWLQJWKHYDULDWLRQRI)[DQG )\ZLWKERWK DQG 7KHLQLWLDOµ6¶VKDSHRIWKH)\YV
FXUYHV DWVPDOO GXH
WRWKHYHUWLFDOVKLIWIXQFWLRQLVFOHDUO\YLVLEOH )RU SRVVLEOH LPSURYHPHQW RI WKH JHQHUDO WHQGHQF\ RI WKH PRGHO DW ODUJHU OHYHOVRIFRPELQHGVOLSEH\RQGWKHUDQJHRIDYDLODEOHWHVWGDWDRQHPLJKWLQFOXGH DGGLWLRQDOµIDEULFDWHG¶GDWDZKLFKDUHGHULYHGIURPVLPLODULW\PHWKRGUHVXOWVDW ODUJHUYDOXHVRIWKHVOLSDQJOH$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\9V RIWKHSRLQW6 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
QRPLQDO UDWHG 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 VOLSTXDQWLWLHV RU 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\EDFNZDUGVUXQQLQJ 9F[ 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'
,QWKHIROORZLQJ6HFWLRQZKHUHWKHLQIOXHQFHRIVSLQ WXUQVOLSDQGFDPEHU LVGHVFULEHGLQWKHH[WHQGHG0DJLF)RUPXODPRGHOWKHSURSHUH[SUHVVLRQVZLOO EHJLYHQIRUWKHVHIDFWRUVDQGDGGLWLRQDOHTXDWLRQVZLOOEHLQWURGXFHG
6(0,(03,5,&$/7<5(02'(/6
8VHU6FDOLQJ)DFWRUV 7KH IROORZLQJVHWRIVFDOLQJIDFWRUV LVDYDLODEOH7KHGHIDXOWYDOXHRIWKHVH IDFWRUVLVVHWHTXDOWRRQH H[FHSW
9
ZKLFKHTXDOV]HURLIQRWXVHG :HKDYH
SXUHVOLS )]R
QRPLQDO UDWHG ORDG
[\
SHDNIULFWLRQFRHIILFLHQW
9
ZLWKVOLSVSHHG9VGHFD\LQJIULFWLRQ
.[
EUDNHVOLSVWLIIQHVV
.\
FRUQHULQJVWLIIQHVV
&[\
VKDSHIDFWRU
([\
FXUYDWXUHIDFWRU
+[\
KRUL]RQWDOVKLIW
9[\
YHUWLFDOVKLIW
.\
FDPEHUIRUFHVWLIIQHVV
.]
FDPEHUWRUTXHVWLIIQHVV
W
SQHXPDWLFWUDLO HIIHFWLQJDOLJQLQJWRUTXHVWLIIQHVV
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\VXUIDFH ZHW ZLWKIULFWLRQGHFD\LQJZLWKLQFUHDVLQJ VOLS 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)RUFH SXUHORQJLWXGLQDOVOLS )
[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)RUFH SXUHVLGHVOLS )
\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
)
(
$OLJQLQJ7RUTXH SXUHVLGHVOLS 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)RUFH FRPELQHGVOLS )
[
*
'
*
@)
[
' 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)RUFH FRPELQHGVOLS )
\
*
'
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/RDG VHHDOVR(T )
]
'
@ )N 5 @ PD[ & ' M) M ' S S
]
]R
R
)]
]
]
]
@
]
)
&]
$
&]
N 5
]R
R
(
2YHUWXUQLQJ&RXSOH VHHDOVR6HF 0
[
'
) 5 ]
@ T & T
R
V[
V[
(%T
V[
) ) \
N @
]R
0[
(
5ROOLQJ5HVLVWDQFH0RPHQW VHHDOVR(TV
0
\
' &)
]
5
R
@ ^T
V\
DUFWDQ 9 9 U
R
%T
V\
) ) [
N `@
]R
(
0\
$OLJQLQJ7RUTXH FRPELQHGVOLS 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\EHFRPHDVKLJKDVFD IRUVRPHWUXFNW\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
VKRZQLQWKHOHIWKDQGGLDJUDP DW
DOVRLQGLFDWHGLQWKHORZHUOHIWGLDJUDP
RI )LJG FDQ EH IDEULFDWHG E\ VLGHZD\V VKLIWLQJ RI WKH VLGH IRUFH FXUYH
n
EHORQJLQJWR]HURVSLQ WXUQVOLS
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 YVVOLSDQJOHFXUYHZKLFKDULVHVGXHWRWKHYHUWLFDOVKLIWRIWKHFXUYH ZKLFKIRU 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] @ VJQ 9F[ (+\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>DUFWDQ T
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)LJGWKHSHDNWRUTXH WKDWZLOO 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[LPXPYDOXH LI&'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>DUFWDQ T
%U
R
7KHIDFWRU&UFRQWUROVWKHDV\PSWRWLFOHYHOZKLFKFRUUHVSRQGVWRWKHWRUTXHDW
E
,QWKDWVLWXDWLRQWKHWXUQVOLS
n ' & 0 _9 _ ' 5 ZKLFKFRUUHVSRQGVWR F\
WKHGHILQLWLRQJLYHQE\(T 7KHPRPHQW0]n WKDWLVJHQHUDWHGZKHQWKH 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,&$/7<5(02'(/6
ZLWK
%[n ' S'[n % S'[n GI] FRV >DUFWDQ S'[n
:LWK WKH IDFWRUV
@
GHWHUPLQHG DQG VXEVWLWXWHG LQ WKH HTXDWLRQV
( WKHGHVFULSWLRQRIWKHVWHDG\VWDWHIRUFHDQGPRPHQWJHQHUDWLRQKDV EHHQFRPSOHWHG7RVKRZWKDWWKHIRUPXODHSURGXFHTXDOLWDWLYHFRUUHFWUHVXOWV DFROOHFWLRQRIFRPSXWHGFXUYHVKDYHEHHQSUHVHQWHGLQWKHGLDJUDPVRI)LJ ,Q7DEOHWKHYDOXHVRIWKHPRGHOSDUDPHWHUVKDYHEHHQOLVWHG7KHGLDJUDPV VKRZWKDWWKHIRUPXODHDUHSHUIHFWO\FDSDEOHWRDWOHDVWTXDOLWDWLYHO\DSSUR[LPDWH WKHFXUYHVWKDWKDYHEHHQFRPSXWHGZLWKWKHEUXVKVLPXODWLRQPRGHORI&KDSWHU FI)LJ ,WKDVQRWEHHQDWWHPSWHGWRILQGDEHVWILWIRUWKHSDUDPHWHUV
7DEOH
0DJLF)RUPXODK\SRWKHWLFDOSDUDPHWHUYDOXHV3DUDPHWHUVWKDWJRYHUQWKHLQIOXHQFH RIORQJLWXGLQDOVOLSDQGFKDQJHVLQYHUWLFDOORDGKDYHEHHQJLYHQWKHWHQWDWLYHYDOXH ]HUR
)]
1)]R
15R
P5H
5R D
P
S&[
S'[
S'[
S([
S([
S([
S([
S.[
S.[
S.[
S+[
S+[
S9[
S9[
S&\
S'\
S'\
S'\
S(\
S(\
S(\
S(\
S.\
S.\
S.\
S+\
S+\
S+\
S9\
S9\
S9\
S9\
T%]
T%]
T%]
T%]
T%]
T%]
T%]
T&]
T']
T']
T']
T']
T']
T']
T']
T']
T(]
T(]
T(]
T(]
T(]
T+]
T+]
T+]
T+]
U%[
U%[
U&[
U+[
U%\
U%\
U%\
U&\
U+\
U9\
U9\
U9\
U9\
U9\
U9\
VV]
VV]
VV]
VV]
S.\n
S'[n
S'[n
S'\n
S+\n
S+\n
S+\n
S+\n
Sg n
Sg n
T'Wn
T%Un
T'Un
T'Un
T&Un
T&Un
S'\n
S'\n
6(0,(03,5,&$/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 VWUDLJKWDKHDGUXQQLQJ DW
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:LWKWKHLQWURGXFWLRQRIWKH VPDOO HTXLYDOHQWVOLSDQJOH
0SO\ . & &0
SO\ ZLWK
&0 ' W&)
SO\
ZHKDYH
DQG
)SO\ . &)
SO\
$OVRDFFRUGLQJWR/HHWKHUHVLGXDOWRUTXHDW]HURVLGHIRUFH0]ULQ)LJLV VWURQJO\FRUUHODWHGZLWK
0]U . &0
R
DQGWKXVZLWKFRQLFLW\:HILQGIURPWKHGLDJUDP
R
,I ZH PD\ IXUWKHU DVVXPH WKDW IRU WKH FRQLFLW\ IRUFH DQG PRPHQW D VLPLODU FRUUHVSRQGHQFHZLWKFDPEHUUHVSRQVHH[LVWVZHZRXOGKDYHDIWHULQWURGXFLQJDQ HTXLYDOHQWFDPEHUDQJOH
0FRQ . &0
FRQ
FRQ ZLWK
&0 ' W &)
ZKHUHW ! UHSUHVHQWVWKHGLVWDQFHRIWKHSRLQWRIDSSOLFDWLRQRIWKHUHVXOWLQJ FDPEHUWKUXVWLQIURQWRIWKHFRQWDFWFHQWUH WKDWLV QHJDWLYHWUDLO 7KHFRQLFLW\ IRUFHEHFRPHV
0]
)\
no camber:
γ=0
&)α
0]U δo
&0α
α
)LJ7KHVLWXDWLRQQHDU]HURVOLSDQJOHIRUFHDQGPRPHQWGXHWRFRQLFLW\DQGSO\VWHHU
6(0,(03,5,&$/7<5(02'(/6
)FRQ . &)
FRQ
:LWKWKHVHDVVXPSWLRQVLWLVSRVVLEOHWRHVWLPDWHWKHFRQWULEXWLRQVRISO\VWHHU DQGFRQLFLW\LQWKHLQLWLDOYDOXHVRIVLGHIRUFHDQGDOLJQLQJWRUTXH DW
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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,&$/7<5(02'(/6
0]
no camber:
γ =0
)\
0]U
conicity compensated
γ = -γFRQ
)\
0 ]
α
α
SO\
∆αo δo )LJ
α
I
&KDUDFWHULVWLFVRIDW\UHZLWKSO\VWHHUDQGFRQLFLW\VKRZLQJFRQYHUVLRQWRHTXLYDOHQW FDPEHUDQGVOLSDQJOH
7KHHTXLYDOHQWVOLSDQJOHFDQQRZEHREWDLQHGIURP
'
SO\
R
&
&) &)
9
VJQ F[
FRQ
7KHIRXUVOLSDQGFDPEHUVWLIIQHVVHVDUHDYDLODEOHIURP(TV ( 7KHVLJQRIWKHIRUZDUGYHORFLW\KDVEHHQSURSHUO\LQWURGXFHG 7KHQH[WWKLQJZHKDYHWRGRLVH[SUHVVLQJ
R
DQG
R
LQWHUPVRIWKHVKLIWV
DQG WKH LQLWLDO UHVLGXDO WRUTXH DV GHILQHG LQ 6HFWLRQ :H REWDLQ ZLWK (
R
' 6+I
DQGIURP DQG)LJ
R'
)\ 0 ] % ' 0]U &0 &) &0
ZLWK&0 DFFRUGLQJWR(T ( DQGWKHLQLWLDOUHVLGXDOWRUTXHIURP (
0]U ' 'UR ' )] 5R T']%T '] GI] @
0U
)LQDOO\WKHLQLWLDOVLGHIRUFHDQGWRUTXHDUHWREHUHPRYHGIURPWKHHTXDWLRQVE\ SXWWLQJWKHSDUDPHWHUVS+\S+\S9\S9\T ']DQGT']RUWKHVFDOLQJIDFWRUV +\
9\
DQG
0U
HTXDOWR]HURDQGE\UHSODFLQJLQ(TV ( WKHRULJLQDOVLGH
VOLSLQSXWYDULDEOH
(
HII
'
WDQ
%
WDQ
SO\ VJQ
@VJQ9[E\LWVHIIHFWLYHYDOXH
9[
DQG LQ (TV ( WKH RULJLQDO FDPEHU
VLQ E\ WKH HIIHFWLYH WRWDO
6(0,(03,5,&$/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,&$/7<5(02'(/6
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
7KH2YHUWXUQLQJ&RXSOH
7KHRYHUWXUQLQJFRXSOHLVHVSHFLDOO\LPSRUWDQWWRLQYHVWLJDWHWKHYHKLFOHUROORYHU RFFXUUHQFHDQGWKHFXUYLQJEHKDYLRXURIDPRWRUF\FOH,QPRGHUDWHFRQGLWLRQV (T ( PD\ RIWHQ EH VXIILFLHQW WR PRGHO WKH RYHUWXUQLQJ PRPHQW ,Q WKH SUHVHQWVHFWLRQZHZLOOVWXG\FDVHVWKDWUHTXLUHPRUHHODERUDWHPRGHOOLQJ
0[< 0
)\ &
)]
0[ > 0
)\ )] &
)LJ 1DUURZDQGZLGHW\UHVXEMHFWHGWRVLGHVOLSJLYLQJULVHWRDQRYHUWXUQLQJFRXSOHWKDW PD\EHRIDGLIIHUHQWVLJQ
6(0,(03,5,&$/7<5(02'(/6
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
M0[ M DW]HURFDPEHULVSRVLWLYH
6(0,(03,5,&$/7<5(02'(/6
0[
2.2 4.4 1.1
40
0
6.6
[Nm] -40
)]
-80
8.8 kN
0
2
4
6
8
10
α
12
14
16 [o ]
18 20
)LJ 2YHUWXUQLQJFRXSOHFDOFXODWHGZLWKIRUPXODH FI)LJIRUVLPLODU PHDVXUHGFKDUDFWHULVWLFV
7RPRGHOVXFKDVLJQFKDQJLQJVLWXDWLRQWKHRYHUWXUQLQJFRXSOHUHVSRQVHWR VLGH VOLS RI WKH ZKHHO LV WDNHQ DV DQ H[DPSOH :H ZLOO XVH D IRUPXOD WKDW LV HVWDEOLVKHGE\DGGLQJWZRIXQFWLRQV
0
[
'&
)
] \
%
\
ZLWKWKHWZRFRQWULEXWLRQVWRWKHPRPHQWDUP
\
\
'
) 5
'&
R T[
5
)
\
]R
R T[ FRV
T
[ DUFWDQ T[ )] )]R
@ VLQ T
[ DUFWDQ T[ )\ )]R
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
6(0,(03,5,&$/7<5(02'(/6
VWLIIQHVVLVHIIHFWHGE\KRUL]RQWDOW\UHIRUFHVFI5HLPSHOO (T
7DEOH3DUDPHWHUYDOXHVIRURYHUWXUQLQJFRXSOHFRPSXWDWLRQFI(TV
)]R
N1
\ T[
5R
P
&\
T[
(\
T[
S.\
T[
S.\
T[
T[
'XHWR&DPEHU 7KH VHFRQG FDVH WR EH GHDOW ZLWK FRQFHUQV WKH PRWRUF\FOH W\UH $ VSHFLDO SUREOHPFDQEHLGHQWLILHGWKDWFRQFHUQVWKHSRVVLEO\YHU\ODUJHZKHHOLQFOLQDWLRQ DQJOH6LPLODUVLWXDWLRQVPD\EHHQFRXQWHUHGZLWKDFDUW\UHZKHQH[DPLQLQJWKH UROORYHUHYHQWRUZKHQWKHZKHHOUXQVRYHUDORFDOO\VWHHSWUDQVYHUVHVORSH,Q WKHVHFDVHVWKHJHRPHWU\RIWKHFURVV VHFWLRQ RIWKHW\UHZLWKLWVILQLWHZLGWK EHFRPHVRILPSRUWDQFHDQGLVODUJHO\UHVSRQVLEOHIRUWKHRYHUWXUQLQJFRXSOHWKDW
γ U
UR (=5R)
UO
UOR
\F
UF
U\
&
Z
)[
0[
)\
&
)] ]
0]
0\
ρ]
&
\
)] )1 (normal force)
moment arm
)LJ 7KHPRWRUF\FOHZKHHODWYHU\ODUJHFDPEHUDQJOH 7ZRDOWHUQDWLYHDSSURDFKHVWR GHILQH WKH FRQWDFW FHQWUH
&
DQG WKH DVVRFLDWHG µORDGHG UDGLXV¶
FRQVHTXHQFH WKH RYHUWXUQLQJ FRXSOH FRQWRXUPD\EHWDNHQ
0
[
U
O
DQG DV D
,QVWHDG RI WKH FLUFOH WKH DFWXDO VHFWLRQ
6(0,(03,5,&$/7<5(02'(/6
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¶WKDWUHPDLQVHTXDOWRWKHYDOXHWKDWKROGVIRUWKH O XSULJKWZKHHO UORZKLFKLVHTXDOWRWKHIUHHUDGLXVURPLQXVWKHQRUPDOGHIOHFWLRQ ]
,WVKRXOGWKHQEHUHDOLVHGWKDWDFRQFXUUHQW VOLJKW ULVHZNRIWKHHIIHFWLYH
URDGSODQHPD\QRWEHQHJOLJLEOH7KLVPHWKRGWRDFTXLUHWKHRYHUWXUQLQJFRXSOH FRUUHVSRQGVWRWKHSUDFWLFDOSURFHGXUHRIWHQHPSOR\HGZKHQPHDVXULQJRQ 0N [ WKHURDGZKHUHWKHDFWXDOKHLJKWRIWKHZKHHOFHQWUHDERYHWKHURDGVXUIDFHLVQRW PHDVXUHG7KHVHFRQGDOWHUQDWLYHLVSRVVLEO\WKHSK\VLFDOO\PRVWUHDOLVWLFRQH ,WGHILQHVDFRQWDFWFHQWUH&´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
6WDQGDUGGHILQLWLRQXVLQJ& IRUJHQHUDOFRQWRXUDQGIRUFLUFOHVHFWLRQ U
O
' \R % FR WDQ & ] FRV ' R & F % F & [ ' & ] FR FRV & ] WDQ ' & ] F &
0
U
\
)
U
\
U
)
U
U
] FRV ] WDQ
6(0,(03,5,&$/7<5(02'(/6
3UDFWLFDOGHILQLWLRQXVLQJ&¶ U
O
' OR ' R & ] [ ' & ] F & ] VLQ % ' & F & ] & FRV U
U
0
)
Z
U
U
)
\ Z
3K\VLFDOO\PRUHUHDOLVWLFGHILQLWLRQXVLQJ&´
' \ ' R& F% F& O ' [ F ' F & ] VLQ
UO
O
U
U
U
U
] FRV
0 \
U
7KHIRUFHVDFWLQWKHSRLQWV&&¶DQG&´UHVSHFWLYHO\ 7KHRYHUWXUQLQJFRXSOH0[DFFRUGLQJWRWKHVWDQGDUGGHILQLWLRQ LV GLUHFWO\ UHVSRQVLEOH IRU WKH OHDQ DQJOH RI WKH PRWRUF\FOH LQ D VWHDG\ WXUQ WR EHFRPHODUJHUWKDQZKDWZRXOGEHH[SHFWHGDFFRUGLQJWRWKHUDWLRRIFHQWULIXJDO IRUFHDQGYHKLFOHZHLJKW,QWKHSUDFWLFDOGHILQLWLRQLWZRXOGEHWKHFRPELQDWLRQ DQGWKHHIIHFWLYHURDGVXUIDFHULVHZNWKDWFDXVHVWKHLQFUHDVHLQUROODQJOH RI0N [ ZKLOHZLWKWKHWKLUGGHILQLWLRQLWZRXOGEHGXHWRWKHODWHUDOVKLIW\FRIWKHYLUWXDO ZKHHOSODQH ,UUHVSHFWLYHRIWKHGHILQLWLRQHPSOR\HGLWPXVWEHFOHDUKRZWKHPHDVXUHG GDWD KDYH EHHQ SURFHVVHG DQG FRQYHUWHG IURP WKH ZKHHO D[OH V\VWHP RI D[HV ZKHUHLQJHQHUDOWKHIRUFHVDUHPHDVXUHG WRWKHURDGDQGOLQHRILQWHUVHFWLRQ EDVHG D[HV V\VWHP $Q DGGLWLRQDO FRQYHUVLRQ PD\ EH QHFHVVDU\ WR VXLW WKH UHTXLUHPHQWVRIWKHYHKLFOHG\QDPLFLVW,WPD\EHUHDOLVHGWKDWZLWKUHVSHFWWRWKH VWDQGDUGGHILQLWLRQRIWKHORDGHGUDGLXVUO(T WKHHIIHFWLYHUROOLQJUDGLXV UHIROORZVDTXLWHGLIIHUHQWFRXUVH$OVRDWODUJHFDPEHUDQJOHVUH ZLOOUHPDLQ FORVHWRWKHUDGLXVUOO
U\GHILQHGE\ DQGVKRZQLQ)LJ
7RDFFRXQWIRUWKHHIIHFWRIWKHODWHUDOW\UHGHIOHFWLRQDQGWKURXJKWKDWWKH FKDQJHLQYHUWLFDOSUHVVXUHGLVWULEXWLRQWKHSDUWRI0[DWWULEXWHGWRZKHHOFDPEHU DQGGHILQHGE\RQHRIWKHDERYHRUVLPLODUH[SUHVVLRQVVKRXOGEHH[WHQGHGZLWK WKH SDUW RI WKH RYHUWXUQLQJ FRXSOH GHVFULEHG E\ (TV RU LI DSSURSULDWHE\DVLPSOHUYHUVLRQRIWKDW
1RWHRQWKH$OLJQLQJ7RUTXHDW/DUJH&DPEHU ,WPD\ EH RILQWHUHVWWREHDZDUHRIDQRWKHUXQH[SHFWHGSKHQRPHQRQWKDW DULVHV DW ODUJH FDPEHU DQJOHV 2QH PLJKW PDNH D PLVWDNH ZKHQ GHULYLQJ WKH DOLJQLQJWRUTXHIURPWKHZKHHOD[OHRULHQWHGPHDVXUHGIRUFHVDQGPRPHQWV,I WKHVH ODWWHU TXDQWLWLHV DUH GHQRWHG ZLWK DQ DGGLWLRQDO VXEVFULSW D WKH DOLJQLQJ WRUTXH LV REWDLQHG IURP WKH PRPHQW HTXLOLEULXP DERXW WKH YHUWLFDO D[LV DV
6(0,(03,5,&$/7<5(02'(/6
IROORZV FI)LJ 0
]
'
0
]D FRV
%
0
\D VLQ
&
)
[D U VLQ
RUDOWHUQDWLYHO\IURPWKHPRPHQWHTXLOLEULXPDERXWWKHZKHHOUDGLXV 0
]
'
0
\ WDQ
%
0
]D FRV
,IWKHIDFWLVQRWUHFRJQLVHGWKDWDWFDPEHUDORQJLWXGLQDOIRUFH)[LVJHQHUDWHG HYHQ DW IUHH UROOLQJ ZKHUH 0\D LV ]HUR RU YHU\ VPDOO DQG FRQVHTXHQWO\ )[D LV GLVUHJDUGHGLQ(T WKHFDOFXODWHGDOLJQLQJWRUTXHLVLQFRUUHFW,Q WKHUROOLQJUHVLVWDQFHPRPHQW0\PD\QRWEHQHJOLJLEOHIRUDFDPEHUHGZKHHO UXQQLQJ DW D VLGH VOLS DQJOH 7KHQ WKH UHODWLYHO\ VWLII EHOW PD\ EH D[LDOO\ GHIRUPHG LQ DQ DV\PPHWULF ZD\ 7KLV PD\ H[SODLQ WKDW H[SHULPHQWV WHQG WR LQGLFDWHWKDWWKHWKURXJKFDPEHUJHQHUDWHG)[LVYLUWXDOO\LQGHSHQGHQWRIWKHVOLS DQJOH
&RPSDULVRQZLWK([SHULPHQWDO'DWDIRUD&DUDQGD7UXFN7\UH
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
8
)\
6(0,(03,5,&$/7<5(02'(/6
8
γ =0
)\
] 2000N 4500N 7000N
)
4
[kN]
γ
] 7000N
)
4
=5
o
0
-5
α
5
[kN]
0
0
-4
-8 -15
0
-4
test data MF-fit
-10
-5
0
α
5
[deg.]
10
-8 -15
15
γ =0
0
] 2000N, 4500N, 7000N
)
-2
-2
[kN]
-10
-5
0
[deg.]
γ =0 ] 7000N
α = -10
o
-4
)[
)[ -6
15
)
[kN]
-4
10
-5
-6
2 -2 -8
-8
-1
)\
2
-0.8
-0.6
κ
-0.4
]
)\
7000N
0
-0.6
κ
-0.4
[-] -0.2
0
-2
-2
-2 -4
-5
-0.6
κ
-0.4
[-] -0.2
-5
γ =0
-10
-6
0
[kN]
-2
-4
α = 2o
2
0
0
-0.8
-0.8
o
[kN]
-1
-1
α =2
γ=0 )
0
[-] -0.2
0
-6 0
-10
] 7000N
)
-8
-6
)[
-4
[kN]
-2
0
)LJ )RUFH FKDUDFWHULVWLFV RI D 5 FDU W\UH 0DJLF )RUPXODFRPSXWHGUHVXOWV FRPSDUHGZLWKGDWDIURPPHDVXUHPHQWV GRWWHGFXUYHV FRQGXFWHGZLWKWKH'HOIW 7\UH7HVW7UDLOHU
6(0,(03,5,&$/7<5(02'(/6
200
0
250
0
γ =0
]
) ]
100
[Nm]
γ
]
2000N 4500N 7000N
=5
o ) ]
7000N
0 -5
[Nm]
0
0
-100
-200 -15
-10
-5
0
α
5
10 [deg.]
γ =0
200
0
) ]
]
15
-5
-10
-5
0
α
5
10 [deg.]
200
7000N
-2
100
-250 -15
0
15
-5
]
-2
100
[Nm]
-10
0
-10
[Nm]
0
0 0
α = 2o
-100 -1
-0.8
-0.6
κ
γ =0
-100
-0.4 [-] -0.2
) ]
0
α =2
o
7000N
-8
-6
)[
-4
[kN]
-2
0
)LJ $OLJQLQJ WRUTXH FKDUDFWHULVWLFV RI D 5 FDU W\UH 0DJLF)RUPXODPRGHO FRPSXWHGUHVXOWVFRPSDUHGZLWKGDWDIURPPHDVXUHPHQWV GRWWHGFXUYHV FRQGXFWHG ZLWKWKH'HOIW7\UH7HVW7UDLOHU
2.2
)] = 8.9 kN
100
390
0[
UO
4.4
6.7 0
4.4
[Nm]
2.2
380 6.7 [mm]
)] = 8.9 kN
370
-100 o
-20
o
-10
0
α
o
10
o
20
o
-20
o
-10
0
α
o
10
o
20
)LJ 2YHUWXUQLQJFRXSOHDQGORDGHGUDGLXVRIDSDVVHQJHUFDUW\UHPHDVXUHGRQ)RUG¶V 076)ODW7UDF,,,W\UHWHVWLQJPDFKLQH 9G-DJW &RPSDUHZLWK)LJ
6(0,(03,5,&$/7<5(02'(/6
30 20
)\
10
30
γ =0
20
0 [kN]
-10
-10
-20
-20
0
0
o
-2
10
0 [kN]
0 -10
=4
)\
] 16.5 kN 29.0 kN 41.0 kN
)
-30 -15
γ
] 41 kN
)
-5
0
α
5
[deg.]
10
15
4 -2
-30 -15
γ =0
0
] 11kN, 22kN, 33kN
)
-2
-10
-5
0
α
γ =0
)[
15
o
] 11 kN
)
-4
)[
10
-5 2 1 -2 -1,0
[kN]
-20
[deg.]
α = -8
[kN]
-10
5
-6
α = -8
o
-8
-30 -1
4
)\
-0.8
-0.6
κ
-0.4
[-] -0.2
0
0
-10 -1
-0.8
-0.6
κ
-0.4
4
γ=0
2
] 11 kN
)
0
0
1 0
0 [kN]
-1
[kN]
2
)\
1
-1
-2 -4
-2 -4
-5 o
-0.8
-0.6
κ
-0.4
[-] -0.2
-5
γ=0
α = -8 -8 -1
0
[-] -0.2
α = -8
o
] 11 kN
)
0
-8 -10
-8
-6
)[
-4 [kN] -2
0
)LJ )RUFHFKDUDFWHULVWLFVRID5WUXFNW\UH0DJLF)RUPXODFRPSXWHGUHVXOWV FRPSDUHGZLWKGDWDIURPPHDVXUHPHQWVFRQGXFWHGZLWKWKH&DOVSDQIODWWUDFNW\UH WHVWIDFLOLW\ GRWWHGFXUYHV
6(0,(03,5,&$/7<5(02'(/6
800
800
γ =0
0
]
)
]
400
γ
0
o
=4
[Nm]
41.0 kN
)
]
] 400
16.5 kN 29.0 kN 41.0 kN
0
[Nm] 0 0
0 4 -400
-400
-2 -15
-10
-5
0
γ =0
200
0
)
]
α
5
10 [deg.]
-800 -15
15
-10
-5
0
α = -8,-5,-2,-1,0,1,2 o
α
5
10 [deg.]
15
α = -8,-5,-2,-1,0,1,2 o 200
11 kN
0]
] 0
0
α = -8o
[Nm]
α = -8o
[Nm]
2
-200
o
2
-200
γ =0 ) ]
-400 -1
-0.8
κ
-0.6
-0.4
[-] -0.2
0
-400 -10
-8
-6
)[
11 kN
-4 [kN] -2
0
)LJ $OLJQLQJ WRUTXH FKDUDFWHULVWLFV RI D 5 WUXFN W\UH0DJLF)RUPXOD FRPSXWHG UHVXOWV FRPSDUHG ZLWK GDWD IURP PHDVXUHPHQWV FRQGXFWHG ZLWK WKH &DOVSDQIODWWUDFNW\UHWHVWIDFLOLW\ GRWWHGFXUYHV
([HUFLVH $VVHVVPHQW RI RIIQRPLQDO W\UH VLGHIRUFHFKDUDFWHULVWLFVDQG FRPELQHGVOLSFKDUDFWHULVWLFVZLWK
)[DVLQSXWTXDQWLW\
&RQVLGHUWKHGLDJUDPVRI)LJ)RUWKHRULJLQDO QRPLQDO VLGHIRUFHFKDUDFWHULVWLF
)
\R
& )
DQGIRUWKHFRUQHULQJVWLIIQHVVYVZKHHOORDGFKDUDFWHULVWLF
)
]
HPSOR\WKH
IRUPXODHDVJLYHQE\(TV XVLQJWKHGDWDLQGLFDWHGLQWKHILJXUH7KHYDULDWLRQ RI FDPEHU VWLIIQHVV ZLWK ZKHHO ORDG
& ) )
]
LV DVVXPHG WR EH OLQHDU 7R FRYHU WKH
&
FRPELQHGVOLSVLWXDWLRQXVH(TV QRWH
)
LVH[SUHVVHGLQ1GHJUHHFRQYHUW
WKLVILUVWLQWR1UDGWRJHWLQOLQHZLWK(T ,Q(T D FKRRVH QRPLQDOFRQGLWLRQZHKDYHSDUDPHWHUYDOXHV
)
]R
1
R
&
)
R
1GHJ
8VH(TV DQGGRWKHIROORZLQJ
&
) R
1GHJ
Q
,QWKH
6(0,(03,5,&$/7<5(02'(/6
)
) ]
)]R
) [
) \o 2400 [N]
γ =0
) = 3000N, = 0.8 ) ]R
'\=2400N, (\=-2, &\=1.2 o 0
6
α
o
20
\o \ \o = 400N/
& %&'
)α ο
o
o
F1F 2 = 500/) , F 2 = 2
]R
&
Q 4
)α
-1
)α
Eq.(4.42) (N/rad!)
&
µ
0
) [
400 [N/ o ] 1
) [ )]
0 &
)γ
&
)γ
3000
0
500
6000 [N]
)
)]
) ]R ) [
0
Eq.(4.44)
-1
µ
0
) [ )]
1
0
50 [N/ o ] 3000 6000 [N]
)]
)LJ 2QWKHDVVHVVPHQWRIRIIQRPLQDODQGFRPELQHGVOLSW\UHVLGHIRUFHFKDUDFWHULVWLFV ZLWKORQJLWXGLQDOIRUFHDVLQSXWTXDQWLW\XVLQJWKHVLPLODULW\PHWKRG ([HUFLVH
)
'HULYHDQGSORWWKHIXQFWLRQ
\
IRUWKHIROORZLQJSDUDPHWHUYDOXHVGHVFULELQJ
WKHW\UHDWDGLIIHUHQWFRQGLWLRQ
)
]
1
FRQGLWLRQ
) ) ) )
E
3ORW WKH JUDSK IRU R
\
]
[
)
[
1
IRU WKH FRQVWDQW VOLS DQJOHV
]R
DQG
E
DQG DW WKH
([HUFLVH $VVHVVPHQW RI IRUFH DQG PRPHQW FKDUDFWHULVWLFV DWSXUHDQG FRPELQHGVOLSXVLQJWKH0DJLF)RUPXODDQGWKHVLPLODULW\PHWKRG ZLWK
DVLQSXW
*LYHQDUHWKHSXUHVOLSIRUFHDQGPRPHQWFKDUDFWHULVWLFVDWQRPLQDOYHUWLFDOORDG)]R 1DQGWKHFRUQHULQJVWLIIQHVVYVYHUWLFDOORDG FI)LJVDE 7KHSUREOHPV DUHIRUPXODWHGDVIROORZV
'HWHUPLQHIRUWKHRULJLQDOFXUYHVWKDWLVDWWKHQRPLQDOORDGWKHYDOXHVRIWKH FRHIILFLHQWV % & ' DQG ( IURP WKH GDWD LQGLFDWHG LQ WKH ILJXUHV EHORZ 7KH YHUWLFDO DQG KRUL]RQWDO VKLIWV DUH GLVUHJDUGHG 8VH WKH IRUPXOD IRU WKH ORQJLWXGLQDOIRUFHWKHVLGHIRUFHDQGDOVRIRUWKHPRPHQW&RQVLGHU)LJ DQG XVHWKHIRUPXODH DQG 'UDZWKHUHVXOWLQJFXUYHV
6(0,(03,5,&$/7<5(02'(/6
)[R
)\R
longitudinal force
4400
0
lateral force
4000
3400
κ
0.07
0
3400
tanα
0.15
)LJD2ULJLQDOW\UHIRUFHFKDUDFWHULVWLFVDWSXUHVOLS ([HUFLVH
[R
\R
)[SHDN
1
)\SHDN
1
)[D
1
)\D
1
WDQ
&)
&)
P
&)
&
)α
R
cornering stiffness
&
4000
&)
aligning torque
0
]R
110kN/rad
)αR
0
P
R
100 Nm )]
8000N
0
20
tanα
0.07
)LJE&RUQHULQJVWLIIQHVVDQGRULJLQDOPRPHQWFKDUDFWHULVWLFV ([HUFLVH
S
N1UDG
0]SHDN
1P
S
1
0]D
1P
IXUWKHU
WDQ &) )]
)]>1@
&0
P
R
&0
HIIHFWLYHFRPSOLDQFHSDUDPHWHU
W )]
î
IRU0](T
&0 )]
W&) )] >1PUDG@
F
F )RU)]
î )]RDR DQG1ZLWK[
SORW WKH FXUYHV IRU SXUH VOLS )[
[RDQG\
)\
\RZKLOH
DQG 0]
)]>P@
FRPSXWHDQG
(PSOR\ WKH VLPLODULW\
(TV
)RU)]
1ZLWK[
[R\
\RDQG
FRPSXWHDQGSORWWKHFRPELQHG
VOLSFXUYHV)\ )[ DQG0] )[ DWWZRVOLSDQJOHV
WKHORQJLWXGLQDOVOLSPD\UDQJHIURP
E
E
DQG 7KHYDOXHVRI
WR8VH(TV DQG
&KDSWHU 12167($'<67$7(2872)3/$1( 675,1*%$6('7<5(02'(/6
,QWURGXFWLRQ
7KHWUDQVLHQWDQGRVFLOODWRU\G\QDPLFEHKDYLRXURIWKHW\UHZLOOEHGLVFXVVHGLQ WKLV DQG WZR HQVXLQJ FKDSWHUV 7KH SUHVHQW FKDSWHU LV GHYRWHG WR WKH PRGHO GHYHORSPHQWRIWKHW\UHDVDQLQWHJUDOFRPSRQHQW7KHVWUHWFKHGVWULQJPRGHOLV FKRVHQ DV WKH EDVLV IRU WKH SK\VLFDO GHVFULSWLRQ RI WKH RXWRISODQH DQWL V\PPHWULF EHKDYLRXU7KLVPRGHOH[KLELWVDILQLWHFRQWDFWOHQJWKWKDWDOORZVWKH VWXG\RIVKRUWSDWKZDYHOHQJWKSKHQRPHQD7KHPRGHOLVUHODWLYHO\VLPSOHLQ VWUXFWXUHDQGLQWHJUDWHVWKHFDUFDVVFRPSOLDQFHDQGFRQWDFWSDWFKVOLSSURSHUWLHV )RUWKHPRPHQWUHVSRQVHWR\DZYDULDWLRQVWKHILQLWHZLGWKRIWKHFRQWDFWSDWFK QHHGV WR EH LQWURGXFHG ZKLFK LV DFFRPSOLVKHG E\ FRQQHFWLQJ WR WKH VWULQJ WKH EUXVKPRGHOIHDWXULQJRQO\IRUHDQGDIWWUHDGHOHPHQWFRPSOLDQFH,QWKHPRUH DGYDQFHG VWULQJ PRGHO WKH DGGHG WUHDG HOHPHQWV DUH DOORZHG WR DOVR GHIOHFW VLGHZD\V 7KH EHKDYLRXU RI WKLV PRUH FRPSOH[ PRGHO LV H[SHFWHG WR EH PRUH UHDOLVWLF 7KLV EHFRPHV HVSHFLDOO\ DSSDUHQW LQ WKH WUHDWPHQW RI WKH VLGH IRUFH UHVSRQVHWRDFRQVWDQWVOLSDQJOHZKHQWKHZKHHOUXQVRYHUDQXQGXODWHGURDG VXUIDFH7KHLQHUWLDRIWKHW\UHLVRILPSRUWDQFHZKHQUXQQLQJDWKLJKHUVSHHGV J\URVFRSLFFRXSOH DQGZKHQWKHIUHTXHQF\RIODWHUDODQG\DZH[FLWDWLRQFDQQR ORQJHU EH FRQVLGHUHG VPDOO 6HYHUDO DSSUR[LPDWLRQV RI WKH NLQHPDWLF DQG G\QDPLFPRGHOZLOOEHGLVFXVVHG,Q&KDSWHUWKHWKHRU\ZLOOEHDSSOLHGLQWKH DQDO\VLV RI WKH ZKHHO VKLPP\ SKHQRPHQRQ ,Q &KDSWHU WKH PRGHO ZLOO EH VLPSOLILHGWRWKHVLQJOHSRLQWFRQWDFWPRGHOWKDWUHVWULFWVWKHDSSOLFDWLRQWRORQJHU ZDYHOHQJWKVLWXDWLRQVEXWHQDEOHVWKHH[WHQVLRQRIWKHDSSOLFDWLRQWRORQJLWXGLQDO DQGFRPELQHGWUDQVLHQWVOLSVLWXDWLRQV&KDSWHU WUHDWVWKHPRUHFRPSOH[PRGHO WKDW LQFOXGHV DQ DSSUR[LPDWH UHSUHVHQWDWLRQ RI WKH HIIHFW RI WKH ILQLWH FRQWDFW OHQJWK WKH FRPSOLDQFH RI WKH FDUFDVV DQG WKH LQHUWLD RI WKH EHOW 7KLV PRUH YHUVDWLOHPRGHOLVDEOHWRFRQVLGHUERWKRXWRISODQHDQGLQSODQHW\UHG\QDPLF EHKDYLRXUWKDWFDQEHH[WHQGHGWRWKHQRQOLQHDUVOLSUDQJHDOVRDWUHODWLYHO\VKRUW ZDYHOHQJWKV0RUHRYHUUROOLQJRYHUURDGXQHYHQQHVVHVLVLQFOXGHG
12167($'<67$7(675,1*%$6('7<5(02'(/6
5HYLHZRI(DUOLHU5HVHDUFK
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
12167($'<67$7(675,1*%$6('7<5(02'(/6
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|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
12167($'<67$7(675,1*%$6('7<5(02'(/6
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
7KH6WUHWFKHG6WULQJ0RGHO
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
12167($'<67$7(675,1*%$6('7<5(02'(/6
wheel plane
[ string
\
)\
0]
path
[
tread elements
0]
]
\
)\ ]
)LJ 7\UHPRGHOZLWKVLQJOHVWUHWFKHGVWULQJDQGPRGHOH[WHQGHGZLWKPRUHSDUDOOHOVWULQJV SURYLGHGZLWKWUHDGHOHPHQWVZKLFKDUHIOH[LEOHLQWKHORQJLWXGLQDOGLUHFWLRQ
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
2\ DQGDERXWWKHD[LVQRUPDOWRWKHURDG \DZDQJOH
7KHVHPRWLRQVFRQVWLWXWH
WKHLQSXWWRWKHV\VWHP7KHH[FLWDWLRQIUHTXHQF\LVGHQRWHGE\
Q :LWK
9UHSUHVHQWLQJWKHDVVXPHGFRQVWDQWVSHHGRIWUDYHOZHREWDLQIRUWKHVSDWLDORU SDWKIUHTXHQF\
V
V
9DQGWKHZDYHOHQJWKRIWKHSDWKRIFRQWDFWSRLQWV
7KHGLVWDQFHWUDYHOOHGEHFRPHVV
9Q
9W
$OWHUQDWLYHLQSXWTXDQWLWLHVPD\EHFRQVLGHUHGZKLFKDUHQRWUHODWHGWRWKH SRVLWLRQ RI WKH ZKHHO ZLWK UHVSHFW WR WKH URDG EXW WR LWV UDWH RI FKDQJH FKDUDFWHUL]HGE\WKHVOLSDQJOH
G
2
G\GVDQGWKHSDWKFXUYDWXUHRUWXUQVOLS
GV7KHIRUFH)\DQGWKHPRPHQW0]ZKLFKDFWIURPWKHJURXQGRQWKH
W\UHLQWKH\GLUHFWLRQDQGDERXWWKH]D[LVUHVSHFWLYHO\IRUPWKHUHVSRQVHWRWKH LPSRVHGZKHHOSODQHPRWLRQ
12167($'<67$7(675,1*%$6('7<5(02'(/6
d[
[
[
&
Y
FY
F
d[
'
M d M
' ' [ [
E 6
1
6
MY
6
1 M[
1
E 6
1
d
T [ \
\
MY M2Y M[ + M[2 d[
)LJ/DWHUDOHTXLOLEULXPRIGHIOHFWHGWUHDGEDQGHOHPHQW
0RGHO'HYHORSPHQW
7RREWDLQDQH[SUHVVLRQIRUWKHODWHUDOGHIOHFWLRQYRIWKHVWULQJZHFRQVLGHU WKHODWHUDOHTXLOLEULXPRIDQHOHPHQW RI WKHWUHDGEDQGVKRZQLQ)LJ7KH HOHPHQWLVRIIXOOWUHDGZLGWKDQGFRQWDLQVHOHPHQWVRIWKHSDUDOOHOVWULQJVZLWK WKHUXEEHULQEHWZHHQ,QWKHODWHUDOGLUHFWLRQ \ WKHHTXLOLEULXPRIIRUFHVDFWLQJ RQWKHHOHPHQWRIOHQJWKG[DQGZLGWKEUHVXOWVLQWKHIROORZLQJHTXDWLRQ
T \G[ & FFY G[ & ' % ' % M' G[ & 6 MY % 6 M[ M[
MY
M[
%
MY
M[
[ '
G
ZKHUH FF GHQRWHV WKH ODWHUDO FDUFDVV VWLIIQHVV SHU XQLW OHQJWK 6 WKH FLUFXP IHUHQWLDO LQWKH[GLUHFWLRQ FRPSRQHQWRIWKHWRWDOWHQVLRQIRUFHDFWLQJRQWKHVHW RIVWULQJVDQG'WKHVKHDUIRUFHLQWKHFURVVVHFWLRQRIWKHWUHDGEDQGDFWLQJRQ WKHUXEEHUPDWUL[7KHVKHDUIRUFHLVDVVXPHGWREHDOLQHDUIXQFWLRQRIWKHVKHDU DQJOHDFFRUGLQJWRIRUPXOD
' ' 6 MY M[
:LWKWKHLQWURGXFWLRQRIWKHHIIHFWLYHWRWDOWHQVLRQ6 (T
6 6 ZHGHGXFHIURP
12167($'<67$7(675,1*%$6('7<5(02'(/6
_
_
[
[
2
V
_
\
D
9
ψ
_
\
&
Y2
D
[ Y
0] _
\
σ
Y1
α
[
)\
\
)LJ 7RSYLHZRIWKHVLQJOHVWULQJPRGHODQGLWVSRVLWLRQZLWKUHVSHFWWRWKHIL[HGIUDPH
6 M Y & F FY ' &T\ M[
,Q WKH SDUW RI WKH W\UH QRW PDNLQJ FRQWDFW ZLWK WKH URDG WKH FRQWDFW SUHVVXUH YDQLVKHVVRWKDWT\
6 M Y & F FY ' M[
DQG
[
LI * * !
D
7KHODWHUDOEHKDYLRXURIWKHPRGHOZLWKVHYHUDOSDUDOOHOVWULQJVDQGRIWKHPRGHO ZLWK D VLQJOH VWULQJ ZLOO EH LGHQWLFDO LI SDUDPHWHU 6 LV WKH VDPH ,Q )LJ D WRSYLHZRIWKHVLQJOHVWULQJPRGHOLVGHSLFWHG7KHOHQJWK GHVLJQDWHGDVWKH UHOD[DWLRQOHQJWKKDVEHHQLQGLFDWHGLQWKHILJXUH7KHUHOD[DWLRQOHQJWKHTXDOV
6 FF
'
:LWKWKLVTXDQWLW\LQWURGXFHG(T IRUWKHIUHHSRUWLRQRIWKHVWULQJEHFRPHV
MY
M[
&Y'
[
LI * * !
D
,IZHFRQVLGHUWKHFLUFXPIHUHQFHRIWKHW\UHWREHPXFKORQJHUWKDQWKHFRQWDFW OHQJWKZHPD\DVVXPHWKDWWKHGHIOHFWLRQYDWWKHWUDLOLQJHGJHKDVDQHJOLJLEOH HIIHFWRQWKHGHIOHFWLRQYDWWKHOHDGLQJHGJH7KHGHIOHFWLRQVRIWKHIUHHVWULQJ QHDUWKHFRQWDFWUHJLRQPD\WKHQEHFRQVLGHUHGWREHWKHUHVXOWRIWKHGHIOHFWLRQV RIWKHVWULQJDWWKHOHDGLQJHGJHDQGDWWKHWUDLOLQJHGJHUHVSHFWLYHO\DQGQRWRI
12167($'<67$7(675,1*%$6('7<5(02'(/6
9 9J2 )2
9J1 )1
)LJ 7ZR VXFFHVVLYH SRVLWLRQV RI WKH VWULQJ PRGHO 9DQLVKLQJ UHJLRQV RI VOLGLQJDWWKH
9 DQG IULFWLRQDO )UHTXLUHGWRPDLQWDLQDSRVVLEOHNLQNZKLFKDSSDUHQWO\FDQRQO\H[LVWDWWKH
OHDGLQJ DQG WUDLOLQJ HGJHV DQG WKH FRPSDWLELOLW\ RI VOLGLQJ VSHHG IRUFHV
J
WUDLOLQJHGJH
D FRPELQDWLRQ RI ERWK 7KH GHIOHFWLRQV LQ WKHVH UHVSHFWLYH UHJLRQV QRZ UHDG DSSUR[LPDWHO\
Y ' & H &[
Y' & H
[
LI
LI
[!D
[ &D
ZKLFKFRQVWLWXWHWKHVROXWLRQRI(T FRQVLGHULQJWKHVLPSOLI\LQJERXQGDU\ FRQGLWLRQWKDWIRUODUJH*[*WKHGHIOHFWLRQWHQGVWR]HUR$WWKHHGJHV[ Y
YDQGY
MY M[ MY M[
'&
Y
'
YUHVSHFWLYHO\ZHKDYHIRUWKHVORSH
Y
DZKHUH
IRU
[9D
IRU
[ 8&D
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
12167($'<67$7(675,1*%$6('7<5(02'(/6
MY
M[
'&
Y
LI
[ 8D [ ' D [ 9D
7KLVHTXDWLRQFRQVWLWXWHVDQLPSRUWDQWUHODWLRQVKLSIRUWKHGHYHORSPHQWRIWKH XOWLPDWHH[SUHVVLRQIRUWKHGHIOHFWLRQRIWKHVWULQJDVZHZLOOVHHDOLWWOHODWHU ,Q &KDSWHU ZH KDYH GHULYHG WKH JHQHUDO GLIIHUHQWLDO HTXDWLRQV IRU WKH ORQJLWXGLQDODQGODWHUDOVOLGLQJYHORFLW\RIDUROOLQJERG\ZKLFKLVVXEMHFWHGWR ODWHUDO VOLS DQG VSLQ 7KHVH HTXDWLRQV EHFRPH LI WKH VOLGLQJ VSHHG HTXDOV]HURDVLVRXUDVVXPSWLRQKHUH
MX
& MX ' \n M[ MV MY & MY ' & & [n M[ MV
,QWKHVHHTXDWLRQVVGHQRWHVWKHGLVWDQFHWUDYHOOHGE\WKHZKHHOFHQWUH RUEHWWHU WKH FRQWDFW FHQWUH DQG [ DQG \ WKH FRRUGLQDWHV WKH FRQVLGHUHG SDUWLFOH ZRXOG KDYH ZLWK UHVSHFW WR WKH PRYLQJ D[HV V\VWHP & [ \ LQ WKH KRUL]RQWDOO\
XQGHIRUPHGVWDWH 7KHVHSDUWLDOGLIIHUHQWLDOHTXDWLRQVZLOOEHVROYHGE\XVLQJ/DSODFHWUDQV IRUPDWLRQ7KHWUDQVIRUPVZLOOEHZULWWHQLQFDSLWDOV7KHWUDQVIRUPDWLRQZLOOQRW EHFRQGXFWHGZLWKUHVSHFWWRWLPHEXWZLWKUHVSHFWWRWKHGLVWDQFHWUDYHOOHGV 9WZKHUHWKHVSHHG9LVDVVXPHGWREHDFRQVWDQW7KH/DSODFHWUDQVIRUPRID YDULDEOHTXDQWLW\JHQHUDOO\LQGLFDWHGE\TLVGHILQHGWKURXJK
/ >T V @ ' 4 S '
m
4 &SV H T V GV
ZKHUHSLVWKH/DSODFHYDULDEOH :LWKLQLWLDOFRQGLWLRQVX [
Y [
DWV
ZHREWDLQIURP
WKHWUDQVIRUPHGHTXDWLRQV
8 & S8 ' \ G[ G9 & S9 ' &$ &[ G[ G
7KHVROXWLRQVRIWKHVHRUGLQDU\ILUVWRUGHUGLIIHUHQWLDOHTXDWLRQVUHDG
8 ' &X H S[ & S \
9 ' &Y H S[ % S $ % S
S
%[
,Q(TV WKHFRHIILFLHQWV&XDQG&YDUHFRQVWDQWVRILQWHJUDWLRQ7KH\DUH IXQFWLRQVRISDQGGHSHQGRQWKHW\UHFRQVWUXFWLRQWKDWLVWKHVWUXFWXUHRIWKH
12167($'<67$7(675,1*%$6('7<5(02'(/6
PRGHO)RURXUVWULQJPRGHOZLWKWUHDGHOHPHQWVZKLFKFDQEHGHIRUPHGLQWKH ORQJLWXGLQDOGLUHFWLRQRQO\ZHKDYHWKHERXQGDU\FRQGLWLRQVDWWKHOHDGLQJHGJH [
D
X'
8'
RU
DQG
MY M[
'&
Y
RU
9' 9 & G[
G
ZLWKWKHODWWHUHTXDWLRQV FRUUHVSRQGLQJWR(T 7KHFRQVWDQW&XQRZREYLRXVO\EHFRPHV
&X ' S \ H &SD
)RUWKHGHWHUPLQDWLRQRI&YZHKDYHWRGLIIHUHQWLDWH(T ZLWKUHVSHFWWR[
9 ' & SH S[ % Y S G[
G
:LWK ZHREWDLQ
&Y ' & S
9 % S
H &SD
ZKLFKZLWK \LHOGVIRUWKHGHIOHFWLRQDWWKHOHDGLQJHGJH
9 '
% S
$%D
DQGIRUWKHGHIOHFWLRQLQWKHFRQWDFW]RQH
9'
& $ % % D % S H S [&D % $ % [ % S % S S
7KHWHUPVFRQWDLQLQJH
S [D
SRLQWWRDUHWDUGDWLRQDOEHKDYLRXUZKLFKFRUUHVSRQGV
WRGHOD\WHUPVLQWKHRULJLQDOH[SUHVVLRQV1RWHWKDWDPHPRU\HIIHFWH[LVWVGXH WR WKH IDFW WKDW WKH QRQVOLGLQJ FRQWDFW SRLQWV UHWDLQ LQIRUPDWLRQ DERXW WKHLU
2
2
ORFDWLRQZLWKUHVSHFWWRWKHLQHUWLDOV\VWHPRID[HV [2\ DVORQJWKH\DUHLQ WKHFRQWDFW]RQH (TXDWLRQ WUDQVIRUPHGEDFN\LHOGVWKHILUVWRUGHUGLIIHUHQWLDOHTXDWLRQ IRUWKHGHIOHFWLRQRIWKHVWULQJDWWKHOHDGLQJHGJH
Y % Y ' % Dn GV
G
' & G \2 & GV
D
G
GV
7KLVHTXDWLRQZKLFKLVRIIXQGDPHQWDOLPSRUWDQFHIRUWKHWUDQVLHQWEHKDYLRXURI WKH W\UH PRGHO QRWH WKH SUHVHQFH RI WKH UHOD[DWLRQ OHQJWK
LQ WKH OHIWKDQG
12167($'<67$7(675,1*%$6('7<5(02'(/6
6 σY
6
2
6 σY
σ
D
D
Y1
Y2
σ
1
6
[
\
]
0
\
)
)LJ(TXLOLEULXPRIIRUFHVLQWKHFRQWDFWUHJLRQ
PHPEHU PLJKW KDYH EHHQ IRXQG PRUH HDVLO\ E\ FRQVLGHULQJ D VLPSOH WUDLOLQJ ZKHHOV\VWHPZLWKWUDLOHTXDOWR GLVWDQFH
DQGVZLYHOD[LVORFDWHGLQWKHZKHHOSODQHD
DLQIURQWRIWKHZKHHOD[LVFI)LJ7KHHTXDWLRQPD\DOVREH
IRXQGLPPHGLDWHO\IURP(T E\WDNLQJ [
DDQGFRQVLGHULQJFRQGLWLRQ
(TXDWLRQ PD\DOVREHWUDQVIRUPHGEDFNZKLFKSURGXFHVWKHGHOD\ WHUPVPHQWLRQHGEHIRUH+RZHYHUZHSUHIHUWRPDLQWDLQWKHH[SUHVVLRQLQWKH WUDQVIRUPHGVWDWHVLQFHZHZRXOGOLNHWRREWDLQWKHUHVXOWWKDWLV WKHIRUFHDQG PRPHQWUHVSRQVHLQWKHIRUPRIWUDQVIHUIXQFWLRQV
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
12167($'<67$7(675,1*%$6('7<5(02'(/6
$FFRUGLQJWR7HPSOH¶VPHWKRGZHREWDLQIRUWKHODWHUDOIRUFH )
'
\
F
D
m&
F
%
Y G[
%
6 Y
Y
D
DQGIRUWKHPRPHQWGXHWRODWHUDOGHIRUPDWLRQV
N'
0
F
]
m&
D
%
Y [ G[
F
6 D
%
Y
&
Y
D
ZLWKYDQGYGHQRWLQJWKHGHIOHFWLRQVDW[
DDQGDUHVSHFWLYHO\DQG6
FF
DFFRUGLQJWR(T )RUDILUVWDSSOLFDWLRQRIWKHWKHRU\ZHUHIHUWRH[HUFLVH
7KHPRPHQW0 ]GXH WR ORQJLWXGLQDOGHIRUPDWLRQVZKLFKLQRXUPRGHODUH SHUIRUPHGE\RQO\WKHWUHDGHOHPHQWVLVGHULYHGIURPWKHGHIOHFWLRQXGLVWULEXWHG RYHUWKHFRQWDFWDUHD:LWKFS[
GHQRWLQJWKHORQJLWXGLQDOVWLIIQHVVRIWKHHOHPHQWV SHUXQLWDUHDZHREWDLQ 0
( ]
'&
D
E
D
E
m& m&
F
S[
X \ G[ G\
%\DGGLQJXSERWKFRQWULEXWLRQVWKHWRWDOPRPHQWLVIRUPHG 0
]
'
N%
0
0
]
(
]
7KH/DSODFHWUDQVIRUPVRI) \ 0 ]DQG0 ]LVQRZUHDGLO\REWDLQHGXVLQJ(TV DQG ZLWK DQGWKHWUDQVIRUPHGYHUVLRQVRI(TV DQG ,QJHQHUDOWKHWUDQVIRUPHGUHVSRQVHVPD\EHZULWWHQDV /^)
`
\
' '
+
$
+
<
)
) \
% %
n
+
)
+
)
HWF7KHFRHIILFLHQWVRIWKHWUDQVIRUPHGLQSXWYDULDEOHVFRQVWLWXWHWKHWUDQVIHU IXQFWLRQV 7KH IRUPXODH IRU FRQYHQLHQFH ZULWWHQ LQ YHFWRULDO IRUP IRU WKH
nDQG
UHVSRQVHVWR
2
UHDG VLQFH
2
G\GVH[SUHVVLRQVIRUWKHUHVSRQVHWR\
KDYHEHHQRPLWWHG
+
)
n
S
&
+
0
&
%
D
S
S
n
H
&
F
'
%
S S
F
F
D
& %
S
%
S SD
D
S
& H
S
%
DQGIXUWKHUPRUH
%
S
'
SD
F
D
% & % D
S
S
% %
% %
D
D
&
H
& %
D S
& %
S
&
SD
S
% % & %
D
S
D S
12167($'<67$7(675,1*%$6('7<5(02'(/6
wheel plane
ψ
steering axis
σ
D
9
\
ψDσ contact line )LJ6WHDG\VWDWHUHVSRQVHWKDWRFFXUVZKHQVZLYHOOLQJDERXWWKH
+
( 0
n
S
'
( DS
&
DS
&
H
&
SD
&
S
D
SRLQW
LQZKLFKWKHSDUDPHWHU
('
D
E
F
S[
KDVEHHQLQWURGXFHG
2
7KH WUDQVIHU IXQFWLRQV RI WKH UHVSRQVHV WR \ DQG
DUH REWDLQHG E\
FRQVLGHULQJWKHUHODWLRQVEHWZHHQWKHWUDQVIRUPHGTXDQWLWLHV
'& ' &
S
$
S<
DQG LQVHUWLQJ WKHVH LQ :H ILQG LQ JHQHUDO IRU WKH WUDQVIHU IXQFWLRQ FRQYHUVLRQ + +
\
'& ' &
S+
+
n
S+
%\WUDQVIRUPLQJEDFNWKHH[SUHVVLRQVVXFKDV WKHGHIOHFWLRQWKHIRUFH DQGWKHPRPHQWFDQEHIRXQGDVDIXQFWLRQRIGLVWDQFHWUDYHOOHGVIRUDJLYHQ YDULDWLRQRI
DQG
nRURI\2DQG
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
D
12167($'<67$7(675,1*%$6('7<5(02'(/6
SRLQWLVFRPSRVHGRIWKH\DZDQJOH
2
DQGWKHODWHUDOGLVSODFHPHQW\
2
D
DQGIXUWKHUPRUHWKDWWKHUHVSRQVHWRG\GVLVHTXDOWR WKH UHVSRQVHWR ILQGHJIRUWKHWUDQVIHUIXQFWLRQRI)\WR +
)
%
D
'
'
&
)
% %
+
)
D S +
)
ZH
VD
:LWKWKHDLGRI WKHIXQGDPHQWDOUHODWLRQVKLSEHWZHHQWKHUHVSRQVHVWR DQG
n FDQEHDVVHVVHG:HKDYHLQJHQHUDO
n'
+
&
+
S
+
VV
% %
D +
ZKHUHIRUWKHUHVSRQVHVWR)\DQG0 ]ZHKDYHWKHVWHDG\VWDWHUHVSRQVHIXQFWLRQV GHQRWHGE\+ +
)
%
D
'
DQG +
0
%
VV
D
+
)
'
VV
+
0
'
VV
&
)
'&
&
0
7KHLPSRUWDQWFRQFOXVLRQLVWKDWZHPD\VXIILFHZLWKHVWDEOLVKLQJDVLQJOHSDLU RI WUDQVIHU IXQFWLRQV HJ + IRU )\ DQG 0 ] DQG GHULYH IURP WKDW WKH RWKHU IXQFWLRQVE\XVLQJWKHUHODWLRQV WRJHWKHUZLWK 6LQFHLQ SUDFWLFHWKHIUHTXHQF\UHVSRQVHIXQFWLRQVDUHRIWHQDVVHVVHGH[SHULPHQWDOO\E\ SHUIRUPLQJ\DZRVFLOODWLRQWHVWVZHJLYHEHORZWKHFRQYHUVLRQIRUPXODHWREH GHULYHGIURPWKHWUDQVIHUIXQFWLRQV+ /DWHURQZHZLOODGGUHVVWKHSUREOHPRI
ILUVWVXEWUDFWLQJ0 ]IURPWKHPHDVXUHGWRWDOPRPHQW0]WRUHWULHYH0 ]IRUZKLFK WKHFRQYHUVLRQLVYDOLG
'&
+
n'
+
+
\
%
D S
+
S
'&
&
+
&
+
VV
+
S+
6WULFWO\ VSHDNLQJ WKH DERYH FRQYHUVLRQ IRUPXODH RQO\ KROG H[DFWO\ IRU RXU PRGHO7KHDFWXDOW\UHPD\EHKDYHGLIIHUHQWO\HVSHFLDOO\UHJDUGLQJWKHHIIHFWRI
WKH PRPHQW 0 ] WKDW LQ UHDOLW\ PD\ VOLJKWO\ URWDWH WKH FRQWDFW SDWFK DERXW WKH YHUWLFDO D[LV DQG WKXV DIIHFWV WKH VOLS DQJOH VHHQ E\ WKH FRQWDFW SDWFK $V D FRQVHTXHQFHWKHREVHUYDWLRQGHSLFWHGLQ)LJPD\QRWEHHQWLUHO\WUXHIRUWKH UHDOW\UH ,QWKHIROORZLQJILUVWWKHVWHSUHVSRQVHIXQFWLRQVZLOOEHDVVHVVHGDQGDIWHU WKDWWKHIUHTXHQF\UHVSRQVHIXQFWLRQV
12167($'<67$7(675,1*%$6('7<5(02'(/6
6WHSDQG6WHDG\6WDWH5HVSRQVHRIWKH6WULQJ0RGHO
$QLPSRUWDQWFKDUDFWHULVWLFDVSHFWRIWUDQVLHQWW\UHEHKDYLRXULVWKHUHVSRQVHRI WKHODWHUDOIRUFHWRDVWHSZLVHYDULDWLRQRIWKHVOLSDQJOH 7KHLQLWLDOFRQGLWLRQ DWV
UHDGV Y [
IRUV!WKHVOLSDQJOHEHFRPHV
R
)URP(T
ZHREWDLQIRUVPDOOVOLSDQJOHVE\LQYHUVHWUDQVIRUPDWLRQIRUWKHODWHUDOGHIOHF WLRQRIWKHVWULQJLQWKHFRQWDFWUHJLRQ
' & %
Y
D
[
&
H
& & % V
[
D
IRU [
!
D
&
V
R
ZKLOH IRU WKH ROG SRLQWV ZKLFK DUH VWLOO RQ WKH VWUDLJKW FRQWDFW OLQH WKH VLPSOH H[SUHVVLRQKROGV
'
Y
IRU [
V
# & D
V
R
([SUHVVLRQ LV FRPSRVHG RI D SDUW D [
R
ZKLFK LV WKH ODWHUDO
GLVSODFHPHQW RI WKH ZKHHO SODQH GXULQJ WKH GLVWDQFH UROOHG D [ DQG DQ H[SRQHQWLDO SDUW 7KH SRLQW ZKLFK DW WKH LQVWDQW V FRQVLGHUHG LV ORFDWHG DW FRRUGLQDWH [ ZDV WKH SRLQW DW WKH OHDGLQJ HGJH ZKHQ WKH ZKHHO ZDV UROOHG D GLVWDQFH D[ DJR WKDW LV DW VD[ $W WKDW LQVWDQW ZH KDG D GHIOHFWLRQ DW WKH OHDGLQJ HGJH Y VD[ 7KH QHZ Y
Y V HTXDOV WKH ROG Y
Y VD[ SOXV WKH
2EYLRXVO\WKHH[SRQHQWLDO R
VXEVHTXHQWODWHUDOGLVSODFHPHQWRIWKHZKHHO D[
SDUWRI LVWKHYDWWKHGLVWDQFHUROOHGVD[7KLVFDQHDVLO\EHYHULILHGE\ VROYLQJ(T IRUY :LWK ILQDOO\WKHH[SUHVVLRQIRUWKHIRUFHFDQEHFDOFXODWHGIRUWKH WZRLQWHUYDOVZLWKDQGZLWKRXWWKHROGFRQWDFWSRLQWV ) )
\
\
' '
)
R
)
R
' '
F
F
% %
^
FF
&ò &
D V
D
V
H
`
R
& & V
D R
#D
LIV!D
#D
LIV
)RUWKHDOLJQLQJWRUTXHZHREWDLQE\XVLQJ
]
N'
0N
R
N'
0N
R
0
0
]
'
F
F
'&
V
F
ZKHUH WKH TXDQWLWLHV
F
&
D
%
%
D
D V
% & D
R
DH
& & V
D R
LIV
LIV!D
V GHVLJQDWH WKH XQLW VWHS UHVSRQVH IXQFWLRQV 7KHVH
IXQFWLRQV FRUUHVSRQG WR WKH LQWHJUDO RI WKH LQYHUVH /DSODFH WUDQVIRUPV RI WKH WUDQVIHUIXQFWLRQV JLYHQDERYH 7KHJUDSKRI)LJVKRZVWKHUHVXOWLQJYDULDWLRQRI)\DQG0 ]YVWUDYHOOHG GLVWDQFHV$VKDVEHHQLQGLFDWHGWKHFXUYHVDUHFRPSRVHGRIDSDUDEROD RIWKH
12167($'<67$7(675,1*%$6('7<5(02'(/6
0]
,
)\
steady-state level
1
0]VV )\VV
)\
parabola
exponential 0]
2 nd 3d
B
0
2D
[
B
σ
\
9
D
D
σ [
B
9
α D
Y2
\
Y1
Y
0
path [
D
\
[
B
[
\
B
.V
[
)LJ
) DQGWKHDOLJQLQJWRUTXH0 WRDVWHSLQSXWRIWKH D
7KHUHVSRQVHRIWKHODWHUDOIRUFH
\
]
VOLSDQJOH FDOFXODWHGIRUDUHOD[DWLRQOHQJWK
VHFRQG DQG WKLUG GHJUHH UHVSHFWLYHO\ DQG DQ H[SRQHQWLDO IXQFWLRQ 7KH VWHS UHVSRQVHVKDYHEHHQSUHVHQWHGDVDUDWLRWRWKHLUUHVSHFWLYHVWHDG\VWDWHYDOXHV 7KH VWHDG\VWDWH YDOXHV RI )\ DQG 0 ] RU 0] DUH GLUHFWO\ REWDLQHG IURP DQG E\FRQVLGHULQJWKHVKDSHRIWKHGHIOHFWHGVWULQJDWVWHDG\VWDWH VLGHVOLS )LJ LHDVWUDLJKWFRQWDFWOLQHDWDQDQJOH ZLWKWKHZKHHOSODQH DQGDGHIOHFWLRQDWWKHOHDGLQJHGJHY
WKURXJKZKLFKWKHFRQGLWLRQWRDYRLG
DNLQNLQWKHVWULQJDWWKDWSRLQWLVREH\HGRUIURP(TV E\OHWWLQJV DSSURDFKLQILQLW\:HKDYH
)\ ' &) 0] ' & &0
ZLWKWKHFRUQHULQJDQGDOLJQLQJVWLIIQHVVHV
D
σ
α
Y
Y
9
D
σ α
[
1
2
\
)LJ6WHDG\VWDWHGHIOHFWLRQRIDVLGHVOLSSLQJVWULQJW\UHPRGHO FRPSOHWHDGKHVLRQ
12167($'<67$7(675,1*%$6('7<5(02'(/6
&) ' FF % D &0 ' FF D ^ % D % D
`
DQGWKHSQHXPDWLFWUDLO
D^ & W' 0 ' &)
% D % D
% D
`
7KHYDULDWLRQRIWKHVHTXDQWLWLHVDQGRIWKHSQHXPDWLFWUDLOW
&0 &) DQGDOVR
RIWKHUHOD[DWLRQOHQJWKZLWKZKHHOORDG)]WKHODWWHUEHLQJDVVXPHGWRYDU\SUR
SRUWLRQDOO\ ZLWK D WXUQV RXW WR EH TXLWH XQUHDOLVWLF ZKHQ FRPSDUHG ZLWK H[SHULPHQWDOHYLGHQFH$YDULDWLRQPXFKFORVHUWRUHDOLW\ZRXOGEHREWDLQHGLI WUHDGHOHPHQWVZHUHDWWDFKHGWRWKHVWULQJ)RUPRUHGHWDLOVDOVRFRQFHUQLQJWKH QRQOLQHDUFKDUDFWHULVWLFVDQGWKHQRQVWHDG\DQDO\VLVRIWKDWHQKDQFHGEXWPXFK PRUHFRPSOLFDWHGPRGHOZHUHIHUWR6HFWLRQ 7KHXQLWVWHSUHVSRQVHV
2
WRWKHRWKHULQSXWYDULDEOH
n DQGWKHDVVRFLDWHG
DUH RI LQWHUHVW DV ZHOO 7KH\ PD\ EH GHULYHG E\ LQYHUVH
YDULDEOHV \ DQG
WUDQVIRUPDWLRQ DQG VXEVHTXHQW LQWHJUDWLRQ RI WKH WUDQVIHU IXQFWLRQV RUGLUHFWO\E\FRQVLGHULQJWKHDVVRFLDWHGVWULQJGHIOHFWLRQVVLPLODUWR RUIURPWKHXQLWVWHSUHVSRQVHVWRWKHVOLSDQJOHFRUUHVSRQGLQJWRWKH FRHIILFLHQWV RI
R
VKRZQ LQ E\ PDNLQJ XVH RI WKH IROORZLQJ
UHODWLRQVKLSVDQDORJRXVWR(TV
n'
I &+
\'
&
'
VV G
G G
V
&
V % % D
G G
V
n
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
12167($'<67$7(675,1*%$6('7<5(02'(/6
α path of
&
9
&
α
wheel plane
&
&
9
&
α
string wheel plane contact line steady-state path of contact points
_
\
&
wheel plane
&
contact line
string
&
&
&
path of wheel plane
ϕ
& path of
9
&
steady-state transient path of contact points
&
ϕ
&
_
\
&
&
contact line wheel plane string
ψ wheel plane
& wheel plane
D ψ
&
path of
9
&
σ
steady-state path of contact points
contact line
string
)LJ 7UDQVLHQWUHVSRQVHRIVWULQJGHIOHFWLRQWRVWHSFKDQJHLQPRWLRQYDULDEOHV
12167($'<67$7(675,1*%$6('7<5(02'(/6
1
)\ &)αα o )\ &)αψo
)\ &
)\
\o
)\
ϕ )ϕ o
&
0
0
2
4
1
0
&
]
0ααo
ϕo
&
]
0
ψo
0α
&
ψo
]
0
ψ
0
10
0
ϕ
0]* &
8
*
]
0
6
VD
0αϕo
D&
D0
]
0α \o
&
0
0
2
4
VD
6
8
10
) DOLJQLQJWRUTXH0¶DQGWUHDGZLGWKPRPHQW0 ]WR VLGHVOLS ODWHUDOGLVSODFHPHQWO \WXUQVOLSnDQG\DZDQJOH &RPSXWHGIRUVWULQJ PRGHOZLWKGHIOHFWLRQUHOD[DWLRQOHQJWK D
)LJ 6WHSUHVSRQVHRIVLGHIRUFH
\
]
WHUPVRIWKHPRGHOSDUDPHWHUV /DWHUDOVWLIIQHVVRIWKHVWDQGLQJW\UH &
)\
'
FF
%
D
&RUQHULQJRUODWHUDOVOLSVWLIIQHVV &
)
'
FF
%
D
$OLJQLQJVWLIIQHVV &
0
'
FF D ^
%
%
D
D
`
7RUVLRQDOVWLIIQHVVRI WKLQ VWDQGLQJW\UH &
0
'
FF D ^
%
D
%
D
`
12167($'<67$7(675,1*%$6('7<5(02'(/6
7XUQVOLSVWLIIQHVVIRUWKHIRUFH
n ' F
&
)
F
D^
%
D
%
D
`
n HTXDOV]HUR7KHUHVSRQVHVWRVLGH
1RWHWKDWWKHVWHDG\VWDWHUHVSRQVHRI0 ]WR
VOLSKDYHDOUHDG\EHHQSUHVHQWHGLQ)LJ,WQRZDSSHDUVWKDWWKHUHVSRQVHRI
n LV LGHQWLFDO WR WKH UHVSRQVH RI 0 WR
WKH VLGH IRUFH WR WXUQ VOLS
]
7KLV
UHFLSURFLW\SURSHUW\LVDOVRUHIOHFWHGE\WKHHTXDOLW\RIWKHVOLSVWLIIQHVVHVJLYHQ E\ ,WIXUWKHUPRUHDSSHDUVWKDWWKHUHVSRQVHVRI)\WR 0 ]WR
nDQGRI
DQG
WHQGWRDSSURDFKWKHVWHDG\VWDWHFRQGLWLRQDWWKHVDPHUDWH7KLVZLOOEH
VXSSRUWHGE\WKHODWHUILQGLQJWKDWWKHFRUUHVSRQGLQJIUHTXHQF\UHVSRQVHVDWORZ IUHTXHQFLHV ODUJHZDYHOHQJWKV DUHVLPLODU7KHIUHTXHQF\UHVSRQVHVDWVKRUW ZDYHOHQJWKVDUHPDLQO\JRYHUQHGE\WKHVWHSUHVSRQVHEHKDYLRXUVKRUWO\DIWHUWKH VWHS FKDQJH KDV FRPPHQFHG $V DSSHDUV IURP WKH JUDSKV DW GLVWDQFHV UROOHG VPDOOHUWKDQWKHFRQWDFWOHQJWKODUJHGLIIHUHQFHVLQWUDQVLHQWEHKDYLRXURFFXU$V
ODQG RI 0 ] WR H[SHFWHGWKHLQLWLDOUHVSRQVHVRI)\WR\
DUHLPPHGLDWHDQG
DVVRFLDWHGZLWKWKHUHVSHFWLYHVWLIIQHVVHV
7KHUHVSRQVHRIWKHPRPHQW0 ]GXHWRWUHDGZLGWKPRGHOOHGZLWKWKHEUXVK PRGHO WKDW GHIOHFWV RQO\ LQ WKH ORQJLWXGLQDO GLUHFWLRQ PD\ EH GHULYHG E\ FRQVLGHULQJWKH/DSODFHWUDQVIRUPRIWKHORQJLWXGLQDOGHIOHFWLRQXDFFRUGLQJWR (T ZLWK 7KURXJKLQYHUVHWUDQVIRUPDWLRQRUVLPSO\E\LQVSHFWLRQ RIWKHGHYHORSPHQWRIWKLVGHIOHFWLRQZKLOHWKHHOHPHQWPRYHVWKURXJKWKHFRQWDFW UDQJHWKHIROORZLQJH[SUHVVLRQVDUHREWDLQHG
' & &
X
n
[ \
D
&
IRU [
!
IRU [
# &
D
V
V
R
'&
X
n
\V
D
R
%\XVLQJ(T WKHIROORZLQJH[SUHVVLRQVIRUWKHVWHSUHVSRQVHRI0 ]WR
n
UHVXOW 0 0
n ' (n n '
]
('
0
R
('
0
R
]
(n
F
S[
F
S[
DV D
&
E
V
n
R
E
n
R
#D
D
LIV!D
E
LIV
7KHJUDSKLFDOUHSUHVHQWDWLRQRIWKHVHIRUPXODHLVJLYHQLQ)LJ7KHVOLSDQG VWLIIQHVVFRHIILFLHQWVHPSOR\HGUHDG 7XUQVOLSVWLIIQHVVIRUWKHPRPHQWFI E
n'
&
0
('
F
S[
D
E
64
7RUVLRQDOVWLIIQHVVRIVWDQGLQJW\UHGXHWRWUHDGZLGWKFI ZLWKS &
0
(
'
D
0n '
&
S[ D E
F
1RWHWKDWWKHVWHDG\VWDWHUHVSRQVHRI0 ]WR
nHTXDOV]HUR
12167($'<67$7(675,1*%$6('7<5(02'(/6
5
1
)\
α = 2o
[kN]
step in side slip
0]
step in side slip
α =-2o
0.5 o
0
0
[kNm]
o
0
0
-2 o -5
0
0.4
V [m] 1.2
0.8
2o -0.5
0
0.4
D
V [m] 1.2
0.8 E
)LJ 6WHSUHVSRQVHRIVLGHIRUFH)\DQGDOLJQLQJWRUTXH0]WRVOLSDQJOH DVPHDVXUHG RQDQDLUFUDIWW\UHZLWKYHUWLFDOORDG)] N1 )URP%HVVHOLQN WHVWGDWDLV SURYLGHGE\0LFKHOLQ$LUFUDIW7LUH&RUSRUDWLRQ
0
1200
0]
)\ 800
-20
[N]
[Nm]
400
0
step in side slip step in yaw angle
0
0.5
V [m]
1.0
D
α = 1o ψ = 1o
-40
-60 1.5
)\
α = 1o ψ = 1o
step in side slip step in yaw angle
0
0.5
1.0
E
V [m]
1.5
2.0
0]
20
1.5
[N]
[Nm] 1.0
10
step in turn slip ϕ = 1/115 [1/m]
step in turn ϕslip = 1 /115 [1/m] o 6 5 = 115 [m] o
0.5
o
6 5o= 115 [m]
0
0
)LJ
0.5
V [m]
1.0
F
1.5
0
0
0.5
G
1.0
V [m]
1.5
6WHSUHVSRQVHRIVLGHIRUFH)\DQGDOLJQLQJWRUTXH0]WRVOLSDQJOH \DZDQJOH DQGWXUQVOLS
n
DVPHDVXUHGRQDSDVVHQJHUFDUW\UHDWORDG)]
N17HVWV
ZHUHFRQGXFWHGRQWKHIODWSODQNPDFKLQHRI78'HOIW7XUQVOLSUHVXOWVKDYHEHHQ REWDLQHGE\LQWHJUDWLRQRIWKHUHVSRQVHWRWXUQVOLSLPSXOVH GLYLVLRQE\5
)URP+LJXFKL
VWHS\DZDQJOHDQG
12167($'<67$7(675,1*%$6('7<5(02'(/6
$VDUHVXOWRIDVWHSFKDQJHLQWXUQVOLSORQJLWXGLQDOVOLSDWERWKVLGHVRIWKH FRQWDFWSDWFKRFFXU7KHWUDQVLHQWUHVSRQVHH[WHQGVRQO\RYHUDGLVWDQFHUROOHG HTXDOWRWKHFRQWDFWOHQJWKDWWKHHQGRIZKLFKWKHVWHDG\VWDWHUHVSRQVHKDVEHHQ UHDFKHG$VLQGLFDWHGLQWKHJUDSKWKHDSSURDFKFXUYHKDVDSDUDEROLFVKDSH 7KHUHVSRQVHWRDVWHSFKDQJHLQ\DZDQJOHLVLPPHGLDWH OLNHWKDWRI0 ] DIWHU
ZKLFKDGHFOLQHRFFXUVZKLFKIRU0 ]LVOLQHDU GHULYDWLYHRIUHVSRQVHWR
n
,Q)LJVH[SHULPHQWDOO\REWDLQHGVWHSUHVSRQVHVKDYHEHHQSUHVHQWHG 7KHGLDJUDPVVKRZYHU\ZHOOWKHH[SRQHQWLDOQDWXUHRIWKHIRUFHUHVSRQVHWRVLGH VOLS(VSHFLDOO\WKHDLUFUDIWW\UHH[KLELWVWKHµGHOD\HG¶UHVSRQVHRIWKHDOLJQLQJ WRUTXHWRVLGHVOLSDVSUHGLFWHGE\WKHWKHRU\ )LJ )LJXUHVKRZVD VLPLODU GHOD\ LQ WKH UHVSRQVHV RI )\ WR \DZ DQG WR WXUQ VOLS DOVR IRXQG LQ WKH WKHRUHWLFDOUHVXOWV7KHSHFXOLDUUHVSRQVHRIWKHPRPHQWWR\DZDQGWXUQVOLSLV
FOHDUO\IRUPHGE\WKHVXPRIWKHUHVSRQVHVRI0 ]DQGRI0 ]DOWKRXJK ZLWKD UDWLRWKDWGLIIHUVIURPWKHDVVXPSWLRQDGRSWHGLQ)LJ *UDSKVRIVWHSUHVSRQVHIXQFWLRQVPD\VHUYHWRFRPSDUHWKHSHUIRUPDQFHRI GLIIHUHQW PRGHOV DQG DSSUR[LPDWLRQV ZLWK HDFK RWKHU 7KLV ZLOO EH GRQH LQ 6HFWLRQ)LUVWZHZLOOGLVFXVVWKHIUHTXHQF\UHVSRQVHIXQFWLRQV
)UHTXHQF\5HVSRQVH)XQFWLRQVRIWKH6WULQJ0RGHO
7KHIUHTXHQF\UHVSRQVHIXQFWLRQVIRUWKHIRUFHDQGWKHPRPHQWFRQVWLWXWHWKH UHVSRQVH WR VLQXVRLGDO PRWLRQV RI WKH ZKHHO DQG FDQ EH HDVLO\ REWDLQHG E\ UHSODFLQJLQWKHWUDQVIHUIXQFWLRQV WKH/DSODFHYDULDEOHSE\L 7KHSDWKIUHTXHQF\
V UDGP LVHTXDOWR
V
ZKHUH GHQRWHVWKHZDYHOHQJWK
λ 64 Dλ = 0 φ)ψ = 0 λ = 24D Dλ .0.042 φ)ψ. -56o λ = 12D Dλ .0.084 φ)ψ. -90o )LJ 7KHVWULQJPRGHODWVWHDG\VWDWHVLGHVOLSDQGVXEMHFWHGWR\DZRVFLOODWLRQVZLWKWZR GLIIHUHQWZDYHOHQJWKV
12167($'<67$7(675,1*%$6('7<5(02'(/6
RIWKHVLQXVRLGDOPRWLRQRIWKHZKHHO)LJXUHLOOXVWUDWHVWKHPDQQHULQZKLFK WKHVWULQJGHIOHFWLRQYDULHVZLWKWUDYHOOHGGLVWDQFHZKHQWKHPRGHOLVVXEMHFWHG WRD\DZRVFLOODWLRQZLWKGLIIHUHQWZDYHOHQJWKV 7KHIUHTXHQF\UHVSRQVHIXQFWLRQVVXFKDV+) L RI RXWSXW HJ )\ DQG LQSXW HJ
V DUHWKHFRPSOH[UDWLRV
,Q )LJV WKH YDULRXV IUHTXHQF\
UHVSRQVHIXQFWLRQVKDYHEHHQSORWWHGDVDIXQFWLRQRIWKHQRQGLPHQVLRQDOSDWK ò
IUHTXHQF\D
V D 7KHIXQFWLRQVDUHUHSUHVHQWHGE\WKHLUDEVROXWHYDOXH
*+*DQGWKHSKDVHDQJOHkRIWKHRXWSXWZLWKUHVSHFWWRWKHLQSXW LIQHJDWLYHWKHQ RXWSXWODJVEHKLQGLQSXW HJ
+ ' _+ _ H k) ' L
)
)
)_
_
H k) L
\
_
_
,IWKHYDULDEOHVDUHFRQVLGHUHGDVUHDOTXDQWLWLHVRQHJHWV *)\*FRV
IRULWVUHVSRQVH )\
V
_ _FRV
V
V DQG
k ,QWKHILJXUHWKHDEVROXWHYDOXHVKDYH
V
EHHQPDGHQRQGLPHQVLRQDOE\VKRZLQJWKHUDWLRWRWKHLUYDOXHVDW
V
WKH
VWHDG\VWDWHFRQGLWLRQ7KUHHGLIIHUHQWZD\VRISUHVHQWDWLRQKDYHEHHQXVHGHDFK ZLWKLWVRZQDGYDQWDJH 7KHIRUFHUHVSRQVHWRVOLSDQJOHYHU\PXFKUHVHPEOHVDILUVWRUGHUV\VWHP EHKDYLRXUDVFDQEHVHHQLQWKHXSSHUJUDSKZLWKDORJORJVFDOH7KHFXWRII IUHTXHQF\ WKDW LV IRXQG E\ FRQVLGHULQJ WKH VWHDG\VWDWH UHVSRQVH DQG WKH DV\PSWRWLFEHKDYLRXUDWODUJHSDWKIUHTXHQFLHVDSSHDUVWREHHTXDOWR
V)
FR
'
%D
+RZHYHUWKHSKDVHODJDWIUHTXHQFLHVWHQGLQJWR]HURLVQRWHTXDOWR
V
D
DVRQHZRXOGH[SHFWIRUDILUVWRUGHUV\VWHPEXWVRPHZKDWVPDOOHU$QDO\VLV UHYHDOVWKDWWKHSKDVHODJWHQGVWR
&k 6 )
)
V
IRU
V
6
ZLWKWKHUHOD[DWLRQOHQJWKIRUWKHVLGHIRUFHZLWKUHVSHFWWRWKHVOLSDQJOH )
' %D&W
ZKLFKZLWK EHFRPHVHTXDOWRDLI
D7KHSKDVHODJGRHVDSSURDFK
E IRU IUHTXHQFLHV JRLQJ WR LQILQLW\ 7KH ILUVWRUGHU DSSUR[LPDWLRQ ZLWK WKH VDPH FXWRII IUHTXHQF\ KDV EHHQ DGGHG LQ WKH JUDSK IRU FRPSDULVRQ 7KH FRUUHVSRQGLQJIUHTXHQF\UHVSRQVHIXQFWLRQUHDGV
+
)
'
%L
&
)
V
% D
7KH IUHTXHQF\ UHVSRQVH RI WKH IRUFH WR \DZ VKRZV D ZDY\ FXUYH IRU WKH DPSOLWXGHDWKLJKHUIUHTXHQFLHV DWZDYHOHQJWKVVPDOOHUWKDQFDWZRWLPHVWKH FRQWDFWOHQJWK 7KHGHFOLQHRIWKHSHDNVRFFXUVDFFRUGLQJWRWKHVDPHDV\PSWRWH
12167($'<67$7(675,1*%$6('7<5(02'(/6
)\
)\
α )α
&
1
)\ _
&
)\
0.4
α
α
)
) 1
\ 0.1
)\
ϕ )ϕ
&
0.04
ψ
)
)\
ψ )α
0.01
&
D 2π σ +D
0.004
0.001 0.01
0.02
ϕ
D π o W σ +D
)
0.04
0.08 0.1
0.2
Dλ
0.4
0.8 1
2
0.8 1
2
1.5 )\
α )α
&
)\ &
ϕ
)ϕ
)\ &
)\ )\
&
_
\
α
)
ψ
0.5
)α
ϕ
)
α
) 1
ϕ
α
ψ
) 1
)
α
)
)
ψ
)
0 90
)\
0
)\
α
α α
)
-90
φ
)\
)\
1
ψ
)
[o] -180
) 1 )
ϕ
)
ψ
)
ϕ
)
-270 -360 0.01 0
)LJ
0.050.020.1
0.04
0.08 0.1
Dλ
0.2
0.4
6WULQJPRGHOIUHTXHQF\UHVSRQVHIXQFWLRQVIRUWKHVLGHIRUFHZLWKUHVSHFWWRYDULRXV PRWLRQLQSXW YDULDEOHV7KHQRQGLPHQVLRQDOSDWKIUHTXHQF\LVKDOIWKHFRQWDFW OHQJWKGLYLGHGE\WKHZDYHOHQJWKRIWKHPRWLRQ
D
)RUWKHUHVSRQVHWR
V
DILUVWRUGHUDSSUR[LPDWLRQZLWKWKHVDPHFXWRIIIUHTXHQF\ EHHQDGGHG )
V
D KDV
7KUHHZD\VRISUHVHQWDWLRQKDYHEHHQXVHG ORJORJOLQORJDQG
OLQOLQ7KHPRGHOSDUDPHWHU
DZKLFKOHDGVWRW
D
12167($'<67$7(675,1*%$6('7<5(02'(/6
]
0
α
0α
&
1
]_
ϕ
0
0.4
D0
0α \
0
&
* ψ
]
ϕ
0α
D&
0 \
ψ
0.1
0
0
α
0.04
]
0
0
0.01
ψ
0α
&
D
0.004
*
π
]
D0
ψ
0ϕ
&
]
]_
0
D0
α
0α
&
0.001 0.01
0α \
0.02
0.04
0.08 0.1
oW σ +D
Dλ
0.2
0.4
0.8 1
2
0.8 1
2
1.5
&
1
ϕ
0
]
0
ϕ
0α
D&
]
α
0
0
ψ
0α
&
0
*
]
ψ
0ϕ
&
0 90
* ψ
0 \
ψ
0
0
ϕ
0
0
0
[o] -180
α
0 \
ψ
0
0
-270 0.050.020.1 0.1
α
ψ
0
-360 0.01 0
*ψ ϕ
0
0
-90
φ
ϕ
0
ψ
0 \
D0
α
0
ψ*
0
0.5
0.04
0.08 0.1
0.2
Dλ
0.4
)LJ 6WULQJPRGHOIUHTXHQF\UHVSRQVHIXQFWLRQVIRUWKHDOLJQLQJWRUTXHZLWKUHVSHFWWR YDULRXV PRWLRQ LQSXW YDULDEOHV 6DPH FRQGLWLRQV DV LQ )LJ 7KH UHVSRQVH
0
IXQFWLRQVRIWKHPRPHQWGXHWRWUHDGZLGWK
]KDYHEHHQDGGHG
12167($'<67$7(675,1*%$6('7<5(02'(/6
DVIRXQGIRUWKHVOLSDQJOHUHVSRQVH&RQVHTXHQWO\WKHVDPHFXWRIIIUHTXHQF\ DSSOLHV
FR
V)
'
%D
:KHQDQDO\VLQJWKHEHKDYLRXUDWVPDOOIUHTXHQFLHVLWDSSHDUVWKDWKHUHWKHSKDVH ODJGRHVWHQGWR
&k 6 )
)
V
IRU
V
6
ZLWKWKHUHOD[DWLRQOHQJWKIRUWKHVLGHIRUFHZLWKUHVSHFWWRWKH\DZDQJOH )
' %D
)XUWKHUDQDO\VLVUHYHDOVWKDWZKHQGHYHORSLQJWKHIUHTXHQF\UHVSRQVHIXQFWLRQ +)ψLQDVHULHVXSWRWKHVHFRQGGHJUHHLQL
+ L )
V
'& ' &
)
)
& &
^
)
)
L E
V
)
V
&E L & L )
V
% %
`
V
)
V
DQGVXEVHTXHQWO\HPSOR\LQJWKHIXQGDPHQWDOUHODWLRQVKLS EHWZHHQ
n UHVSRQVHVWKHIUHTXHQF\UHVSRQVHIXQFWLRQ+
EHFRPHV
+ L )
V
'&
)
&
)
L
V
% ' &
)
)
DQG
XSWRWKHILUVWGHJUHHLQL
&E L )
V
%
V
ZKLFKVKRZVWKDW
)
'E
)
ZKLFK LV DQ LPSRUWDQW UHVXOW LQ YLHZ RI DVVHVVLQJ
)
IURP \DZ RVFLOODWLRQ
PHDVXUHPHQWGDWDDQGFKHFNLQJWKHFRUUHVSRQGHQFHZLWK 7KHDOLJQLQJWRUTXH 0 ])LJ VKRZVDUHVSRQVHWRWKHVOLSDQJOHZKLFK LVFORVHUWRDVHFRQGRUGHUV\VWHPZLWKDSKDVHODJWHQGLQJWRDYDULDWLRQDURXQG
E
DQGD DV\PSWRWLFVORSHRIWKHDPSOLWXGHZLWKDFXWRIIIUHTXHQF\HTXDO WR
V0
FR
'
W % D
ZKHUHWGHQRWHVWKHSQHXPDWLFWUDLOFI $JDLQWKHUHVSRQVHRI)\WR WXUQVRXWWREHWKHVDPHDVWKHUHVSRQVHRI0 ]WR
VKRZVWKHDPSOLWXGHRI0 ]DVDUHVSRQVHWR\DZRVFLOODWLRQV GLSDW ZLWKSDUDPHWHU
D DZDYHOHQJWK
-
n
$VWKHJUDSKRI)LJ H[KLELWVDFOHDU
D 7KLV FRQGLWLRQFRUUHV
SRQGVWRWKHVLWXDWLRQGHSLFWHGLQ)LJ WKLUGFDVH DQGLVUHIHUUHGWRDVWKH PHDQGHULQJSKHQRPHQRQRUDVNLQHPDWLFVKLPP\ZKLFKRFFXUVLQSUDFWLFHZKHQ WKHZKHHOLVDOORZHGWRVZLYHOIUHHO\DERXWWKHYHUWLFDOD[LVWKURXJKWKHZKHHO FHQWUHDQGWKHV\VWHPLVVORZO\PRYHGIRUZDUGV7KHQHDUO\V\PPHWULFVWULQJ
12167($'<67$7(675,1*%$6('7<5(02'(/6
Im
φ
ψ
ψ
Re
0
]
0
ωV
φ
= φ
ψ
0
*
]
0
\
)
ψ
)
]
0
)LJ&RPSOH[UHSUHVHQWDWLRQRIVLGHIRUFHDQGPRPHQWUHVSRQVHWR\DZRVFLOODWLRQV
GHIRUPDWLRQH[SODLQVZK\WKHDPSOLWXGHRIWKHDOLJQLQJWRUTXHDOPRVWYDQLVKHV DW WKLV ZDYHOHQJWK $W KLJKHU IUHTXHQFLHV WKH DPSOLWXGH UHPDLQV ILQLWH DQG
64 WKH VDPHYDOXHDVLWKDGDW WKDWLV 0 & 7KH SKDVH DQJOH DSSURDFKHV E ,W LV LQWHUHVWLQJ WKDW DQDO\VLV DW
DSSURDFKHVDW &0¶
V
V
]
0
IUHTXHQFLHVDSSURDFKLQJ]HURVKRZVWKDWWKHSKDVHODJERWKIRUWKHUHVSRQVHRI 0 ]WR
DQGWR
DQGWKXVDOVRIRU)\WR
n DSSURDFKWKHYDOXH
DOVRDSSHDUHGWREHWUXHIRUWKHUHVSRQVHRI)\WR
'
0
0
'
n'
)
)
' %D
V
D WKDW
6RZHKDYH
([SUHVVLRQV HTXLYDOHQW WR DSSHDU WR KROG IRU WKH HQKDQFHGPRGHOZLWKWUHDGHOHPHQWVDWWDFKHGWRWKHVWULQJFI6HFWLRQ
7KHWRUTXHGXHWRWUHDGZLGWK0 ]VKRZV D UHVSRQVHWR\DZDQJOH
WKDW
LQFUHDVHVLQDPSOLWXGHZLWKSDWKIUHTXHQF\DQGVWDUWVRXWZLWKDSKDVHOHDGRI
E
$WORZIUHTXHQFLHVWKHPRPHQWDFWVOLNHWKHWRUTXHRID
ZLWKUHVSHFWWR
YLVFRXVURWDU\GDPSHUZLWKGDPSLQJUDWHLQYHUVHO\SURSRUWLRQDOZLWKWKHVSHHG RIWUDYHO9:HILQGZLWK(T
(0
0( ' & ]
IRU
9
$WKLJKIUHTXHQFLHV
V
V
6
64WKDWLVDWYDQLVKLQJZDYHOHQJWK 6ZKHUHWKHW\UH
LV VWDQGLQJ VWLOO WKH W\UH DFWV OLNH D WRUVLRQDO VSULQJ DQG WKH PRPHQW 0 ] DSSURDFKHV&0
V0
(nFR
'
&0n DFI 7KHFXWRIIIUHTXHQF\DSSHDUVWREHFRPH
D
7KHWRWDOPRPHQWDERXWWKHYHUWLFDOD[LVLVREWDLQHGE\DGGLQJWKHFRPSRQHQWV GXHWRODWHUDODQGORQJLWXGLQDOGHIRUPDWLRQV 0
]
'
0
]
%
0
( ]
)RUWKHVWDQGLQJW\UHRQHILQGVIURPD\DZWHVWWKHWRWDOWRUVLRQDOVWLIIQHVV&0 ZKLFK UHODWHV WR WKH DOLJQLQJ VWLIIQHVV DQG WKH VWLIIQHVV GXH WR WUHDG ZLGWK DV IROORZV
12167($'<67$7(675,1*%$6('7<5(02'(/6
4
0
Im
)
\
&
)α
ψ
0.12
-0.5
0
-φ)ψ
1
0.01
)\
0.02
0.08 0.06
0
Re
Dλ
0.03
0.04
1
)
\
&
)α
ψ
)LJ 1\TXLVWSORWRIWKHQRQGLPHQVLRQDOIUHTXHQF\UHVSRQVHIXQFWLRQRIWKHVLGHIRUFH)\ ZLWKUHVSHFWWRWKHZKHHO\DZDQJOH
&
0
'
D
n%&
&
0
D
3DUDPHWHU
0
2EYLRXVO\WKLVUHODWLRQVKLSRIIHUVDSRVVLELOLW\WRDVVHVVWKHWXUQVOLSVWLIIQHVV IRUWKHPRPHQW&0n
'XHWRWKHDFWLRQRI0 ]WKHSKDVHODJRI0]ZLOOEHUHGXFHG7KLVDSSHDUV WREHLPSRUWDQWIRUWKHVWDELOLVDWLRQRIZKHHOVKLPP\RVFLOODWLRQV FI&KDSWHU 7KHWRWDOPRPHQWDQGLWVFRPSRQHQWVFDQEHVWEHSUHVHQWHGLQD1\TXLVWSORW ZKHUHWKHPRPHQWFRPSRQHQWVDQGWKHUHVXOWLQJWRWDOPRPHQWDSSHDUDVYHFWRUV LQDSRODUGLDJUDP)LJXUHGHSLFWVWKHYHFWRUGLDJUDPDWDORZYDOXHRIWKH SDWKIUHTXHQF\
V
%\FRQVLGHULQJWKHYDULRXVSKDVHDQJOHVDWORZIUHTXHQFLHVZHPD\EHDEOH WR H[WUDFW WKH PRPHQW UHVSRQVH GXH WR WUHDG ZLGWK IURP WKH WRWDO PHDVXUHG UHVSRQVH DQG ILQG WKH UHVSRQVH RI WKH PRPHQW IRU D µWKLQ¶ W\UH 6LQFH DW ORZ
IUHTXHQFLHV WKH PRPHQW YHFWRU 0 ] WHQGV WR SRLQW XSZDUGV ZH ILQG ZKLOH FRQVLGHULQJ DQG
(
_0] _
'
_0]_
:LWK NQRZQ
)
)
DQG
V
k
% k&
0
0
(
'
_
V
_
WR EH GHWHUPLQHG IURP WKH PHDVXUHPHQW DW ORZ
IUHTXHQF\WKHPRPHQWWXUQVOLSVWLIIQHVV&0n
(
PD\EHDVVHVVHGLQWKLVZD\
,Q WKH GLDJUDPV RI )LJV WKH QRQGLPHQVLRQDO IUHTXHQF\ UHVSRQVH IXQFWLRQV+) L DQG+0
L
V
&) DQG+0 L
V
&0 ZLWKLWVFRPSRQHQWV+0¶ L
V
&0
&0 KDYHEHHQSUHVHQWHGDVDIXQFWLRQRIWKHQRQGLPHQVLRQDO V
SDWKIUHTXHQF\D 7KHSDUDPHWHUYDOXHVDUH
DDQG&0n
D&0
7KHGLDJUDPRI)LJFOHDUO\VKRZVWKHLQFUHDVHLQSKDVHODJDQGGHFUHDVH RIWKHDPSOLWXGHRI)\ZLWKGHFUHDVLQJZDYHOHQJWK 7KHZDY\EHKDYLRXUDQG
12167($'<67$7(675,1*%$6('7<5(02'(/6
1.5
0
]
0 0
]
*
with tread width (wide tyre)
]
0
]
1
Im
zero tread width (thin tyre)
]
0
0α
&
ψ
0
]
0.5
0.12 0
]
0.10
4
0
φ
0
-
0.08
]
φ
-
0
*
]
0
4
1
1
0.01
φ*
0.06
-0.5
0.02 0.03
0.04 0
1
Dλ 0
]
Re
0α
&
2
ψ
)LJ 1\TXLVW SORW RI WKH QRQGLPHQVLRQDO IUHTXHQF\ UHVSRQVH IXQFWLRQ RI WKH DOLJQLQJ PRPHQW0]ZLWKUHVSHFWWRWKHZKHHO\DZDQJOH
7KHFRQWULEXWLQJFRPSRQHQWV0 ]
GXHWRODWHUDO GHIRUPDWLRQVDQG0 ]GXHWRWUHDGZLGWKDQGDVVRFLDWHGORQJLWXGLQDO GHIRUPDWLRQV3DUDPHWHUV
DDQG&0n
D&0
E
WKHDVVRFLDWHGMXPSVIURP WRRIWKHSKDVHDQJOHGLVSOD\HGLQ)LJ EHFRPHFOHDUZKHQYLHZLQJWKHORRSVWKDWDSSHDUWRRFFXUZKHQWKHZDYHOHQJWK EHFRPHVVPDOOHUWKDQDERXWWZRWLPHVWKHFRQWDFWOHQJWK 7KH DOLJQLQJ WRUTXH YHFWRU )LJ IRU WKH µWKLQ¶ W\UH WXUQV RYHU
E
EHIRUHIURPDZDYHOHQJWKRIDERXWWKHFRQWDFWOHQJWKWKHORRSVEHJLQWRVKRZXS $W DERXW
D WKH FXUYH JHWV FORVHVW WR WKH RULJLQ 7KLV FRUUHVSRQGV WR WKH
IUHTXHQF\ZKHUHWKHGLSRFFXUVLQ)LJDQGLVLOOXVWUDWHGDVWKHODVWFDVHRI )LJ )RU YDOXHV RI
(
VXIILFLHQWO\ ODUJH WKH WRWDO PRPHQW FXUYH GRHV QRW FLUFOH
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
12167($'<67$7(675,1*%$6('7<5(02'(/6
W\UHLQHUWLDDQGHVSHFLDOO\WKHJ\URVFRSLFFRXSOHGXHWRW\UHODWHUDOGHIRUPDWLRQ UDWHVLVQRORQJHUQHJOLJLEOH,Q6HFWLRQDQG&KDSWHU WKHVHPDWWHUVZLOOEH DGGUHVVHG
([HUFLVH6WULQJPRGHODWVWHDG\WXUQVOLS
&RQVLGHUWKH VLQJOH VWUHWFKHGVWULQJW\UHPRGHOUXQQLQJDORQJDFLUFXODUSDWKZLWK
5DQWLFORFNZLVH n
UDGLXV
5 DQGZLWKRXWVLGHVOLS
DVGHSLFWHGLQ)LJ
5
wheel axis path
ϕ
&
wheel plane
string
D
D
9
[ σ
)\
\
)LJ 7KHVWULQJW\UHPRGHOLQVWHDG\WXUQLQJVWDWH
'HULYH WKH H[SUHVVLRQ IRU WKH ODWHUDO IRUFH
) DFWLQJ XSRQ WKH PRGHO XQGHU WKHVH Y [ \
VWHDG\VWDWH FLUFXPVWDQFHV )LUVW ILQG WKH H[SUHVVLRQ IRU WKH ODWHUDO GHIOHFWLRQ
XVLQJ(T ZKLFKOHDGVWRDTXDGUDWLFDSSUR[LPDWLRQRIWKHFRQWDFWOLQH7KHQXVH (T IRUWKHFDOFXODWLRQRIWKHVLGHIRUFH 1RZFRQVLGHULQDGGLWLRQVRPHVLGHVOLSDQGGHWHUPLQHWKHYDOXHRI UHTXLUHGWR QHXWUDOL]HWKHVLGHIRUFHJHQHUDWHGE\WKHSDWKFXUYDWXUH
50DNHDVNHWFKRIWKH
UHVXOWLQJVWULQJGHIRUPDWLRQDQGZKHHOSODQHRULHQWDWLRQZLWKUHVSHFWWRWKHFLUFXODU SDWKIRUWKHIROORZLQJYDOXHVRIUDGLXVDQGUHOD[DWLRQOHQJWK
5
D
DQG
D
$SSUR[LPDWLRQVDQG2WKHU0RGHOV
,QWKHSUHVHQWVHFWLRQDSSUR[LPDWLRQVWRWKHH[DFWWKHRU\ZLOOEHWUHDWHGWRPDNH WKHWKHRU\PRUHDFFHVVLEOHWRDSSOLFDWLRQV6XEVHFWLRQGLVFXVVHVDQXPEHU RIRWKHUPRGHOVNQRZQLQWKHOLWHUDWXUH$IWHUWKDWLQ6XEVHFWLRQDPRUH FRPSOH[PRGHOVKRZLQJWUHDGHOHPHQWVIOH[LEOHDOVRLQWKHODWHUDOGLUHFWLRQZLOO EHWUHDWHGWRSURYLGHDUHIHUHQFHPRGHOWKDWLVFORVHULQSHUIRUPDQFHWRWKHUHDO W\UH
12167($'<67$7(675,1*%$6('7<5(02'(/6
Y2
Y1
path Von Schlippe exact (Segel) Smiley straight tangent
4
)LJ6HYHUDODSSUR[LPDWLYHVKDSHVRIWKHFRQWDFWOLQHRIWKHVWULQJPRGHO
$SSUR[LPDWH0RGHOV
,QWKHOLWHUDWXUHVHYHUDOVLPSOHUPRGHOVKDYHEHHQSURSRVHG1RWDOORIWKHVHDUH EDVHG RQ WKH VWULQJ PRGHO EXW PDQ\ GR )LJXUH GHSLFWV D QXPEHU RI DSSUR[LPDWHGFRQWDFWOLQHVDVSURSRVHGE\VHYHUDODXWKRUV7KHPRVWZHOONQRZQ DQGDFFXUDWHDSSUR[LPDWLRQLVWKDWRI9RQ6FKOLSSH ZKRDSSUR[LPDWHG WKHFRQWDFWOLQHE\IRUPLQJDVWUDLJKWFRQQHFWLRQEHWZHHQWKHOHDGLQJDQGWUDLOLQJ HGJHVRIWKHH[DFWFRQWDFWOLQH.OXLWHUV JDYHDIXUWKHUDSSUR[LPDWLRQE\ LQWURGXFLQJD3DGp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¶V DQG 6PLOH\¶V VHFRQGRUGHU DSSUR[LPDWLRQDVZHOODVWKHVWUDLJKWWDQJHQWDQGSRLQWFRQWDFWDSSUR[LPDWLRQV 7KHSHUIRUPDQFHRIWKHVHPRGHOVZLOOEHVKRZQLQFRPSDULVRQZLWKWKHH[DFW µEDUH¶VWULQJPRGHODQGZLWKWKHHQKDQFHGPRGHOZLWKODWHUDOO\FRPSOLDQWWUHDG HOHPHQWV)LJXUHVJLYHVWKHUHVXOWVLQWHUPVRIVWHSUHVSRQVHIUHTXHQF\ UHVSRQVH%RGHSORWVDQG1\TXLVWGLDJUDPV )RU VRPH RI WKH RWKHU DSSUR[LPDWH PRGHOV 5RJHUV .OXLWHUV .HOG\VK DQG0RUHODQG RQO\WKHJRYHUQLQJHTXDWLRQVZLOOEHSURYLGHG ZLWKVRPHFRPPHQWVRQWKHLUEHKDYLRXU)RUPRUHLQIRUPDWLRQZHUHIHUWRWKH RULJLQDOSXEOLFDWLRQVRUWRWKHH[WHQVLYHFRPSDULWLYHVWXG\RI%HVVHOLQN
12167($'<67$7(675,1*%$6('7<5(02'(/6
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
9RQ6FKOLSSH¶VVWUDLJKWFRQQHFWLRQPRGHO
7KLVPRGHOZKLFKVKRZVUHVXOWVWKDWFDQKDUGO\EHGLVWLQJXLVKHGIURPWKHH[DFW RQHVRQO\UHTXLUHVWKHVWULQJGHIOHFWLRQVDWWKHOHDGLQJDQGWUDLOLQJHGJHVYDQG Y:HILQGIRUWKH/DSODFHWUDQVIRUPVRIWKHVHGHIOHFWLRQVDVGHULYHGIURPWKH H[SUHVVLRQV LQYHFWRULDOIRUP
+
Y
+
Y
n
n
S
S
'
'& S
&
D
%
S H
DS
&
DS
% % & %
%
S
S
%
D S
D
S
& % D
DS
S
ZKHUHWKHH[SRQHQWLDO IXQFWLRQUHIHUVWRWKHUHWDUGDWLRQHIIHFWRYHUDGLVWDQFH HTXDOWRWKHFRQWDFWOHQJWK7KHWUDQVIHUIXQFWLRQVWRWKHDOWHUQDWLYHVHWRILQSXW PD\EHREWDLQHGE\XVLQJWKHFRQYHUVLRQIRUPXODH
YDULDEOHV \
7KHUHVSRQVHVWRDVWHSFKDQJHLQVOLSDQJOHEHFRPHFI
' ' '
Y
V
Y
V
Y
V
Y
R
Y
R
Y
R
' ' '
V
R
&
H
V R
%
D
&
&
LI
V
LI
V
& & V
H
D R
#
D
! D
UHVSRQVHVWRVWHSFKDQJHVRIRWKHULQSXWYDULDEOHVPD\EHGHWHUPLQHGE\XVLQJ WKHFRQYHUVLRQIRUPXODH 7KHVLGHIRUFHDQGWKHDOLJQLQJWRUTXHDUHREWDLQHGDVIROORZV
)
\
'
N'
0
]
F
F
F
F
%
D Y
%
% % D
Y
D
'
Y
&
)
Y
&
Y
% %
'
Y
D
Y &
0
&
Y
D
7KHILUVWHTXDWLRQVKRZVWKDWWKHIRUFHLVREWDLQHGE\PXOWLSO\LQJWKHDYHUDJH
12167($'<67$7(675,1*%$6('7<5(02'(/6
ODWHUDOGHIOHFWLRQZLWKWKHODWHUDOVWLIIQHVVFI ZLWK DQGWKHPRPHQW E\PXOWLSO\LQJWKHVORSHRIWKHFRQQHFWLRQOLQHZLWKWKHWRUVLRQDOVWLIIQHVVRIWKH µWKLQ¶ W\UH RU ,W PD\ EH QRWHG WKDW GXH WR WKH DGRSWHG VWUDLJKW FRQWDFWOLQHWKHWXUQVOLSVWLIIQHVVIRUWKHIRUFHEHFRPHVVOLJKWO\OHVVWKDQWKH H[DFWRQH
n'
&
F
)
F
D
%
D
7KH YDULRXV GLDJUDPV VKRZ WKDW RQO\ LQ VRPH SDUWLFXODU FDVHV D VPDOO GLIIHUHQFHEHWZHHQµH[DFW¶DQGµ9RQ6FKOLSSH¶FDQEHREVHUYHG7KHPRGHOFDQ < \ < FRRUGLQDWHV EHHDVLO\XVHGLQYHKLFOHVLPXODWLRQVWXGLHVE\UHPHPEHULQJWKH [
RIWKHOHDGLQJHGJHZLWKUHVSHFWWRWKHJOREDOD[LVV\VWHP FI)LJ DQGXVH WKHVHFRRUGLQDWHVDJDLQDIWHUWKHZKHHOLVUROOHGDGLVWDQFHDIXUWKHUZKHQWKH WUDLOLQJHGJHKDVDVVXPHGWKLVORFDWLRQ
6PLOH\¶VDQG5RJHU¶VDSSUR[LPDWLRQV
$VVXPLQJ LQ )LJ WKDW WKH ZKHHO PRYHV DORQJ WKH [ < D[LV ZLWK RQO\ VPDOO < RI WKH GHYLDWLRQV LQ WKH ODWHUDO GLUHFWLRQ DQG LQ \DZ WKH ODWHUDO FRRUGLQDWH \
OHDGLQJHGJHIROORZVIURP(T $IWHUUHDOLVLQJWKDW
2
\
' 2% \
D
%
Y
WKHGLIIHUHQWLDOHTXDWLRQIRU\ < EHFRPHV
G\ 2
%2 ' 2% % \
GV
\
D
7KHSUREOHP6PLOH\KDVDGGUHVVHGLVWKHDVVHVVPHQWRIWKHORFDWLRQRIWKHFHQWUH < RIWKLVSRLQWPD\EHDSSUR[LPDWHG RIWKHFRQWDFWOLQH7KHODWHUDOFRRUGLQDWH\
E\ XVLQJ D 7D\ORU VHULHV 6WDUWLQJ RXW IURP WKH SRVLWLRQ RI WKH FHQWUH FRQWDFW SRLQW WKH VORSH DQG WKH FXUYDWXUH HWF RI WKH SDWK RI FRQWDFW SRLQWV WKH UHODWLRQVKLSEHWZHHQWKHODWHUDOFRRUGLQDWHRIWKHIRUHPRVWFRQWDFWSRLQWDQGWKDW RIWKHFHQWUHSRLQWFDQEHZULWWHQDVIROORZV
2
\
'
2
\
%
D
G\ 2 G
V
%
D
G G
2
\
V
% @@@@
,WVGHULYDWLYHEHFRPHV
G\ 2 G
V
'
G\ 2 G
V
%
G
2
\
D
G
V
%
D
G G
2
\
V
% @@@@
$IWHUVXEVWLWXWLRQRIWKHVHVHULHVLQ(T WKHIROORZLQJJHQHULFIRUPXODLV REWDLQHG
12167($'<67$7(675,1*%$6('7<5(02'(/6
Q
%
Q
G
Q
'
L
&
Q
D D
G
Q
2
\
V
Q
%
2
\
' 2% %
D
\
7KHVLGHIRUFHDQGWKHDOLJQLQJWRUTXHFDQEHIRXQGE\PXOWLSO\LQJWKHODWHUDO \ < \ < DQG WKH WRUVLRQ DQJOH
GHIOHFWLRQ Y
G\ < GV
ZLWK WKH UHVSHFWLYH
VWLIIQHVVHV&)\DQG&0¶ FI(TV :HREWDLQ )
'
\
0
]
&
)\
'&
\ 2
& 2
\
G\ 2 &
&
0
G
V
8VLQJWKHFRQYHUVLRQIRUPXODH WKHWUDQVIHUIXQFWLRQVZLWKUHVSHFWWR
nFDQEHREWDLQHG)RUWKHRUGHUQRI HTXDOWRWKHPRGHOEHFRPHV
DQG
RIWKHVHFRQGRUGHUDQGRQHILQGVWKHIROORZLQJWUDQVIHUIXQFWLRQVLQYHFWRUIRUP
+
)
n
S
'
&
%ò
D )\
D
D S
% %
%ò
D
D S
%ò
D
% %
D S
D
%
%
D
DQG
+
0
n
S
'&
&
&
0
D
D
%ò %ò
D S
D S
%ò % % % %
D
D
D
S
S
,WPD\EHQRWHGWKDWWKHVLGHIRUFHWXUQVOLSVWLIIQHVVIRUWKLVDSSUR[LPDWHPRGHO LVVRPHZKDWODUJHUWKDQDFFRUGLQJWRWKHH[DFWH[SUHVVLRQ:HKDYHIRU6PLOH\¶V PRGHO &
n'
)
F F D
%ò
D
%
D
7KHVWHSUHVSRQVHVRI)LJVKRZUHDVRQDEOHWRYHU\JRRGFRUUHVSRQGHQFHIRU WKHGLIIHUHQWLQSXWV$OVRWKHIUHTXHQF\UHVSRQVHIXQFWLRQVVKRZQLQ)LJ LQGLFDWH WKDW WKH DFFXUDF\ RI WKLV UHODWLYHO\ VLPSOH 6PLOH\ DSSUR[LPDWLRQ LV TXLWHJRRG7KHGLSLQWKH0 ]UHVSRQVHWR
LVZHOOUHSUHVHQWHGDQGLVORFDWHGDW
WKHµPHDQGHULQJ¶SDWKIUHTXHQF\ ]HURRI D
V
D
%^D
òD `
7KHDSSUR[LPDWLRQPD\EHFRQVLGHUHGWREHDFFHSWDEOHIRUZDYHOHQJWKVODUJHU WKDQDERXWWLPHVWKHFRQWDFWOHQJWK D UHFRPPHQGVWRXVHWKHRUGHUQ
,QKLVSXEOLFDWLRQ6PLOH\
7KHLQLWLDOHPSLULFDOO\DVVHVVHGIRUPXODHRI5RJHUV DUHDOPRVWWKH VDPHDVWKHIXQFWLRQV WKHWHUPVòDGRQRWDSSHDULQKLVH[SUHVVLRQV EXW LQ WKH QXPHUDWRU RI WKH PRPHQW UHVSRQVH WR WXUQ VOLS VHFRQG HOHPHQW RI
g
WKH HPSLULFDOO\ DVVHVVHG WHUP LV DGGHG $OVR D FRQQHFWLRQ ZLWK .OXLWHUV¶DSSUR[LPDWLRQDSSHDUVWRH[LVW
12167($'<67$7(675,1*%$6('7<5(02'(/6
.OXLWHUV¶DSSUR[LPDWLRQ
.OXLWHUV FI%HVVHOLQN DGRSWHGD3DGpILOWHUWRDSSUR[LPDWHWKH SRVLWLRQRIWKHWUDLOLQJHGJH7KHWUDQVIHUIXQFWLRQRI\ZLWKUHVSHFWWR\UHDGV LIWKHRUGHURIWKHILOWHULVWDNHQHTXDOWR
+
\ \
S
:LWK
2
\
Y
' 2% % ' 2 &2% \
\
D D
' D
\
D
S
D
S
& %
DS DS
n
Y
Y
D
ZHREWDLQZLWK
+
% %
S
'& S
D
S
% D
%
D
S
& % D
&
D
% % %
DS
S
DS
% % & %
D S
D
S
%
DS
S
S
7KHGHIOHFWLRQDWWKHIRUHPRVWSRLQWLVRIFRXUVHJRYHUQHGE\WUDQVIHUIXQFWLRQ 7KHFRUUHVSRQGHQFHRIWKLVWKLUGRUGHUPRGHOZLWK9RQ6FKOLSSH¶VPRGHO LVYHU\JRRGIRUZDYHOHQJWKVODUJHUWKDQDERXWWLPHVWKHFRQWDFWOHQJWK7KH DFFXUDF\ LV EHWWHU WKDQ HYHQ 6PLOH\¶V WKLUGRUGHU DSSURDFK ,W WXUQV RXW WKDW
g
ZKHQDILUVWRUGHU3DGpILOWHULVXVHGWKHIRUPXODHRI5RJHUVZLWKRXW DULVH
6WUDLJKWWDQJHQWDSSUR[LPDWLRQ
)RU WKLV YHU\VLPSOHDSSUR[LPDWLRQWKHFRQWDFWOLQHLVVROHO\JRYHUQHGE\WKH GHIOHFWLRQYDWWKHOHDGLQJHGJH7KHDSSUR[LPDWHGVKDSHRIWKHGHIOHFWHGVWULQJ FRUUHVSRQGV WR WKH VWHDG\VWDWH GHIOHFWLRQ GHSLFWHG LQ )LJ ZLWK GHIOHFWLRQ
N
DQJOH
Y 7KHVLGHIRUFHDQGDOLJQLQJWRUTXHDUHIRXQGE\PXOWLSO\LQJWKH
GHIOHFWLRQ DQJOH ZLWK WKH FRUQHULQJ VWLIIQHVV DQG WKH DOLJQLQJ VWLIIQHVV UHVSHFWLYHO\ 8VLQJ ZHREWDLQIRUWKHWUDQVIHUIXQFWLRQV
+
)
+
0
n
n
S
'
S
&
)
'&
%
&
D S
&
0
%
DS
&
D S
DS
7KH VWHS UHVSRQVHV HTXDO WKH UHVSHFWLYH VOLS VWLIIQHVVHV PXOWLSOLHG ZLWK WKH UHVSRQVHVRIY
DFFRUGLQJWR $VH[SHFWHGWKHDFFXUDF\EHFRPHVPXFK
OHVVDQGIURPWKHIUHTXHQF\UHVSRQVHIXQFWLRQVZHPD\FRQFOXGHWKDWDFFHSWDEOH
12167($'<67$7(675,1*%$6('7<5(02'(/6
DJUHHPHQWLVDWWDLQHGZKHQWKHSDWKIUHTXHQF\LVOLPLWHGWRDERXW D
RU
DZDYHOHQJWKODUJHUWKDQDERXWWLPHVWKHFRQWDFWOHQJWK7KHUHVSRQVHRIWKH DOLJQLQJWRUTXHWRSDWKFXUYDWXUHDSSHDUVWREHIDURIIZKHQFRPSDUHGZLWKWKH H[DFWUHVSRQVHV6LQFHWKLVSDUWLFXODUUHVSRQVHLVWKHOHDVWLPSRUWDQWLQUHDOLVWLF VLWXDWLRQVWKHVWUDLJKWWDQJHQWPRGHOPD\VWLOOEHDFFHSWDEOH,QWKHDQDO\VLVRI WKHVKLPP\SKHQRPHQRQWKLVZLOOEHGHPRQVWUDWHGWREHWUXHIRUVSHHGVRIWUDYHO ZKLFKDUHQRWWRRORZ ZKHUHWKHZDYHOHQJWKEHFRPHVWRRVKRUW )LJXUHV FOHDUO\ VKRZ WKDW DW OHDVW XS WR D
WKH SKDVH ODJ
FORVHO\IROORZVWKHH[DFWYDULDWLRQZLWKIUHTXHQF\7KHDPSOLWXGHKRZHYHULV
WRRODUJHHVSHFLDOO\IRUWKHPRPHQW$FRPELQDWLRQZLWKDILFWLYH0 ] ILUVWRUGHU DSSUR[LPDWLRQVHHIXUWKHURQ ZLWKSURSHUO\ FKRVHQSDUDPHWHU
(
&0nPD\
KHOSWRUHGXFHWKHDPSOLWXGHRIWKHPRPHQWWRDPRUHDFFHSWDEOHOHYHODOVRIRU QRQGLPHQVLRQDO SDWK IUHTXHQFLHV KLJKHU WKDQ )LJXUH VKRZV D FRQVLGHUDEOHLPSURYHPHQWLIZHZRXOGFKRVH&0n
&0
'LIIHUHQWLDO HTXDWLRQ WR EH VKRZQ ODWHU RQ JRYHUQV WKH VWUDLJKW WDQJHQW DSSUR[LPDWLRQ %HFDXVH ZH KDYH D GHIOHFWLRQ VKDSH HTXDO WR WKDW RFFXUULQJDWVWHDG\VWDWHVLGHVOLSPRWLRQWKHH[WHQVLRQRIWKHPRGHOWRQRQOLQHDU ODUJH VOLS FRQGLWLRQV LV HDV\ WR HVWDEOLVK :H PD\ HPSOR\ WKH VWHDG\VWDWH FKDUDFWHULVWLFVHJ0DJLF)RUPXODHDQGREWDLQ )
\
'
)
\
0
]
'
0
]
ZLWKWKHGHIOHFWLRQDQJOH
'
Y
7KHDPSOLWXGHRIWKHVHOIH[FLWHGVKLPP\RVFLOODWLRQDSSHDUVWREHFRPHOLPLWHG GXHWRWKHQRQOLQHDUGHJUHVVLYHFKDUDFWHULVWLFVRIWKHVLGHIRUFHDQGWKHDOLJQLQJ WRUTXHYHUVXVVOLSDQJOH
6LQJOHSRLQWFRQWDFWPRGHO
7KLVVLPSOHVWDSSUR[LPDWLRQGLVUHJDUGVWKHLQIOXHQFHRIWKHOHQJWKRIWKHFRQWDFW OLQH7KHODWHUDOGHIOHFWLRQDWWKHFRQWDFWFHQWUHDWVWHDG\VWDWHVLGHVOLSVKRXOG EHWDNHQHTXDOWRWKDWRIWKHH[DFWPRGHO7KLVUHTXLUHVDPRGHOUHOD[DWLRQOHQJWK
HTXDO WR WKH VXP RI WKH VWULQJ GHIOHFWLRQ UHOD[DWLRQ OHQJWK
DQG KDOI WKH
FRQWDFW OHQJWK D 7KH FRUUHVSRQGLQJ WUDQVIHU IXQFWLRQ IRU WKH GHIOHFWLRQ Y EHFRPH
+
Y
ZLWK
n
S
'
%
S
12167($'<67$7(675,1*%$6('7<5(02'(/6
' %
D
$SSDUHQWO\WKHUHVSRQVHWRWXUQVOLSYDQLVKHV$GHIOHFWLRQDQJOHPD\VWLOOEH
N
GHILQHG
Y
$VDUHVXOWUHVSRQVHIXQFWLRQVIRUWKHIRUFHDQGWKHPRPHQW
DUHREWDLQHG
+
n
)
+
n
0
S
'
&
)
'&
S
%
&
0
S
%
S
7KLVIXQFWLRQFRUUHVSRQGVWRWKHIUHTXHQF\UHVSRQVHIXQFWLRQ WKDWKROGV IRUDILUVWRUGHUV\VWHPZLWKWKHVDPHFXWRIIIUHTXHQF\DVWKHRQHIRUWKHH[DFW PRGHO:LWKUHOD[DWLRQOHQJWK
DWKHPRGHOSURGXFHVDFRUUHFWSKDVHODJ
IRUWKHUHVSRQVHWR\DZRVFLOODWLRQVDWODUJHZDYHOHQJWKVDVZDVDOUHDG\DVVHVVHG EHIRUHFI ([FHSWIRUWKHPRPHQWUHVSRQVHWR\DZWKHDPSOLWXGHVDSSHDU WREHFRPHVRPHZKDWWRRVPDOOLQWKHSUREDEO\DFFHSWDEOHZDYHOHQJWKUDQJH
!
FD D ,Q WKH 1\TXLVW SORW RI )LJ WKH FXUYH IRU WKH PRPHQW UHVSRQVH FRLQFLGHVZLWKWKDWRI6PLOH\¶VDSSUR[LPDWLRQ QRWWKHIUHTXHQF\PDUNV 7KH UHVSRQVHRIWKHVLGHIRUFHWRWXUQVOLSPD\EHDUWLILFLDOO\LQWURGXFHGE\SXWWLQJ DQµD¶LQVWHDGRIWKHµ¶LQ WKH LQSXWYHFWRURI 7KLVZRXOGKRZHYHU UHTXLUHDQDGGLWLRQDOILUVWRUGHUGLIIHUHQWLDOHTXDWLRQIRUWKHVLGHIRUFH 'LIIHUHQWLDOHTXDWLRQ JRYHUQVWKHVLQJOHSRLQWFRQWDFWPRGHO
$SSUR[LPDWLRQVRIWUHDGZLGWKPRPHQWUHVSRQVH
7KHWUDQVIHUIXQFWLRQ IRU0 ]PD\EHVLPSOLILHGE\IROORZLQJWKHDSSURDFK RI9RQ6FKOLSSHEXWQRZIRUWKHORQJLWXGLQDOGHIOHFWLRQVWKDWLV E\DVVXPLQJD OLQHDULQWHUSRODWLRQRIWKHORQJLWXGLQDOGHIOHFWLRQXEHWZHHQWKHH[DFWGHIOHFWLRQV DWWKHOHDGLQJDQGWUDLOLQJHGJHV$VWKHGHIOHFWLRQDWWKHILUVWSRLQWLVHTXDOWR ]HURWKHOLQHDULQWHUSRODWLRQZRXOGOHDGWRDQDYHUDJHGHIOHFWLRQHTXDOWRKDOIWKH GHIOHFWLRQ DW WKH UHDU PRVW SRLQW X 7KH FRUUHVSRQGLQJ WUDQVIHU IXQFWLRQ EHFRPHV
+
( 0
n
S
'
( DS
&
H
&
SD
&
S
)LJXUHV VKRZ WKH SHUIRUPDQFH RI WKLV OLQHDU LQWHUSRODWLRQ PRGHO WRJHWKHUZLWKWKHH[DFWUHVSRQVHDQGRWKHUDSSUR[LPDWHGPRGHOUHVSRQVHV7KH DSSURDFKPD\EHXVHGLQFRQQHFWLRQZLWKWKH9RQ6FKOLSSHODWHUDOGHIOHFWLRQ DSSUR[LPDWLRQ ZKHUH WKH ORFDWLRQ RI WKH FRQWDFW SRLQW DW WKH OHDGLQJ HGJH LV UHPHPEHUHG RYHU D GLVWDQFH UROOHG HTXDO WR WKH FRQWDFW OHQJWK ZKHQ WKLV LQIRUPDWLRQLVXVHGWRFDOFXODWHWKHGHIOHFWLRQDWWKHUHDUHGJH
12167($'<67$7(675,1*%$6('7<5(02'(/6
([SDQVLRQRIWKHH[DFWUHVSRQVHIXQFWLRQ LQDVHULHVRISRZHUVRIS \LHOGVZKHQOLPLWLQJWRWKHVHFRQGSRZHU IRUWKHUHVSRQVHWR
+
(
n
0
ZKHUH
S
E
( &
'
D S
&
S
&0n7KHILUVWRUGHU RQO\ILUVWWHUPLQ DQGWKHVHFRQGRUGHU
ERWK WHUPV DSSUR[LPDWLRQV VKRZ UHVSRQVHV DV SUHVHQWHG LQ WKH ILJXUHV 7KH ILUVWRUGHUUHVSRQVHKDVEHHQLQGLFDWHGLQFRPELQDWLRQZLWKWKHVWUDLJKWWDQJHQW
UHVSRQVHIRU0 ]LQ)LJ7KHYHFWRUVIRU0 ]SRLQWXSZDUGV YHUWLFDOO\ 7KH FDVH RI &0n
D&0 JHQHUDWHV D UHVSRQVH ZKLFK LPSURYHV RQ WKH VWUDLJKW
WDQJHQW EDUH VWULQJ UHVSRQVH (VSHFLDOO\ FRPSDUHG ZLWK WKH HQKDQFHG PRGHO SURYLGHGZLWKWUHDGHOHPHQWVWKHDJUHHPHQWLVVDWLVIDFWRU\7RPRGHOWKHIRRW SULQW&KDSWHU XVHVDILUVWRUGHUGLIIHUHQWLDOHTXDWLRQZLWKWKHUHOD[DWLRQOHQJWK HTXDOWRD7KLVFRUUHVSRQGVWRWKHFXWRIIIUHTXHQF\RIWKHH[DFWPRGHO
'LIIHUHQWLDOHTXDWLRQV
)URP WKH YDULRXV DSSUR[LPDWH WUDQVIHU IXQFWLRQV WKH JRYHUQLQJ GLIIHUHQWLDO HTXDWLRQVFDQEHHDVLO\HVWDEOLVKHGE\UHSODFLQJWKH/DSODFHYDULDEOHSZLWKWKH GLIIHUHQWLDWLRQRSHUDWRUG GV7KLVLVQRWWKHFDVHZLWKWKH H[DFW DQGWKH9RQ
6FKOLSSH WUDQVIHU IXQFWLRQV VLQFH WKHVH DUH RI LQILQLWH RUGHU :H PD\ H[SDQG WKHVHIXQFWLRQVLQVHULHVRISRZHUVRISDQGWUXQFDWHDWDFHUWDLQSRZHUDIWHU ZKLFK S LV UHSODFHG E\ G GV 7UXQFDWLRQ DIWHU WKH ILUVW GHJUHH RI WKH VHULHV
H[SDQVLRQRIWKHH[DFWIXQFWLRQV \LHOGVWKHGLIIHUHQWLDOHTXDWLRQV ZULWWHQLQWHUPVRILQSXWYDULDEOHV
G )\ GV
% '
G 0 ]
G
V
)
%
\
0
]
%
&
)
'&
&
0
%
D
&
DQG
nn&&
)
G
D&
0
0
%
W
WGHQRWLQJWKHSQHXPDWLFWUDLO
G
&
GV
n%
G
nD
&
)
GV
@@@
%
GV
D &
n
&
D
)
% %
D &
0
&
D
n%
G
&
)\
GV
@@@
ZLWKFRHIILFLHQWVDFFRUGLQJWR :KHQXVLQJWKHFRQYHUVLRQIRUPXODH
2
HTXDWLRQVLQWHUPVRILQSXWYDULDEOHV\DQG
G )\ GV G 0 ] G
V
% %
)
\
'
0
]
&
&
'&
)
&
0
G\ 2 GV
&
&
G\ 2 GV
D
G
D
GV
&
D
G
% &
G GV
%
D
W
G
2
\
GV
2
\
GV
DUHREWDLQHG
% @@@
% @@@
)URP(T ZHILQDOO\REWDLQWKHGLIIHUHQWLDOHTXDWLRQVIRU0 ]
12167($'<67$7(675,1*%$6('7<5(02'(/6
0
( ]
'
&
n&
n
0
n %
G
D
GV
@@@
RULQWHUPVRIWKH\DZDQJOH
0
(
'&
]
G
n
&
0
&
GV
G
D
GV
%@@@
7KHVWUDLJKWWDQJHQWDSSUR[LPDWLRQUHPDLQVZKHQWKHODVWWHUPRIWKHULJKWKDQG PHPEHURIERWK(TV LVRPLWWHG7KHVDPHUHVXOWLVREWDLQHGZKHQ WKHVLQJOHILUVWRUGHUGLIIHUHQWLDOHTXDWLRQ LVXVHGLQFRPELQDWLRQZLWKWKH
N DQGWKHFRUQHULQJDQGDOLJQLQJVWLIIQHVVHV:HKDYH
GHIOHFWLRQDQJOH
WKHIROORZLQJVHWRIHTXDWLRQVIRUWKHVWUDLJKWWDQJHQWDSSUR[LPDWLRQ
G Y
%
GV
'
)
'
\
Y
'
G\ 2
&
&
GV
G
D
GV
' % n D
Y
&
)
0
]
'&
&
0
RULQFDVHRIODUJHUVOLSDQJOHV )
'
\
)
\
0
]
'
0
]
7KHVHWPD\EHXVHGLQFRPELQDWLRQZLWKWKHILUVWRUGHUYHUVLRQRI 0
( ]
'&
n
&
0
G
'
GV
RU LQ WKH QRQOLQHDU FDVH
&
nn
0
n PD\ EH XVHG DV LQSXW LQ WKH VWHDG\VWDWH PRGHO
GHVFULSWLRQH[WHQGHGZLWKWXUQVOLS FI6HFWLRQ 7KH VLQJOH SRLQW FRQWDFW PRGHO ZLWK WUDQVIHU IXQFWLRQV LV JRYHUQHGE\WKHVLQJOHILUVWRUGHUGLIIHUHQWLDOHTXDWLRQ
G Y GV
%
Y
'
&
G\ 2 GV
'
ZLWKWKHUHOD[DWLRQOHQJWK
' % '
&
)
D
&
)\
7KHµWUDQVLHQW¶VOLSDQJOH
'
Y
FRPSOHWHVWKHGHVFULSWLRQWRJHWKHUZLWK(TV RU
12167($'<67$7(675,1*%$6('7<5(02'(/6
%\FRQVLGHULQJRQO\WHUPVXSWRWKHILUVWGHJUHHLQSRIWKH WUDQVIRUPHG HTXDWLRQV WKHIROORZLQJLQGLYLGXDOILUVWRUGHUGLIIHUHQWLDOHTXDWLRQV FDQEHGHILQHGZKLFKKROGH[DFWO\IRU
6 4 S 6 LIXVHLVPDGHRIWKHQRWDWLRQ
IRUWKHLQGLYLGXDOUHOD[DWLRQOHQJWKV DQG
G )\ )
% '
GV
)
G 0 ] 0
G
n
V
G )\
)
GV
%
&
\
0
]
'&
% ' )
)
\
&
&
nn
0
)
DQGLQWHUPVRIWKHDOWHUQDWLYHVHWRILQSXWV
G )\ )
GV
% '& )
G 0 ] 0
G
V
G )\ )
GV
G
0
]
'
% '
G 0 ] 0
%
\
V
)
%
\
0
]
G\ 2
&
)
&
GV G\ 2
0
GV
&
)
'&
&
0
ZKHUHWKHUHVSHFWLYHLQGLYLGXDOUHOD[DWLRQOHQJWKVUHDG DFFRUGLQJWRWKHH[DFW EDUHVWULQJPRGHO )
0
' % & ' ' D
W
n'
0
)
)
' %
D
,IDOVRWKHUHOD[DWLRQOHQJWKIRU)\ZLWKUHVSHFWWR
LVWDNHQHTXDOWR
D
WKHZLWKUHVSHFWWRWXUQVOLSLPSURYHGVLQJOHFRQWDFWSRLQWDSSUR[LPDWLRQDULVHV 7DEOHSUHVHQWVWKHYDOXHVRIWKHLQGLYLGXDOUHOD[DWLRQOHQJWKVDVGHILQHG E\(TV HWF DQGDVVHVVHGQXPHULFDOO\IRUWKHYDULRXVPRGHOVGLVFXVVHG DERYHZLWK
D)RUWKHFRUUHVSRQGLQJHQKDQFHGPRGHOZLWKWUHDGHOHPHQWV
ZHUHIHUWR6HFWLRQ)RUDOOPRGHOV
)
&) &)\KDVEHHQNHSWWKHVDPH
,QSUDFWLFDOFDVHVVKLPP\RFFXUVDWSDWKIUHTXHQFLHVEHORZWKHPHDQGHULQJ IUHTXHQF\ HYHQ EHORZ DERXW KDOI WKLV YDOXH 7KLV FRUUHVSRQGV WR D SUDFWLFDO XSSHUOLPLWRID
RUIRUWKHZDYHOHQJWK DORZHUOLPLWRIFDWLPHV
WKHFRQWDFWOHQJWK D ,WWXUQVRXWWKDWLQWKLVSUDFWLFDOUDQJHWKHDJUHHPHQW EHWZHHQWKHYDULRXVPRGHOVLVJRRGRUUHDVRQDEOH
12167($'<67$7(675,1*%$6('7<5(02'(/6
1 0
0
]
αo
]
s.t.
0α
&
1
α
a0
s.t.
0α o
&
Sm2
a0
ex.tr.el.
Sm2
ex.tr.el. ex. vSch.
ex. vSch.
DE
0
0
2
4
VD
6
8
0
10
1
0
2
4
VD
6
8
10
1
)\ &)ϕϕo
0]
ex.tr.el.
ex. vSch.
s.t.
ϕ
s.t.
0α o
D&
0
]
ex.tr.el. Sm2
Sm2 ex. FG vSch. a0 0 0 0 2 4 6 8 10 0 VD
ex.tr.el. a0 2
8
10
6
8
10
6
8
10
]
0
0αψo
a0
&
ex.tr.el.
a0
ex.
Sm2 ex.,vSch.
vSch. Sm2
0
0
HI
s.t.
s.t.
ex.tr.el.
-0.4
0
1.4
2
4
VD
6
8
0
10
]
D0
s.t. Sm2
)\
4
VD
s.t.
0α \o
ex.tr.el.
&
ex. vSch.
vSch. ex.tr.el.
\o
2
0.4
1
)\
6
)\ &)αψo
&
VD
1
1
-0.4
4
Sm2 ex. vSch.
0.2 a0
a0
JK
0
0
2
4
VD
6
8
10
0
0
2
4
VD
)LJ 6WHS UHVSRQVHV RI VLGH IRUFH DQG DOLJQLQJ WRUTXH WR VHYHUDO LQSXWV IRU H[DFWDQG DSSUR[LPDWHPRGHOV
ex. H[DFWµEDUH¶VWULQJPRGHO vSch. 9RQ6FKOLSSH Sm2 6PLOH\VHFRQGRUGHU s.t. VWUDLJKWWDQJHQW a0 SRLQWFRQWDFW ex.tr.el. H[DFWZLWKWUHDGHOHPHQWV
12167($'<67$7(675,1*%$6('7<5(02'(/6
2 nd 1 0
1st
* ] ϕo
D0
ex.
0ϕ
&
*
]
ex.
ψ
0ϕ o
&
s.c. 0.5
s.c.
0.5
0 )LJ
1st, 2 nd
1
2
VD
4
0
2
VD
4
6WHSUHVSRQVHRIWKHPRPHQWGXHWRWUHDGZLGWKWRWXUQVOLS SDWKFXUYDWXUH DQG\DZ DQJOHIRUH[DFWDQGDSSUR[LPDWLRQV
ex. H[DFW EUXVKPRGHO s.c. VWUDLJKWFRQQHFWLRQ OLQHDULQWHUSRODWLRQ
2nd VHFRQGRUGHUDSSUR[LPDWLRQ 1st ILUVWRUGHUDSSUR[LPDWLRQ 1
)\ &)αα
ex. vSch. Sm2 s.t.
0.8 0.6
a0 ex.tr.el.
0.4
a0 ex.tr.el.
ex. vSch. Sm2
0.2 0 0
φ)α [o]
s.t.
-45
a0 -90 00
s.t.
ex.tr.el. a0 ex. vSch. 0.05 0.05
0.10.1
0.04
ex.tr.el.
s.t.
Sm2
vSch. D
0.08 0.1
Dλ
0.2
0.4
0.8 1
1 0.8
]
0
0αα
&
0.6 0.4
ex.tr.el. a0
0.2 0 0
φ
ex. vSch. s.t.
a0
s.t.
s.t.
s.t. Sm2
a0
α
[o]
-180
0
ex.tr.el.
Sm2 ex. vSch.
Sm2
ex. vSch. -360 0 0.01
0.050.02 0.1
0.04
0.08 0.1
Dλ
0.2
0.4
ex.tr.el. 0.8 1 E
)LJDEFGHI LQFOXGLQJQH[WSDJH )UHTXHQF\UHVSRQVHIXQFWLRQVRIVLGHIRUFHDQGDOLJQLQJWRUTXHWRVHYHUDOLQSXWV IRUH[DFWDQGDSSUR[LPDWHPRGHOV
ex. H[DFWµEDUH¶VWULQJPRGHO vSch. 9RQ6FKOLSSH Sm2 6PLOH\VHFRQGRUGHU s.t. VWUDLJKWWDQJHQW a0 SRLQWFRQWDFW ex.tr.el. H[DFWZLWKWUHDGHOHPHQWV
12167($'<67$7(675,1*%$6('7<5(02'(/6
1
)\ &)ϕϕ
0.8
ex. vSch. s.t.
0.6
ex.tr.el.
0.4 0.2 0 0
s.t.
Sm2
Sm2
a0
ex.tr.el.
ex. vSch.
a0
s.t.
ex.tr.el.
φ)ϕ [o] -180
a0
s.t.
Sm2
ex. F
vSch. -360 0 0.01
0.050.02 0.1
0.04
0.08 0.1
Dλ
0.2
0.4
0.8 1
1 0.8 s.t.
]
0
ϕ
0α
D&
Sm2
0.4
ex.tr.el.
0.2
a0
vSch. ex.
Sm2
0.6
ex.tr.el. a0
s.t.
0 180
φ0ϕ [o]
s.t.
90
s.t. ex.tr.el.
a0
0
G
Sm2 -90 0 0.01
ex., vSch. 0.050.02 0.1
0.04
0.08 0.1
Dλ
0.2
0.4
0.8 1
1
)\ &)αψ
0.8 0.6
s.t.
a0 Sm2
s.t.
a0
0.4
ex.tr.el.
vSch. ex.
0.2 0 0
a0
φ)ψ [o]
ex.tr.el. a0
s.t.
-90
ex. vSch. -180 0 0.01
0.050.02 0.1
s.t. H
Sm2 0.04
0.08 0.1
Dλ
0.2
0.4
0.8 1
1.5
]
0
ψ
0α
&
1
φ
ψ
Sm2
0.5 0 0
vSch. ex.
s.t. a0 ex.tr.el.
[o] -180
s.t. Sm2
ex.tr.el.
a0 s.t.
a0
ex. vSch.
s.t.
I
0
Sm2 -360 0 0.01
0.050.02 0.1
0.04
0.08 0.1
Dλ
0.2
0.4
0.8 1
12167($'<67$7(675,1*%$6('7<5(02'(/6
0
Im
\
)α
)LJ
ψ
-0.5
0
-φ
vSch.
1 )ψ 0.12 a0 )\ 0.08 0.06
1 ex.
) &
)LJFRQW¶G
4 4
VLGH IRUFH
0.01
0
\
)α
ψ
0α
ψ 0
4 1
-0.5
0.10
4
0.12
φ
0.08
ψ
)LJ1\TXLVWSORWRI
]
0.16 0.12
0.06
0.02
0.04
0.03 ]
0
Re
0α
Dλ
IXQFWLRQRIWKHDOLJQLQJ WRUTXH
0 ¶]
\DZDQJOH
1
ZLWK
UHVSHFW WR WKH ZKHHO &XUYHD
LVKLGHGE\6P6DPH
ψ
FRQGLWLRQV
DV
LQ
)LJ
2 nd
1st
ψ
0ϕ
WKHIUHTXHQF\UHVSRQVH
0
0.1 ]
1 0.01
&
&
0
0
0.08 0
D0 *
vSch. ex.
1
s.t.
Im
3DUDPHWHU
0.12
]
0
&
D
ex.tr.el.
Sm2
0.16
0.5
Im
1
) &
WKH
ex. H[DFW µEDUH¶ VWULQJ PRGHO vSch. 9RQ6FKOLSSH Sm2 6PLOH\VHFRQGRUGHU s.t. VWUDLJKWWDQJHQW a0 SRLQWFRQWDFW ex.tr.el. H[DFWZLWKWUHDGHOPQWV
Dλ
0.04
Re
RI
0.03 Sm2
SORW
)\ ZLWK UHVSHFW WR WKH
ZKHHO\DZDQJOH
0.02
s.t.
1\TXLVW
IUHTXHQF\UHVSRQVHIXQFWLRQRIWKH
0.12
0.5
ex.
0.08
0
s.c.
Dλ
0.04 0.02
)LJ1\TXLVWSORWRI WKHIUHTXHQF\UHVSRQVH
4
0
IXQFWLRQ RI WKH WRUTXH
0 ]
GXHWRWUHDGZLGWK ZLWK
UHVSHFW
WR
ZKHHO\DZDQJOH
WKH
ex. H[DFW EUXVK s.c. VWUDLJKWFRQQHFWLRQ
-0.5 0
Re
D0]*
ψ
0ϕ
&
1
OLQHDU LQWHUSRODWLRQ
nd VHFRQGRUGHUDSS
1st ILUVWRUGHUDSSUR[
12167($'<67$7(675,1*%$6('7<5(02'(/6
1.5
Im
Sm2+2 nd
s.t.+ 1st
ex.+ex. ex.tr.el.+ex.
]
0
0α
&
ψ
0
0 0
]
]
*
]
1 1
0.6
&0ϕ D&0α
0.5
vSch.+s.c.
1
0.16 Sm2 ex.tr.el.
0.12
vSch.
0.16 ex.tr.el.
]
0
]
0
0.12
0
1
ex.
4 *
0
]
0.08
0
s.t.
0.02 0.04
0.12 ] 0.08
0
1
0.01
0.08
-0.5
4
1
0.03
D λ
0.04
0
Re
1
]
0
0α
&
2
ψ
)LJ 1\TXLVW SORW RI WKH QRQGLPHQVLRQDO IUHTXHQF\ UHVSRQVH IXQFWLRQ RI WKH DOLJQLQJ
0]
PRPHQW
ZLWKUHVSHFWWRWKHZKHHO\DZDQJOH
0
&RQWULEXWLQJFRPSRQHQWV
]GXH
WR ODWHUDO GHIRUPDWLRQV DQG 0 ] GXH WR WUHDG ZLGWK DQG DVVRFLDWHG ORQJLWXGLQDO GHIRUPDWLRQV3DUDPHWHUV
DDQG&0n
D&0 ([DFWDQGDSSUR[LPDWHPRGHOV
ex. H[DFWµEDUH¶VWULQJPRGHO vSch. 9RQ6FKOLSSH Sm2 6PLOH\VHFRQGRUGHU s.t. VWUDLJKWWDQJHQW a0 SRLQWFRQWDFW KLGGHQE\6P ex.tr.el. H[DFWZWUHOPWV ex. H[DFW EUXVK s.c. VWUDLJKWFRQQHFWLRQ OLQHDULQWHUSRODWLRQ
2nd VHFRQGRUGHUDSSUR[LPDWLRQ 1st ILUVWRUGHUDSSUR[ DOVR&
n
0
D&0
7DEOH,QGLYLGXDOUHOD[DWLRQOHQJWKVIRUWKHYDULRXVPRGHOVGHILQHGIRU
V 6
H[DFW
Y6FKO
6PLOH\
VWUWDQJ
D
D
D
)nD
D
)
0
) 0¶
D
H[WUHO
12167($'<67$7(675,1*%$6('7<5(02'(/6
2WKHU0RGHOV
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¶V DQG 0RUHODQG¶V VLQJOH SRLQW PRGHOV ZLWK D QRQKRORQRPLF FRQVWUDLQW ILUVWRUGHU GLIIHUHQWLDO HTXDWLRQ UDLVHVWKHRUGHURIWKHGHVFULSWLRQIURPRQHWRWZR
.HOG\VK¶VPRGHO
7KLV PRGHO LV NQRZQ WR EH XVHG LQ WKH 5XVVLDQ DLUFUDIW LQGXVWU\ DQG ZDV SXEOLVKHGLQ *RQFKDUHQNRHPSOR\HGWKHPRGHOIRUKLVUHVHDUFKLQVKLPP\ RI ODQGLQJ JHDUV ,Q DGGLWLRQ WR ODWHUDO DQG WRUVLRQDO VWLIIQHVVHV GDPSLQJ KDV EHHQLQFOXGHGEHWZHHQFRQWDFWSDWFKDQGULP7KHVLGHIRUFHDQGDOLJQLQJWRUTXH EHFRPHH[SUHVVHGLQWHUPVRIODWHUDOGHIOHFWLRQYDQGWRUVLRQDQJOH
) ' N Y0 % F Y \
\
0 'N
\
0
]
%F
:LWKWKHDVVXPSWLRQRIIXOODGKHVLRQWKHIROORZLQJUHODWLRQPXVWKROGUHJDUGLQJ WKHGLUHFWLRQRIWKHSDWKRIWKHFRQWDFWSRLQWRIWKHUROOLQJW\UH
Y % \2 GV
G
% '
5HJDUGLQJWKHSDWKFXUYDWXUHDOLQHDUHTXDWLRQLVLQWURGXFHG
%
G G
V
' &T Y & T % T
,QWHUHVWLQJLVWKDW.HOG\VKGLGWDNHLQWRDFFRXQWWKHZKHHOFDPEHUDQJOH DVDQ DGGLWLRQDOLQSXWYDULDEOH7KHWUDQVIHUIXQFWLRQVZLWKUHVSHFWWR
n
DQG
EHFRPH
DQG
+
)
n
S ' F % N 9S
\
\
S%T T T S %T S%T
12167($'<67$7(675,1*%$6('7<5(02'(/6
+
T &S S %T &T S S ' & F % N 9S S %T S%T
0
n
6XUSULVLQJO\ LW WXUQV RXW WKDW WKHVH WUDQVIHU IXQFWLRQV H[FHSW WKH RQH ZLWK
g
UHVSHFWWR FRUUHVSRQGWRWKHIXQFWLRQVRI5RJHUV ZLWK
(TV
ZLWKò DRPLWWHG LIDV%HVVHOLQNLQGLFDWHGWKHIROORZLQJHTXLYDOHQFHFRQGLWLRQV
KROG
T'
D
T ' D% D
F '& ' \
)\
& D%
)
F '& '&
0
0
)XUWKHUPRUHFRPSDULVRQZLWKWKH9RQ6FKOLSSHDSSUR[LPDWLRQDWVWHDG\VWDWH VKRZVWKDW
T '
& &n )
T'
)
& &n )
)
ZKHUHDVEHIRUH&) GHVLJQDWHVWKHFDPEHUIRUFHVWLIIQHVV
0RUHODQG¶VPRGHO
7KLVPRGHOILUVWSXEOLVKHGLQ LQWKHSDSHU³7KHVWRU\RIVKLPP\´XVHGWR EHDSRSXODUWRROLQWKH86IRUWKHDQDO\VLVRIDLUFUDIWVKLPP\7KHVWUXFWXUH LV VLPLODU WR WKDW RI .HOG\VK¶V PRGHO 7KH GLIIHUHQFHV DUH WKDW WKH GDPSLQJ FRHIILFLHQW N LV RPLWWHG DQG PRUH LPSRUWDQW WXUQ VOLS DQG FDPEHU DUH QRW FRQVLGHUHGDQGLQVWHDGRIHTXDWLRQ WKHIROORZLQJUHODWLRQLVXVHG
G G
W
'&
) % & \
)
7KHWUDQVIHUIXQFWLRQVZLWKUHVSHFWWRWKHVOLSDQJOHWKDWZHUHHVWDEOLVKHGUHDG
N 9 S % N %F 9S % F + S ' & & 9S % N 9%& S % F
\
)
\
)
\
)
+ S ' & & 0
\
0
\
)
\
N 9S % F & 9S % N 9%& S % F \
\
)
\
)
\
7KHIDFWWKDWZHDUHGHDOLQJKHUHZLWKDWLPHFRQVWDQW JDYHULVHWRFULWLFLVPDQG FDXVHGGLIILFXOWLHVLQDWWHPSWVWRILWH[SHULPHQWDOGDWD7RPDNHWKHPRGHOSXUH SDWKGHSHQGHQW VKRXOGEHPDGHLQYHUVHO\SURSRUWLRQDOZLWKVSHHG9 WKLV LQ DGGLWLRQWRRPLWWLQJWKHGDPSLQJFRHIILFLHQWN\,WPD\EHREVHUYHGWKDWZKHQ
12167($'<67$7(675,1*%$6('7<5(02'(/6
FRQVLGHULQJWKHUHODWLRQV WKHVXEVWLWXWLRQIRU
'
D
D% 9
OHDGV WR D WUDQVIHU IXQFWLRQV ZLWK UHVSHFW WR
WKDW LV LGHQWLFDO WR WKH
FRUUHVSRQGLQJIXQFWLRQVHVWDEOLVKHGE\.HOG\VKDQG5RJHUV
(QKDQFHG6WULQJ0RGHOZLWK7UHDG(OHPHQWV
$OWKRXJKWKHVWULQJPRGHOZLWKLWVVLPSOLFLW\DQGHVVHQWLDOIHDWXUHV VHUYHGXV ZHOOLQJDWKHULQJLQVLJKWDQGHVWDEOLVKLQJSURSHUPDWKHPDWLFDOGHVFULSWLRQVDQG XVHIXODSSUR[LPDWLRQVWKHVWULQJPRGHOVKRZVDEHKDYLRXUWKDWLQVHYHUDOYLHZV GLIIHUVFRQVLGHUDEO\IURPWKHSURSHUWLHVRIWKHUHDOW\UH7KHUHOD[DWLRQOHQJWK IRUWKHW\UHGHIOHFWLRQGRHVQRWFKDQJHZLWKYHUWLFDOORDGDQGWKXVWKHUHOD[DWLRQ OHQJWKIRUWKHVLGHIRUFHWR\DZ DSSURDFK ]HUR ZKHQ )]
)
DGRHVQRWYDU\VXIILFLHQWO\DQGGRHVQRW
6 DV ZRXOG RFFXU ZLWK WKH UHDO W\UH 7KH VDPH
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¶VWUDQVLHQWDQGRVFLOODWRU\RXWRISODQHEHKDYLRXUKDYHDOUHDG\EHHQ LQFOXGHGLQWKHYDULRXVJUDSKVRIWKHSUHFHGLQJVHFWLRQ
12167($'<67$7(675,1*%$6('7<5(02'(/6
6WHDG\VWDWHFKDUDFWHULVWLFV
)LJXUHGHSLFWVWKHVWUXFWXUHRIWKHPRGHO7KHWRWDOGHIOHFWLRQYLVPDGHXS RIWKHODWHUDOGHIOHFWLRQRIWKHVWULQJ WKHFDUFDVV YFDQGWKHODWHUDOGHIOHFWLRQRI WKHWUHDGHOHPHQWYS$VVRFLDWHGVWLIIQHVVHVRIFDUFDVVDQGWUHDGHOHPHQWVDUHFF DQGFSERWKSHUXQLWOHQJWKDQGWKHODWWHULQWHJUDWHGRYHUWKHWUHDGZLGWKE7KH VWULQJ HIIHFWLYH WHQVLRQ IRUFH LV 6 FI 6HFWLRQ 7KH IROORZLQJ PRGHO FRQVWDQWVDUHLQWURGXFHG
6 F
'
F
6
'
F %F
F
F
'
F
'
F
F
F %F
S
F
S
$SDUDEROLFYHUWLFDOSUHVVXUHGLVWULEXWLRQLVDVVXPHGDFFRUGLQJWR(T $W WKHWUDLOLQJHGJHDVOLGLQJUDQJHVWDUWVWREXLOWXSZKHQVLGHVOLSLQFUHDVHV$OVR DW WKH OHDGLQJ HGJH VOLGLQJ PD\ RFFXU ZKHQ WKH WUHDG HOHPHQW VWLIIQHVV LV UHODWLYHO\ODUJH7KHEDUHVWULQJPRGHOGRHVVKRZDUHODWLYHO\VKRUWVOLGLQJUDQJH WKDWJURZVDQGXOWLPDWHO\PHHWVWKHUHDUVOLGLQJUDQJHDWLQFUHDVLQJVLGHVOLS:H ZLOODVVXPHWKDWWKHWUHDGHOHPHQWVWLIIQHVVLVVXIILFLHQWO\VPDOOWRGLVUHJDUGIURQW VOLGLQJ 7KHWRWDOGHIOHFWLRQYDWVWHDG\VWDWHVLGHVOLSLQWKHUDQJH[W [ DZKHUHWKH
WLSVRIWKHHOHPHQWVDGKHUHWRWKHJURXQGIROORZVWKHGLIIHUHQWLDOHTXDWLRQ ZLWK
n
Y '& GV
G
LI
[ [#D
W
7KH GHIOHFWLRQ RI WKH VWULQJ RXWVLGH WKH FRQWDFW UDQJH LV JRYHUQHG E\ WKH GLIIHUHQWLDOHTXDWLRQ
D
σ*
α
wheel plane
Y2
FF
YF Y
stretched string
Y1
σ
α
YS
sliding range tread elements
9
D
\
)LJ(QKDQFHGVWULQJPRGHOSURYLGHGZLWKWUHDGHOHPHQWV
[
12167($'<67$7(675,1*%$6('7<5(02'(/6
Y &Y ' G[
G
F
F
LI
*[* ! D
$QGLQWKHFRQWDFW]RQH
F
Y & Y ' & & Y G[
G
F
F
LI
*[* # D
7RVROYHIRUWKHFRRUGLQDWHRIWKHWUDQVLWLRQSRLQW[WDQGWKHILYHFRQVWDQWVRI LQWHJUDWLRQLQWKHVROXWLRQVRI(T DQGRI(T IRUWKHWZR]RQHV DGKHVLRQDQGVOLGLQJZHQHHGVL[ERXQGDU\FRQGLWLRQV7KHVHFRQFHUQ]HURWUHDG HOHPHQWGHIOHFWLRQDWWKHOHDGLQJHGJHDQGFRQWLQXLW\RIGHIOHFWLRQDQGVORSHRI WKH VWULQJDQGFRQWLQXLW\RIWKHHOHPHQWGHIOHFWLRQDWWKHWUDQVLWLRQSRLQW7KH FRQGLWLRQVUHDG
[
D
Y ' Y ' Y
[
[W
OLP [8 [ W
[
D
F
F
F
F
Y Y '& G[
G
F
Y Y Y ' G[ Y
G
Y ' G[
G
F
OLP [9 [ W
F
Y Y Y G[
G
F
F
2QFHH[SUHVVLRQVIRUWKHGHIOHFWLRQYS YYFDQGWKHWUDQVLWLRQSRLQW[WKDYHEHHQ HVWDEOLVKHG WKH IRUFH DQG PRPHQW FDQ EH DVVHVVHG E\ LQWHJUDWLRQ DORQJ WKH FRQWDFWOHQJWKRIT\RYHUWKHVOLGLQJUDQJHDQGRIFSYSRYHUWKHDGKHVLRQUDQJH 7KHUHVXOWLQJIRUPXODHFDQEHIRXQGLQWKHRULJLQDOSXEOLFDWLRQ:HZLOOUHVWULFW RXUVHOYHVKHUHZLWKVKRZLQJWKHFKDUDFWHULVWLFVIRUWKHYDOXHV FF
DDQGFS
)LJXUH GHSLFWVWKHGHIOHFWHGPRGHODWDQXPEHURIVOLSDQJOHV,Q
)LJWKHIRUFHDQGPRPHQWFKDUDFWHULVWLFVKDYHEHHQSUHVHQWHGZLWKDORQJWKH DEVFLVVDWKHTXDQWLW\
F
'
FD )
F
7KHFRPSRVLWHPRGHOSDUDPHWHU
F
LVGHILQHGDV
F
]
)RUWUDQVLHQWW\UHDQDO\VLVWKHUHOD[DWLRQOHQJWKIRUWKHOHDGLQJHGJH IRUWKH EDUHVWULQJPRGHODQG
IRUWKHPRGHOZLWKWUHDGHOHPHQWVKHUHGHILQHGDVWKH
GLVWDQFHEHWZHHQOHDGLQJHGJHDQGWKHSRLQWRILQWHUVHFWLRQRIZKHHOSODQHDQG HORQJDWLRQRIWKHVWUDLJKWFRQWDFWOLQH )LJV LVRILPSRUWDQFH:H KDYH
12167($'<67$7(675,1*%$6('7<5(02'(/6
9 α
[
α stretched string
σ*
sliding range
D
tread elements
D
σ
\ )LJ
7KHHQKDQFHGVWULQJW\UHPRGHODWLQFUHDVLQJVOLSDQJOHVVKRZLQJJURZLQJVOLGLQJ UDQJHDQGGHFUHDVLQJµLQWHUVHFWLRQ¶OHQJWK
(
'
(
Y
7KHYDULDWLRQRIWKLVµLQWHUVHFWLRQOHQJWK¶
(
ZLWKVOLSDQJOHKDVEHHQSORWWHGDV
ZHOO7KHSDUDPHWHUVKDYHEHHQFKRVHQVXFKWKDWDW
ZHKDYH
7KHGLDJUDPDOVRFRQWDLQVWKHFXUYHIRUWKHSQHXPDWLFWUDLOW
WKHYDOXHRIWKHSQHXPDWLFWUDLOEHFRPHVW
(
D
0])\$W
DZKLFKLVPXFKPRUHLQ
DFFRUGDQFHZLWKH[SHULPHQWDOILQGLQJVWKDQ ZKDW ZHKDGZLWKWKHEDUHVWULQJ
σ =3.75DFF 55 S F
1
)\ µ)]
0.5
0
σ* 3D
-100] Dµ)]
W D 0
0.02
α θF
0.04
)LJ6WHDG\VWDWHVLGHVOLSFKDUDFWHULVWLFVRIWKHVWULQJPRGHOZLWKWUHDGHOHPHQWV
12167($'<67$7(675,1*%$6('7<5(02'(/6
PRGHO7KHYDOXH DDOVRVXSSRUWVWKHPDJQLWXGHWKDWKDGEHHQIRXQGIRUWKH UHOD[DWLRQOHQJWKIRUWKHIRUFHWRVLGHVOLS
(
)
D W
DDVLQGLFDWHGLQ
7DEOHFI(T ([SUHVVLRQV IRU WKH VOLS VWLIIQHVVHV WKH SQHXPDWLF WUDLO DQG WKH UHOD[DWLRQ OHQJWK DQG WKHLU UHODWLRQVKLS ZLWK YHUWLFDO ORDG ZLOO EH SUHVHQWHG LQ WKH VXEVHTXHQWVXEVHFWLRQZKHUHWKHOLQHDUPRGHOLVGLVFXVVHG
1RQVWHDG\VWDWHEHKDYLRXUDWYDQLVKLQJVOLGLQJUDQJH OLQHDUDQDO\VLV
:HZLOOUHVWULFWRXUVHOYHVWRWKHGHULYDWLRQRIWKHUHVSRQVHWRWKHVOLSDQJOH DQG XVH FRQYHUVLRQ IRUPXOD ZLWK
(
UHSODFHG E\
WR REWDLQ WKH
nDQG IRUWKHUHVSRQVHVWR\DQG
UHVSRQVHIXQFWLRQVWRWXUQVOLS
)RU
WKHWRWDOGHIOHFWLRQYRUWUDQVIRUPHG 9(TV DQGWKXV KROGLQ JHQHUDO:LWK$UHSUHVHQWLQJWKHWUDQVIRUPHG
9' & H % $' & H %
S[
Y
S[
$
S
S
ZHJHW
$
:LWKWKH/DSODFHWUDQVIRUPHGYHUVLRQRI(T WKHGLIIHUHQWLDOHTXDWLRQ IRU9FLVREWDLQHG
9 & 9 ' & & G[
G
F
F
& H % S $
S[
F
$
ZLWKVROXWLRQ
9 ' &% H
&
D
[
F
F
& && H &
&
D
[
% &
F
& H % & S S[
S$
$
F
7KH WKUHH LQWHJUDWLRQ FRQVWDQWV FDQ EH GHWHUPLQHG E\ XVLQJ WKH ERXQGDU\ FRQGLWLRQVDWWKHOHDGLQJDQGWUDLOLQJHGJHV
& '
% S & H % S
ò
SD
$
F
& '& $
% %%% %& %
ZLWK
% ' % S
%% % %& H % & S H & % S & S & S SD
F
% '
K
H
F
SD
F
D
F
12167($'<67$7(675,1*%$6('7<5(02'(/6
,QWHJUDWLRQRI9S
99FWLPHVWKHVWLIIQHVVFSRYHUWKHFRQWDFWOHQJWK\LHOGVWKH
WUDQVIRUPHG H[SUHVVLRQV IRU WKH IRUFH )\ DQG DIWHU PXOWLSOLFDWLRQ ZLWK [ WKH PRPHQW0 ]7KHIROORZLQJIRUPXODHDUHREWDLQHGIRUWKHLUWUDQVIHUIXQFWLRQVWR
WKHVOLSDQJOH
+
)
S 'F
S
S
& D % & S H &H S & S
0
S 'F
SD
& % $
&H
& S S & S
S
SD
F
%
+
D
F
&% %
&H
&
D
& & D H %H & H &H S S
SD
SD
SD
&&
F
SD
& % $
F
%
D % H
%
D % H&
6 LQ WKH H[SUHVVLRQV IRU
D
(
%\ OHWWLQJ S
F
D
%
F
&
&H
F
F
D
& H&
&% %
F
D
9$ [
&&
F
D DQG WKH WUDQVIHU
IXQFWLRQV WKH UHOD[DWLRQ OHQJWK DQG WKH VOLS VWLIIQHVVHV PD\ EH DVVHVVHG:HILQG
% H % & H& & & D ' % H % & H& % & % & ' F D (% D % & ( % H % & H& & % D
(
D
F
D
F
D
F
F
)
S
D
D
F
F
%
& ' F 0
& &
S
&
(
(
D
% H&
F
D
F
&
D %
% &
& &
& n' & )
H
0
DQGWKHSQHXPDWLFWUDLO
&
&
D %H
D
D %H&
D
F
% F
&
F
&H
F
D
&H&
%
F
D
F
12167($'<67$7(675,1*%$6('7<5(02'(/6
& &
W'
0
)
,WLVRILQWHUHVWWRH[DPLQHKRZWKHVHTXDQWLWLHVYDU\ZLWKYHUWLFDOORDG)])RU WKLVSXUSRVHZHDVVXPHWKDWWKHFRQWDFWOHQJWKYDULHVZLWKWKHVTXDUHURRWRIWKH UDGLDOW\UHFRPSUHVVLRQDQGWKDWWKHORDGLVDOLQHDUIXQFWLRQRIWKHFRPSUHVVLRQ 7KHQZHILQGWKDWLIDWWKHQRPLQDORUVWDWLFORDG)]RKDOIWKHFRQWDFWOHQJWKLV GHQRWHGZLWKDRWKH)]YV
D ' D R
) )
UHODWLRQVKLSLVGHVFULEHGE\
]
]R
7KLVLQWURGXFHGLQWKHDERYHH[SUHVVLRQV\LHOGVWKHFKDUDFWHULVWLFVVKRZQLQWKH GLDJUDPRI)LJ7KHYDULRXVQRQGLPHQVLRQDOTXDQWLWLHVKDYHEHHQSUHVHQWHG ERWKIRUWKHEDUHVWULQJPRGHODQGIRUWKHHQKDQFHGPRGHO$VKDVEHHQH[SHFWHG ZHQRZVHHWKDWWKHUHOD[DWLRQOHQJWKIRUWKHIRUFHDQGPRPHQWUHVSRQVHVWR\DZ
(
D YDQLVKHVZKHQWKHORDGJRHVWR]HUR7KHSQHXPDWLFWUDLO¶VUDWLRWRKDOIWKH
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
D
$Q LPSRUWDQW UHVXOW RI WKH DQDO\VLV LV WKDW H[SUHVVLRQV
IRU UHOD[DWLRQ OHQJWKV DQG FXWRII IUHTXHQFLHV DUH WKH
F
VDPHIRUWKHEDUHVWULQJPRGHODQGWKHHQKDQFHGPRGHOLILQWKHVHH[SUHVVLRQV LVUHSODFHGE\
(
GHILQHGE\
)RU FRPSDULVRQ WKH SORWV IRU WKH VWHS UHVSRQVHV DQG IRU WKH IUHTXHQF\ UHVSRQVHIXQFWLRQVKDYHEHHQSUHVHQWHGLQWKH)LJVWRJHWKHUZLWKUHVXOWV RIRWKHUPRGHOV$OVR7DEOH EHORZ)LJ FRQWDLQVGDWDFRPSXWHGZLWK WKHQHZPRGHO7KHJHQHUDOFRXUVHRIWKHFXUYHVLVVLPLODU,PSRUWDQWGLIIHUHQFHV ZLWK UHVSHFW WR WKH EDUH VWULQJ PRGHO DUH WKH FRQVLGHUDEOH VKLIW RI WKH FXWRII IUHTXHQF\RIWKH0 ]UHVSRQVHWR
DQGVLPLODUO\ RI ) \ WR
nDQGWRJHWKHUZLWK
WKHVH WKHPHDQGHULQJIUHTXHQF\ZKHUHWKHGLSRIWKH0 ]UHVSRQVHWR
RFFXUV
7KHUHGXFWLRQRIWKHSQHXPDWLFWUDLOWLVUHVSRQVDEOHIRUWKHLQFUHDVHRIWKHFXW RIIIUHTXHQF\
12167($'<67$7(675,1*%$6('7<5(02'(/6
σ =3.0DFF 0 with tread elements σ =3.75DFF 55 bare string
5
SF SF
4
(σ * D) D0
3
σ*D0
2 DD
0
1 W D
0 0
1
)]] ) 0
2
)LJ 7KHYDULDWLRQRIWKHUHOD[DWLRQOHQJWKDQGKDOIWKHFRQWDFWOHQJWKDVDUDWLRWRWKH VWDWLFYDOXHRIWKHODWWHUDQGWKHUDWLRRIWKHSQHXPDWLFWUDLOWR KDOIWKHFRQWDFW OHQJWKZLWKQRUPDOLVHGYHUWLFDOORDGERWKIRUWKHEDUHVWULQJPRGHODQGIRUWKH PRGHOZLWKWUHDGHOHPHQWV
σ =3.0DFF 0 with tread elements σ =3.75DFF 55 bare string
25
SF SF
20
2 (2FD F 0)
&
)α
15 3 F 0) 0α (2FD
10
&
5 0 0
1
)]] ) 0
2
)LJ 7KH YDULDWLRQ RI QRQGLPHQVLRQDOLVHG FRUQHULQJ DQG DOLJQLQJ VWLIIQHVVHV ZLWK QRUPDOLVHGYHUWLFDOORDGERWKIRUWKHEDUHVWULQJPRGHODQGWKHPRGHOZLWK WUHDG HOHPHQWV
12167($'<67$7(675,1*%$6('7<5(02'(/6
([SHULPHQWDO UHVXOWV DVVHVVHG RQ D P GLDPHWHU VWHHO GUXP KDYH EHHQ SUHVHQWHGLQ)LJ7KHUHVSRQVHVDUHGHSLFWHGLQWKH1\TXLVWGLDJUDP FI )LJV DVDUDWLRWRWKHLUYDOXHVDVVHVVHGDWVWHDG\VWDWHVLGHVOLS7ZR OLJKW WUXFN W\UHV KDYH EHHQ WHVWHG DW D YHUWLFDO ORDG RI )]
N1 %RWK W\UHV
SRVVHVVWKHVDPHFURVVSO\FRQVWUXFWLRQEXWH[KLELWDGLIIHUHQWWUHDGGHVLJQ7KH RULJLQDOW\UHVKRZVDEORFNWUHDGGHVLJQ FHQWUDOULEZLWKWUDQVYHUVHEORFNVRYDO IRRWSULQW ZKLFKLVUHODWLYHO\ZHDNLQWKHORQJLWXGLQDOGLUHFWLRQDWWKHVLGHVRI WKHFRQWDFWSDWFK7KHPRGLILHGW\UHLVSURYLGHGZLWKFLUFXPIHUHQWLDOULEV WKUHH ULEV DOPRVW UHFWDQJXODU IRRW SULQW PDNLQJ WKH WUHDG UHODWLYHO\ VWLII LQ WKH ORQJLWXGLQDO GLUHFWLRQ $V H[SHFWHG WKH ULEEHG W\UH JHQHUDWHV PRUH UHVLVWDQFH
DJDLQVWWXUQLQJZKLFKLVUHIOHFWHGE\DODUJHUPRPHQWGXHWRWUHDGZLGWK0 ]DQG FRQVHTXHQWO\ D FRQVLGHUDEO\ VPDOOHU SKDVH ODJ RI WKH UHVXOWLQJ VHOIDOLJQLQJ WRUTXH0]7KHVLGHIRUFHGRHVQRWDSSHDUWREHPXFKDIIHFWHGDWOHDVWLQWKH ODUJHU ZDYHOHQJWK UDQJH ,Q 7DEOH SDUDPHWHUV HVWLPDWHG IURP WKH PHDVXUHPHQWVKDYHEHHQOLVWHG7KHLQWHUVHFWLRQOHQJWKRUGHIOHFWLRQUHOD[DWLRQ
(
OHQJWK
(
'
ZDVGHWHUPLQHGIURPWKHIRUPXOD
&) &D &)\
:KHQFRQVLGHULQJWKHUHODWLRQD
DQ9ZLWK9NPK 9PVWKHYDOXHIRUWKHQRQ
GLPHQVLRQDOSDWKIUHTXHQF\DW9
NPKDQGDQH[FLWDWLRQIUHTXHQF\RI+]
EHFRPHVD
ZKLFKURXJKO\FRUUHVSRQGVZLWKWKDWLQGLFDWHGRQWKHWKLQ
W\UHPRGHOFXUYHRI)LJ
7DEOH3DUDPHWHUVIRUWZR W\UHVZLWKGLIIHUHQWWUHDGSDWWHUQV
EORFNV
ULEV
UR
P
P
D
P
E
W
(
EORFNV
ULEV
)]
N1
N1
P
&)
N1UDG
N1UDG
P
P
&0
N1PUDG
N1PUDG
D
D
&)\
N1P
N1P
D
D
&0 D
&0 D
(
12167($'<67$7(675,1*%$6('7<5(02'(/6
)\
α )α
&
)\ &
)\
1 0.4
_
α
) 1
)\
\
α
)
0.1
)\
ϕ )ϕ
&
0.04
)\ &
ψ
)
ψ
0.01
)α
0.004
D 2π σ *+D
0.001 0.01
0.02
ϕ
D π o W σ *+D
)
0.04
0.08 0.1
Dλ
0.2
0.4
0.8 1
2
0.8 1
2
1.5 )\
α )α
&
)\ &
ϕ
)ϕ
)\ &
)\ )\
&
_
\
ψ
0.5
)α
0 90 0 -90
φ
)\
)\
1
[o] -180
)α 1
)α
)ϕ
)α
)α 1
)ψ
)ψ
)\
)\
)α
)α 1 )α )ψ
)ψ )ϕ
)ϕ
-270 -360 0.01 0
)ϕ
0.050.020.1
0.04
0.08 0.1
λ
0.2
0.4
D
)LJ )UHTXHQF\UHVSRQVHIXQFWLRQVRIWKHVWULQJPRGHOZLWKWUHDGHOHPHQWVIRUWKHVLGH IRUFHZLWKUHVSHFWWRYDULRXVPRWLRQLQSXWYDULDEOHV)RUWKHUHVSRQVHWR DILUVW RUGHUDSSUR[LPDWLRQZLWKWKHVDPHFXWRIIIUHTXHQF\
V
(
D
KDVEHHQDGGHG
) 7KUHHZD\VRISUHVHQWDWLRQKDYHEHHQXVHG ORJORJOLQORJDQGOLQOLQ7KH PRGHOSDUDPHWHUVDUH
DFSFF
ZKLFKOHDGVWR
(
DW
D
12167($'<67$7(675,1*%$6('7<5(02'(/6
]
0
α
0α
&
1
ϕ
0.4
]_
D0
0
\
0α
&
ψ
α
0
0.1
]
0
0
ϕ
0α
D&
0
* ψ
0 \
0.04
]
0
0.01
ψ
0α
&
0.004
*
π
]
D0
ψ
0ϕ
&
0.001 0.01
0.02
D o W σ *+D
0.04
0.08 0.1
Dλ
0.2
0.4
0.8 1
2
0.8 1
2
1.5
]
]_
0
D0
α
0α
&
]
0α \
&
]
0
ψ
0α
&
*
]
D0
ψ
0.5
*ψ ϕ
0
ϕ
0 \
0 \
0 90
ψ
ψ
0
0
ϕ
0
0 -90
φ
0
0
0
0ϕ
&
α
α
0
0
ϕ
0α
D&
1
[o] -180
0
* ψ ϕ
α
0
0
α
ψ
0
0
0 \
ψ
0
-270 -360 0.01 0
0.050.020.1
0.04
0.08 0.1
λ
0.2
0.4
D
)LJ )UHTXHQF\UHVSRQVHIXQFWLRQVRIVWULQJPRGHOZLWKWUHDGHOHPHQWVIRUWKHDOLJQLQJ WRUTXH ZLWK UHVSHFW WR YDULRXV PRWLRQ LQSXW YDULDEOHV 6DPH FRQGLWLRQV DV LQ
)LJ7KHUHVSRQVHIXQFWLRQVRIWKHPRPHQWGXHWRWUHDGZLGWK 0 ]KDYHEHHQ DGGHG
12167($'<67$7(675,1*%$6('7<5(02'(/6
Q = 1 Hz
Im
ψo = 0.75 o rib
block 0
0
0]
0]R
10
1 80 60 80 40 30 60 20 40
0
30
)\
80 60 80 40 60 40 30 30 20 20
)\R 10
20
10
10
[km/h]
9
)LJ )UHTXHQF\UHVSRQVHSORWVRI)\DQG0]WR
Re
1
0
[km/h]
9
IRUWZROLJKWWUXFNW\UHVZLWKVDPHELDV
SO\FDUFDVVEXWGLIIHUHQWWUHDGSDWWHUQV0HDVXUHPHQWVZHUHSHUIRUPHGRQDP
E
GUXPDWDYHUWLFDOORDGRIN1D\DZDPSOLWXGHRI DWDIUHTXHQF\Q
+]
7\UH,QHUWLD(IIHFWV
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
V
LQGLFDWHG
E\WKHVXEVFULSWR :HREVHUYHWKDWIRUKLJKHUIUHTXHQFLHVWKHFXUYHVRIWKHPRPHQWUHVSRQVH DSSHDUWRVKLIWXSZDUGV7KHIRUFHUHVSRQVHFXUYHVRQWKHRWKHUKDQGVKRZDQ LQFUHDVHLQDPSOLWXGHZKLOHWKHSKDVHODJUHPDLQVDSSUR[LPDWHO\XQFKDQJHGIRU FRQVWDQWZDYHOHQJWK
9Q
12167($'<67$7(675,1*%$6('7<5(02'(/6
Im
1 Hz 2.16 Hz 4 Hz 8 Hz ψo = 0.75o
10
0 100
30
0
40
0 10
0
]
20 30
1
0
80 80 60 60 60 40 30 40
80
]R
60
40 20
30
20 10 [km/h]
9
)LJ
0 1 Re 80 60 \ 40 \R 80 30 60 20 10 40 10 30 30 80 10 40 20 20 40 60 60 80 100
0
)UHTXHQF\ UHVSRQVH SORWV RI )\ DQG 0] WR
)
)
IRU OLJKW WUXFN W\UH ZLWK EORFNWUHDG
SDWWHUQ0HDVXUHPHQWVZHUHSHUIRUPHGRQDPGUXPDWDYHUWLFDOORDGRIN1
E
D\DZDPSOLWXGHRI DQGDWIRXUGLIIHUHQWWLPHIUHTXHQFLHV
)LUVW$SSUR[LPDWLRQRI'\QDPLF,QIOXHQFH *\URVFRSLF&RXSOH
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
] J\U ' &J\U 9
0
G)\ GW
12167($'<67$7(675,1*%$6('7<5(02'(/6
Ω -γW U
0
)
Y0
]J\U
\
)LJ7KHDYHUDJHWLOWDQJOHRIWKHSHULSKHUDOOLQHRIWKHGHIOHFWHGW\UH
LQ ZKLFK W GHQRWHV WKH WLPH 7KH W\UH FRQVWDQW &J\U LV SURSRUWLRQDO WR WKH W\UH PDVVPWDQGLQYHUVHO\SURSRUWLRQDOWRWKHODWHUDOVWLIIQHVV& )\ RI WKH VWDQGLQJ W\UH:HKDYHZLWKWKHQRQGLPHQVLRQDOTXDQWLW\FJ\ULQWURGXFHG
J\U '
&
P F
J\U
&
W
)\
7KHWRWDOPRPHQWZKLFKDFWVXSRQWKHZKHHOLVQRZFRPSRVHGRIWKHPRPHQW GXHWRODWHUDOW\UHGHIRUPDWLRQ0DQGWKDWGXHWRORQJLWXGLQDODQWLV\PPHWULF
]
WUHDGGHIRUPDWLRQV0 ]DQGILQDOO\WKHJ\URVFRSLFFRXSOH0]J\U 0
]'
( ] % 0 ] % 0]J\U
0
:LWKWKLVVLPSOHDGGLWLRQWRWKHHQKDQFHGVWUHWFKHGVWULQJW\UHPRGHO SURYLGHG ZLWK WUHDG HOHPHQWV GLVWULEXWHG RYHU WKH DVVXPHGO\ UHFWDQJXODU FRQWDFW DUHD FDOFXODWLRQV KDYH EHHQ SHUIRUPHG )LJXUH VKRZV WKH 1\TXLVW SORW RI WKH UHVSRQVHVZLWKDORQJWKHFXUYHV
V D QD 9 D ,WVKRXOGEHQRWHGWKDW
WKHYHFWRURIWKHJ\URVFRSLFFRXSOHLVGLUHFWHGSHUSHQGLFXODUO\WRWKHYHFWRURI WKHODWHUDOIRUFH)RUWKHUDQJHRIIUHTXHQFLHVFRQVLGHUHGWKHVLPSOHDSSUR[LPDWH G\QDPLFH[WHQVLRQWKHRU\JLYHVDJRRGUHSUHVHQWDWLRQRIWKHHIIHFWRQWKHPRPHQW UHVSRQVHFXUYHV+RZHYHUWKHWKHRU\GRHVQRWDFFRXQWIRUWKHG\QDPLFFKDQJHV LQWKHIRUFHUHVSRQVH
6HFRQG$SSUR[LPDWLRQRI'\QDPLF,QIOXHQFH )LUVW+DUPRQLF
0RUHFRPSOH[WKHRULHVDVGHYHORSHGE\3DFHMND DQGE\6WUDFNHUMDQ DQGPRUHUHFHQWO\E\0DXULFH FI&KDSWHU \LHOGEHWWHUUHVXOWV DOVRZLWKUHVSHFWWRWKHIRUFHUHVSRQVHDQGH[WHQGVWRKLJKHUIUHTXHQFLHVFRYHULQJ
12167($'<67$7(675,1*%$6('7<5(02'(/6
J\U
κ =0.25&0α
1 Hz 2 Hz 4 Hz 8 Hz
F =0
κ =0
J\U
140
1.0 28
9 [km/h]
0
] ]R
42 0
0
0J\U
ψψ R
0
0.75
] 0.5
F = 0.12
210 km/h
Im
0 0
83 60
]
0.15 0.35 0.25 42 30 21 15 10.4 7.5
0
)
0.05 0.1
1 Re 0.05
\
ωD
)
1.0
\R
0.1
V
0.75 0.15 0.5
9 [km/h]
0.35
ωD V
0.25
)LJ 7KHLQIOXHQFHRIWKHIUHTXHQF\RIWKH\DZRVFLOODWLRQRQWKHG\QDPLFUHVSRQVHDFFRUG LQJWRWKHVLPSOHDSSUR[LPDWHPRGHOH[WHQVLRQXVLQJWKHJ\URVFRSLFFRXSOHGXHWRWKH ODWHUDOW\UHGHIOHFWLRQWLPHUDWHRIFKDQJH
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
2
ZKHHOSODQHGHILQHGE\\DQG
n DQHIIHFWLYHLQSXWLVLQWURGXFHGWKDW
RU DQG
LV GHILQHG DV WKH ODWHUDO DQG WKH \DZ PRWLRQ RI WKH OLQH RI LQWHUVHFWLRQ RI WKH
2
¶G\QDPLF¶ EHOW SODQH DQG WKH URDG VXUIDFH \H
H )LJXUH GHSLFWV WKH
VLWXDWLRQ7KHG\QDPLFEHOWSODQHLVGHILQHGDVWKHYLUWXDOFHQWUHSODQHRIWKHEHOW RIWKHW\UHWKDWLVGLVSODFHGZLWKUHVSHFWWRWKHULPWKURXJKWKHDFWLRQRIODWHUDO LQHUWLDOIRUFHVRQO\7KHVHLQHUWLDOIRUFHVGLVWULEXWHGDORQJWKHFLUFXPIHUHQFHRI WKHEHOWDUHDSSUR[LPDWHGLQVXFKDZD\WKDWRQO\WKHIRUFHVUHVXOWLQJIURPWKH ]HURWKDQGILUVWPRGHVRIYLEUDWLRQDUHWDNHQLQWRDFFRXQWWKDWLV DVLIWKHEHOW ZRXOGUHPDLQFLUFXODU+DUPRQLFVRIVHFRQGDQGKLJKHURUGHUDUHQHJOHFWHGLQ
12167($'<67$7(675,1*%$6('7<5(02'(/6
γ~W wheel plane
dynamic deflection
Ω
dynamic belt plane ~
\
W
U
ψH ψ [
\H \
~
\U W
0]
effective input \
γ~W
ψ~W
)\
[
kinematic deflection \
]
)LJ µ'\QDPLF¶DYHUDJHSODQHGHIOHFWLRQRIEHOWDQGDVVRFLDWHGHIIHFWLYHLQSXW
WKHG\QDPLFSDUWRIWKHPRGHO:HKDYHIRUWKHDSSUR[LPDWHG\QDPLFODWHUDO FDPEHUDQG\DZGHIOHFWLRQVUHVSHFWLYHO\ × \
W'
&
×
W'
& ^ WW %
×
W'
&^
P
@
W \2
,
,
% ,
@W & \
\
WS
0
% 0 W `
W % W & , WS
&
0 `& W
ZLWKWKHWRWDOW\UHGHIOHFWLRQVFRPSRVHGRIG\QDPLFDQGVWDWLFSDUWV WKHODWWHU UHVXOWLQJIURPH[WHUQDOJURXQGIRUFHV \
W'
× \
W % )\ &\
W'
×
W&
W'
×
) U &
\
W % 0 ] &
DQG VWLIIQHVVHV RI WKH ZKROH EHOW ZLWK UHVSHFW WR WKH ULP &\ & DQG & )XUWKHUPRUH WKH HTXDWLRQV FRQWDLQ HIIHFWLYH W\UH LQHUWLD SDUDPHWHUV PW ,W DQG
12167($'<67$7(675,1*%$6('7<5(02'(/6
SRODU PRPHQW RI LQHUWLD ,WS 7KH HIIHFWLYH LQSXWV WR WKH NLQHPDWLF PRGHO DUH GHILQHGE\
2H ' \2 % ×\W & U × W
\
H'
%
×
W
:LWK WKH DLG RI WKHVH HTXDWLRQV SOXV WKH NLQHPDWLF W\UH HTXDWLRQV HJ 9RQ 6FKOLSSH¶VDSSUR[LPDWHHTXDWLRQV ZLWK DQG IRU
2
2
0 ] ZLWK LQSXWV \
UHSODFHG E\ HIIHFWLYH LQSXWV \H
UHVSRQVHIXQFWLRQVRIWKHJURXQGIRUFH)\DQGPRPHQW0]
2
H
WKH IUHTXHQF\
0] 0 ]ZLWKUHVSHFW
PD\EHUHDGLO\HVWDEOLVKHG
WRWKHDFWXDOLQSXWV \
)RUDSSOLFDWLRQVLQYLEUDWRU\SUREOHPVDVVRFLDWHGZLWKZKHHOVXVSHQVLRQDQG VWHHULQJV\VWHPVLWLVRIJUHDWHULQWHUHVWWRNQRZWKHUHVSRQVHRIWKHHTXLYDOHQW IRUFHDQGPRPHQWWKDWZRXOGDFWRQWKHDVVXPHGO\ULJLGZKHHOZLWKW\UHLQHUWLD LQFOXGHGDWWKHFRQWDFWFHQWUH7KHHTXLYDOHQWIRUFHDQGPRPHQWDUHREWDLQHGDV IROORZV )
\ HT
'
)
] HT '
0
\
&
0
]
W @W
P \
%
] &
(
0
,
W
W%
,
WS
0
W
7KLVUHVXOWVKRXOGFRUUHVSRQGWRWKHIRUFHDQGPRPHQWPHDVXUHGLQWKHZKHHO KXE)\KXEDQG0]KXEDIWHUWKHVHKDYHEHHQFRUUHFWHGIRUWKHLQHUWLDOIRUFHDQG PRPHQWDFWLQJRQWKHDVVXPHGO\ULJLGZKHHOSOXVW\UH LQHUWLDEHWZHHQORDGFHOOV DQGJURXQGVXUIDFH :LWKLQHUWLDSDUDPHWHUVRIZKHHOSOXVW\UHZHILQGIRUWKLV FRUUHFWLRQ )
\ HT
0
'
] HT '
)
\ KXE
%
P
] KXE %
0
ZW
@2
\
,
ZW
:KHQOLIWHGIURPWKHJURXQGWKHHTXLYDOHQWJURXQGWRW\UHIRUFHVYDQLVKLIWKH IUHTXHQF\ RI H[FLWDWLRQ LV QRW WRR KLJK ORZ ZLWK UHVSHFW WR WKH ILUVW QDWXUDO IUHTXHQF\RIWKHW\UH )RU WKH VHW RI SDUDPHWHU YDOXHV PHDVXUHG RQ D UDGLDO SO\ VWHHOEHOWHG W\UH OLVWHG LQ 7DEOH WKH DPSOLWXGH DQG SKDVH RI WKH HTXLYDOHQW PRPHQW DV D UHVSRQVHWR\DZDQJOHKDYHEHHQFRPSXWHGDQGDUHSUHVHQWHGLQ)LJDVD IXQFWLRQRIWKHH[FLWDWLRQIUHTXHQF\Q
([SHULPHQWDOO\REWDLQHGFXUYHV
XSWRDPD[LPXPRI+] RQPGUXP KDYHEHHQDGGHG7KHSKDVHODJWKDW RFFXUVLQWKHORZHUIUHTXHQF\UDQJHLVODUJHO\UHVSRQVLEOHIRUWKHFUHDWLRQRIVHOI
12167($'<67$7(675,1*%$6('7<5(02'(/6
QG
7DEOH3DUDPHWHUVIRUFDUW\UHPRGHOZLWKG\QDPLFEHOWH[WHQVLRQ
%LDVSO\W\UH
D
P
P U
5DGLDOSO\VWHHOEHOWHG
1P
D
P
&\
N1P
P
&\
N1P
P
PW
NJ
U
P
PW
NJ
&)
N1
,W
NJP
&)
N1
,W
NJP
&0
N1P
,WS
NJP
&0
N1P
,WS
NJP
PW\UH
NJ
1P
&
DSSUR[LPDWLRQ
&
1P
&
&
PW\UH
1P
NJ
H[FLWHG ZKHHO VKLPP\ 7KH LQIOXHQFH RI W\UH LQHUWLD LV HYLGHQW WKH SKDVH ODJ FKDQJHV LQWR SKDVH OHDG DW KLJKHU YDOXHV RI VSHHG VLPLODU DV LQ )LJ ,Q FRQWUDVW WR WKH G\QDPLF UHVSRQVH GHSLFWHG LQ )LJ WKH NLQHPDWLF PRGHO UHVSRQVH ZLWKRXWWKHLQFOXVLRQRIW\UHLQHUWLD ZRXOGSURGXFHWKHVDPHUHVSRQVH DWHTXDOZDYHOHQJWKV
9QLUUHVSHFWLYHRIWKHYDOXHRIWKHVSHHGRIWUDYHO9
7KHUHVRQDQFHIUHTXHQFLHVZKLFKDWVSHHGVFORVHWR]HURDSSURDFKDYDOXH VRPHZKDW ODUJHU WKDQ WKH QDWXUDO IUHTXHQF\ RI WKH IUHH QRQURWDWLQJ W\UH QRR DSSHDUWRVHSDUDWHDWLQFUHDVLQJVSHHG9LEUDWLRQH[SHULPHQWVSHUIRUPHGRQD IUHH W\UH QRW FRQWDFWLQJ WKH URDG IRU WKH SXUSRVH RI REWDLQLQJ WKH YDOXHV RI FHUWDLQW\UHSDUDPHWHUVJDYHDFOHDULQGLFDWLRQWKDWDFRQVLGHUDEOHUHGXFWLRQRI WKHORZHVWUHVRQDQFHIUHTXHQF\DULVHVZKHQWKHZKHHOLVURWDWHGDURXQGWKHZKHHO VSLQD[LVDWKLJKHUVSHHGVRIUHYROXWLRQ 1\TXLVWSORWVIRUWKH0]HTDQG)\HTUHVSRQVHVWR
KDYHEHHQ SUHVHQWHGLQ
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
12167($'<67$7(675,1*%$6('7<5(02'(/6
experiments 40 5 20 10
10 1 0 0 90
0]HT
0αψ
&
20
60
100
5
5 Hz 5 40
5
20 40
0
10 20m/s
50 40m/s 60m/s
5
10
20
1 0
0
180
φ0ψ
10
10
20
30
Qoo
Q
40
[Hz]
0
20 90
5
40
[o] 60
0 10m/s
)LJ $PSOLWXGH DQG SKDVH UHVSRQVH RI HTXLYDOHQW DOLJQLQJ WRUTXH DV LI DFWLQJIURP JURXQG RQ ULJLG ZKHHOW\UH WR \DZ DQJOH DV D IXQFWLRQ RI WLPH IUHTXHQF\ RI
0
H[FLWDWLRQ>+]@FRPSXWHGXVLQJ9RQ6FKOLSSHDSSUR[LPDWLRQDQGH[DFW
]H[WHQGHG
ZLWKG\QDPLFWKHRU\DFFRUGLQJWRVHFRQGDSSUR[LPDWLRQZLWKG\QDPLFEHOW0RGHO SDUDPHWHUVDUHOLVWHGLQ7DEOH UDGLDOVWHHOEHOWHGW\UH &XUYHVKROGIRUJLYHQ VSHHGRIWUDYHO 9 >PV@([SHULPHQWDOUHVXOWVDUHJLYHQXSWRDIUHTXHQF\RI+] 7KH\DZQDWXUDOIUHTXHQF\RIWKHIUHHQRQURWDWLQJW\UHZLWKIL[HGZKHHOVSLQGOHLV
QRR
0]HT
ψ
&
0α
Ω =ω
5
50
Qoo
1 0
0
10
0
20
20 40
180
φ0ψ
30 60
40 80
Q [Hz] 9 [m/s]
0 100
90
[o] 0
)LJ $PSOLWXGHDQGSKDVHUHVSRQVHDVDIXQFWLRQRIIUHTXHQF\ZLWKWKHIUHTXHQF\RI H[FLWDWLRQHTXDOWRWKDWRIZKHHOUHYROXWLRQ
12167($'<67$7(675,1*%$6('7<5(02'(/6
bias-ply
44
8
0]HT ]HT
16
24
16 16 8 16 8 4 2
20
2
16
2
8
Im
Re )\HT &
1
9 [m/s]
1
ψ
16
-1
)\HT
4
2
)α
Re
16 8
8
Re
2
9 [m/s]
4
1 16 64 8 32 [m/s] 128 4 64 1 2 32 64 16 32 16 16 64 32
1.8
16
8
9 [m/s]
0.1
24
16
11
0
32
32
1 4 8 1.8
0.2
32
0α 24
16
Re
32
ψ
&
0.1
9
-1
40
0]HT
1
0α
32 32 32
Im
0.2
ψ
&
1 8
64
0.5
64
64 32
0
Im
50
0.5
1
5
8
4
Im
46
radial-ply
WKHRU\
&
ψ
1 Hz 4 Hz 8 Hz 12 Hz 16 Hz
)α
64
32
24 24 64 128 32 64 96 85
128 200 )LJ 7KHRUHWLFDOIUHTXHQF\UHVSRQVH1\TXLVWSORWVRIVLGHIRUFHDQGDOLJQLQJWRUTXHWR \DZDQJOHIRUGLIIHUHQWWLPHIUHTXHQFLHV6DPHFRQGLWLRQVDVLQ)LJ
H[SHULPHQW
bias-ply Im
radial-ply
Im 50
50
40
0]HT
ψ
40
5
40 30
10
5
10
5
2.5
0 30
30 20 10
15 20 15
0
30
0
9 [m/s]
2.5
10
10 15
20 10
20
30
50 40 30 20 50 15 40 30 5
10
20
15
10
2.5
5
Im
Re
5
0
9 [m/s]
20
15
0
30 20
0
Re
9 [m/s]
Re 30
)\HT
20
ψ
9 [m/s]
10
1.1 Hz 4.0 Hz 7.8 Hz
0
30
5 10
0
Re
Im
5
40
50
2.5
10
5 30
15
10 20
15
40 50
20
30 40 50
)LJ ([SHULPHQWDOIUHTXHQF\UHVSRQVH1\TXLVWSORWVRIVLGHIRUFHDQGDOLJQLQJWRUTXHWR \DZRVFLOODWLRQVIRUDUDGLDODQGDELDVSO\FDUW\UHPHDVXUHGRQDPGUXP
12167($'<67$7(675,1*%$6('7<5(02'(/6
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
)\
N ZRXOG EH
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
12167($'<67$7(675,1*%$6('7<5(02'(/6
6LGH)RUFH5HVSRQVHWR7LPH9DU\LQJ/RDG
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
6WULQJ0RGHOZLWK7UHDG(OHPHQWV6XEMHFWHGWR/RDG9DULDWLRQV
)LJXUHVKRZVWKHW\UHPRGHOLQWZRVXFFHVVLYHSRVLWLRQVZKLOHWKHZKHHO PRYHV DW D FRQVWDQW VPDOO VOLS DQJOH RYHU DQ XQGXODWHG URDG VXUIDFH ZKLFK FDXVHV WKH FRQWDFW OHQJWK WR YDU\ ZLWK GLVWDQFH WUDYHOOHG V :KHQ WKH FRQWDFW FHQWUH&LVPRYHGDGLVWDQFH
VWKHIROORZLQJFKDQJHVRFFXUVLPXOWDQHRXVO\
ODWHUDOVKLIWRIWKHZKHHOSODQHIRUZDUGUROOLQJ DQG FKDQJHLQFRQWDFWOHQJWK 7KHVH FKDQJHV LQIOXHQFH WKH SRVLWLRQ RI WKH OHDGLQJ DQG WUDLOLQJ HGJHV RI WKH
2
VWULQJ7KH[FRRUGLQDWHVRIWKHVHSRLQWVDUH
[2 ' V % D [2 ' V & D
WKHFKDQJHVRIZKLFKEHFRPH
[2 ' V % D [2 ' V & D
7KH FKDQJHV LQ IURQW DQG UHDU ODWHUDO VWULQJ GHIOHFWLRQV Y F DUH FRPSRVHG RI
12167($'<67$7(675,1*%$6('7<5(02'(/6
D
D
∆D
D
_
∆[2
∆V
position 2 position 1 Y
9
α
&
∆Yc2
Y
c2
∆D
D
_
∆[1 Y
c1
Y
∆Yc1
Y
c1
[
1
c2
deflected string
_ 2
Y
_ [
2
deflected tread elements
_
contact line
[
2
_
V
[
1
_ \
\
)LJ 7ZRVXFFHVVLYHSRVLWLRQVRIW\UHPRGHOZLWKYDU\LQJFRQWDFWOHQJWKDWFRQVWDQWVOLS DQJOH VOLGLQJQRWFRQVLGHUHG
YDULRXVFRQWULEXWLRQV)RUWKHLQFUHPHQWRI WKHGHIOHFWLRQDWWKHOHDGLQJHGJH YFZHREWDLQ
GXHWRODWHUDOZKHHOGLVSODFHPHQW
V
YF ' $ D @ V
D
GXHWRORVVRIFRQWDFWDWWKHWUDLOLQJHGJH
YF ' & % D
Y & YF
[2
E
F
2
GXHWRORVVRIFRQWDFWDWWKHOHDGLQJHGJH
YF ' & %( D
Y & YF
2
[!
[
[2
F
F
GXHWRORQJLWXGLQDOGLVSODFHPHQWRIOHDGLQJHGJH
YF ' &
YF
2
[
[2
G
6LPLODUH[SUHVVLRQVDUHREWDLQHGIRUWKHFRQWULEXWLRQVWRWKHFKDQJHRIWKHODWHUDO GHIOHFWLRQDWWKHUHDU LQWKHHTXDWLRQVDUH
YF7KHFRQWDFWOHQJWKGHSHQGHQWFRHIILFLHQWVDSSHDULQJ
:
$ D ' & % % H
D
F
& & H&
D
F
3 D
12167($'<67$7(675,1*%$6('7<5(02'(/6
% D ' 3 D %( D ' % H D % & H& D 3 D ' % H D & & H& D & %
F
F
F
3 D
F
7KH\DUHGHULYHGIURPVROXWLRQVRIWKHGLIIHUHQWLDOHTXDWLRQVIRUWKHVKDSHRIWKH VWULQJGHIOHFWLRQ(TV )RUWKHSDUDPHWHUVHPSOR\HGZHUHIHUWRWKH (TV )URP WKH LQFUHPHQWV WKH IROORZLQJ GLIIHUHQWLDO
v
v
HTXDWLRQVDUHHVWDEOLVKHGIRUWKHXQNQRZQVWULQJGHIOHFWLRQV c1DQG c2
Y '& GV
G F
Y % G D F & % D GV
Y & YF & GD % GV F Y & YF G D % % ( D % % $ D GV F
Y ' GV
G F
Y & G D F % % D GV
Y & YF % GD % GV F Y & YF & % ( D & G D % $ D GV F
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
x2 2
SODQH \ IRUWKHLQWHUYDOWKDWFRQWDFWH[LVWV,PPHGLDWHO\DIWHUWKHILUVWSRLQW WRXFKHV WKH URDG WKH FRQWDFW OLQH JURZV LQ WZR GLUHFWLRQV LQWHUYDO , 7KLV JURZWKFRQWLQXHVXQWLOWKHVHFRQGLQWHUYDO,,LVUHDFKHGZKHUHJURZWKRIFRQWDFW WDNHVSODFHRQO\DWWKHIURQWDQGDWWKHUHDUORVVRIFRQWDFWRFFXUV)LQDOO\LQWKH WKLUGLQWHUYDO,,,ORVVRIFRQWDFWDWERWKHQGVWDNHVSODFHXQWLOWKHW\UHOHDYHVWKH URDG :KHQ WKH W\UH GRHV QRW OHDYH WKH URDG DQ DGGLWLRQDO LQWHUYDO ,, RFFXUV EHIRUHLQWHUYDO,LVUHDFKHGDJDLQDQGWKHF\FOHVWDUWVDQHZ,QOHVVVHYHUHFDVHV LQWHUYDOV,DQG,,,PD\QRWRFFXU:HWKHQKDYHWKHUHODWLYHO\VLPSOHVLWXDWLRQRI FRQWLQXRXVJURZWKRIFRQWDFWDWWKHIURQWDQGORVVRIFRQWDFWDWWKHUHDU
2
x2 2
7KHXQNQRZQVYDQGYDQGWKH\FRRUGLQDWHRIWKHFRQWDFWSRLQWVLQWKH \
12167($'<67$7(675,1*%$6('7<5(02'(/6
)]
wheel load variation
0
V
_ \
V
_ [1
_ [2
D
,,,,,,
D
contact length variation
9
,,,,
&
wheel plane [ Y
1
Y
_ \
2
,,
_
_
contact line
α (n1)
\
c2
_
\
c1
\
_
V
2
[
)LJ 2QWKHGHYHORSPHQWRIWKHFRQWDFWOLQHRIWKHW\UHPRGHOZLWKWUHDGHOHPHQWVWKDW SHULRGLFDOO\ORVHVFRQWDFWZLWKWKHURDG
SODQHDUHREWDLQHGDVIROORZV FI)LJ
GD ! [ [ GV Y ' YF Y ' YF & \2F ' V & YF & \2F ' V & YF GD 0[ 2 ! 0[2 ! & GV Y ' YF Y ' V % \2F & \2F ' V & YF GD 0[ & 2 0[2 ! GV Y ' V % \2F Y ' V % \2F
,QWHUYDO,
02 ! 02
,QWHUYDO,,
,QWHUYDO,,,
D
E
F
12167($'<67$7(675,1*%$6('7<5(02'(/6
_
C
wheel plane
\
[
Y Y 1 Y Y c1
string
1
c1
9
2D
string trailing edge
α
A
tread
path of leading edge of string
\ [
Y
c1
B
Y
1
contact lines \
_ 2
_
V
[
)LJ7KHGHIOHFWHGW\UHPRGHOLQWZRSRVLWLRQVGXULQJWKHLQWHUYDORIFRQWDFWZLWKWKHURDG
%\PHDQVRIQXPHULFDOLQWHJUDWLRQ(TV KDYHEHHQVROYHGZLWKWKHDLG RIWKHHTXDWLRQV DQGWKHH[SUHVVLRQV IRUWKHFDVHIRUPXODWHG EHORZLQZKLFKWKHYHUWLFDOW\UHGHIOHFWLRQ KDVEHHQJLYHQDSHULRGLFYDULDWLRQ ZLWKDPSOLWXGH
A
JUHDWHUWKDQWKHVWDWLFGHIOHFWLRQ
R
VRWKDWSHULRGLFORVVRI
FRQWDFWEHWZHHQW\UHDQGURDGRFFXUV7KHVWDWLFVLWXDWLRQZLOOEHLQGLFDWHGZLWK VXEVFULSW R7KHW\UHUDGLXVLVGHQRWHGZLWKU7KHYHUWLFDO GHIOHFWLRQDQGWKH FRQWDFWOHQJWKDUHJRYHUQHGE\WKHHTXDWLRQV
' R & A VLQ VV D ' U &
7KHQXPHULFDOYDOXHVRIWKHSDUDPHWHUVRIWKHV\VWHPXQGHUFRQVLGHUDWLRQDUH
DR' U R' U A ' U V' ' DR ' F' DR F S' FF
D
R
' DR
,Q)LJWKHFDOFXODWHGYDULDWLRQRIFRQWDFWOHQJWKD RYDOFXUYH WKHSDWK RIWKHFRQWDFWSRLQWV ORZHUFXUYH$% DQGWKHFRXUVHRIWKHSRLQWVRIWKHVWULQJ DWWKHOHDGLQJDQGWUDLOLQJHGJH $&DQG%& KDYHEHHQLQGLFDWHG7ZRSRVLWLRQV RIWKHW\UHKDYHEHHQGHSLFWHGRQHLQLQWHUYDO,,DQGWKHRWKHULQLQWHUYDO,,, ,WLVQHFHVVDU\WKDWWKHFDOFXODWLRQVLQWKHQXPHULFDOLQWHJUDWLRQSURFHVVEH
12167($'<67$7(675,1*%$6('7<5(02'(/6
FDUULHGRXWZLWKJUHDWDFFXUDF\$VPDOOHUURUJLYHVULVHWRDUDSLGEXLOGXSRI GHYLDWLRQVIURPWKHFRUUHFWFRXUVH,QWKHFDVHFRQVLGHUHGDERYHWKHLQWHJUDWLRQ WLPHLVOLPLWHGDQGDFFXUDWHUHVXOWVFDQEHREWDLQHG,QFDVHVKRZHYHUZKHUH FRQWLQXRXV FRQWDFW EHWZHHQ W\UH DQG URDG H[LVWV D ORQJ LQWHJUDWLRQ WLPH LV QHHGHGEHIRUHDVWHDG\VWDWHVLWXDWLRQLVDWWDLQHGZKLFKLVGXHWRWKHIDFWWKDWWKH H[DFWLQLWLDOFRQGLWLRQVRIDSHULRGLFORDGLQJF\FOHDUHQRWNQRZQ)RUWKLVVRUW RIVLWXDWLRQWKHH[DFWPHWKRGGHVFULEHGDERYHLVGLIILFXOWWRDSSO\GXHWRVWURQJ GULIWRIYFLQSDUWLFXODU )RUIXUWKHULQYHVWLJDWLRQRIWKHHIIHFWRIDWLPHYDU\LQJORDGZHZLOOWXUQWR DQDSSUR[LPDWHPHWKRGEDVHGRQWKHVWULQJPRGHOZLWKRXWWUHDGHOHPHQWV
$GDSWHG%DUH6WULQJ0RGHO
7KHXVHRIWKLVPRGHOLVPXFKVLPSOHU VLQFHWKHGHIOHFWLRQVDWERWKHGJHVDUH LQGHSHQGHQW RI HDFK RWKHU 'ULIW GRHV QRW RFFXU LQ WKH FDOFXODWLRQ SURFHVV ,Q )LJWKHGHIOHFWHGPRGHOKDVEHHQVKRZQLQLQWHUYDO,,)RUWKHWKUHHLQWHUYDOV WKHIROORZLQJVHWVRIHTXDWLRQVDSSO\
JURZWKRIFRQWDFWDWOHDGLQJHGJH
[
02 !
D !& GV
G
D
Y Y ' & % GD GV GV [2 ' V % D \2F ' & V % Y G
F
E
G
JURZWKRIFRQWDFWDWWUDLOLQJHGJH
[
02
D ! GV
G
Y Y ' & & GD GV GV [2 ' V & D \2F ' & V % Y G
D
E
F G
12167($'<67$7(675,1*%$6('7<5(02'(/6
_ \
9
σ
D
&
σ = 3Do
D
[
Y Y 1
1
Y Y 2
.
2
_
[
.
_
_
[2
\
c2
_
_
>0
\
_
[
2
[1
2
V
\
c
>0
_ \
c2
_
_
[
[
1
)LJ2QWKHGHYHORSPHQWRIWKHFRQWDFWOLQHRIWKHEDUHVWULQJPRGHO
ORVVRIFRQWDFWDWOHDGLQJHGJH
D & GV \2F ' \2F [2 ' V%D Y ' V % \2F G
[
02
D
E F
ORVVRIFRQWDFWDWWUDLOLQJHGJH
D GV \2F ' \2F [2 ' V&D Y ' V % \2F G
[
02 !
D
E F
6ROXWLRQVRIWKHDERYHHTXDWLRQVVKRZFRQVLGHUDEOHGLIIHUHQFHVIURPWKHUHVXOWV REWDLQHG XVLQJ WKH PRUH DGYDQFHG PRGHO ZLWK WUHDG HOHPHQWV 7KH PRVW LPSRUWDQWGLIIHUHQFHLVWKHIDFWWKDWZLWKWKHVLPSOHPRGHOWKHODWHUDOIRUFHGRHV QRWJUDGXDOO\GURSWR]HURDVWKHW\UHOHDYHVWKHJURXQG ,QRUGHUWRJHWEHWWHUDJUHHPHQWZHZLOOLQWURGXFHDQRQFRQVWDQWUHOD[DWLRQ OHQJWK
(
D WKDWLVDIXQFWLRQRIWKHFRQWDFWOHQJWKDDQGWKXVRIWKHYHUWLFDO
ORDG :H WDNH
(
HTXDO WR WKH UHOD[DWLRQ OHQJWK RI WKH PRUH DGYDQFHG PRGHO
DFFRUGLQJWR(T )LJXUHSUHVHQWVWKHYDULDWLRQRIUHOD[DWLRQOHQJWK DQGFRQWDFWOHQJWKDVDIXQFWLRQRIWKHYHUWLFDOORDGDVFDOFXODWHGIRUWKHFXUUHQW SDUDPHWHUYDOXHV
12167($'<67$7(675,1*%$6('7<5(02'(/6
σ = 3D o FF=15 SF
2
σ* Do
1
D Do
0 0
1
)]] )o
2
)LJ 7KHFDOFXODWHGYDULDWLRQRIKDOIWKHFRQWDFWOHQJWKDQGWKHUHOD[DWLRQOHQJWKDVDUDWLR WRWKHVWDWLFKDOIFRQWDFWOHQJWKYHUVXVWKHYHUWLFDOORDGUDWLRIRUWKHPRGHOZLWKWUHDG HOHPHQWVXVLQJ(T
,Q)LJ D FRPSDULVRQLVPDGHEHWZHHQWKHUHVXOWVRIWKHWKUHHPRGHOV ZLWKRXW WUHDG HOHPHQWV ZLWK WUHDG HOHPHQWV H[DFW DQG DFFRUGLQJ WR WKH DSSUR[LPDWLRQZLWKYDU\LQJ
(
D
7KHFDOFXODWLRQVKDYHEHHQFDUULHGRXWIRU
WKH SDUDPHWHU YDOXHV 7KH DSSUR[LPDWH SDWK RI FRQWDFW SRLQWV VKRZV JRRGDJUHHPHQWZLWKWKHSDWKIRUWKHH[DFWPRGHOZLWKWUHDGHOHPHQWV:KHQWKH WUHDG HOHPHQWV DUH RPLWWHG ZH KDYH WKH EDUH VWULQJ PRGHO IRU ZKLFK WKH SDWK EHFRPHVZLGHUDQGWKHODWHUDOGHIOHFWLRQVJUHDWHU
7KH)RUFHDQG0RPHQW5HVSRQVH
7KHODWHUDOIRUFH)\DQGWKHPRPHQW0]ZKLFKDFWRQWKHW\UHFDQEHDVVHVVHG ZLWK JRRG DSSUR[LPDWLRQ WKURXJK WKH IROORZLQJ VLPSOLILHG IRUPXODH ,Q WKHLU GHULYDWLRQVZHKDYHUHSODFHGWKHFRQWDFWOLQHE\WKHVWUDLJKWOLQHFRQQHFWLQJWKH EHJLQQLQJDQGHQGSRLQWVRIWKHFRQWDFWOLQH 9RQ6FKOLSSH¶VDSSURDFK )RUWKH PRGHOZLWKWUHDGUXEEHUHOHPHQWVZHREWDLQ
FF ^D % $ D ` Y F F % FS F F D 0 ] ' & F S ^DD % % D & D ` F F % FS )\ '
F S
DQGIRUWKHPRGHOZLWKRXWWUHDGHOHPHQWV
)\ )\ )\
12167($'<67$7(675,1*%$6('7<5(02'(/6
string with tread el. exact string adapted σ =σ * (D) σ = 3Do string exact
)\
_ \
D
V
α
D
σ = σ * (D) σ = 3Do
9 [
trailing edge
tangent to leading edge
leading edge
\
path string with tread el. exact path string adapted σ =σ *(D) path string exact σ = 3Do _ [
)LJ 3DWKVRIFRQWDFWSRLQWVDQGYDULDWLRQVRIWKHVLGHIRUFHDFFRUGLQJWRWKUHHPRGHOV
)\ ' FF D % Y
0] ' & FFD^DD %
% D `
,QWKHVHH[SUHVVLRQVWKHIROORZLQJTXDQWLWLHVKDYHEHHQLQWURGXFHG
Y '
DQG
Y %Y
ò
'
Y &Y D
ò
& D ' $ D & D % ( D % %
D
% D
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
12167($'<67$7(675,1*%$6('7<5(02'(/6
average cornering stiffness Do 1 = λ 2
_
1
tyre vertical deflection
1 4
0 0.8
&)α &)αo
1 8
ρ
1 16 0
ρo ρ8sin(2π Vλ) σ = 2Do 0
1
0.5
ρ8 ρo
1 16
1 3.15
0
ρ8
)LJ
periodic lift off
ρ8
1 8
1 4
1 2 periodic lift off
Do λ
&RPSXWHGYDULDWLRQRIDYHUDJHFRUQHULQJVWLIIQHVVZLWKDPSOLWXGHRIWKHYHUWLFDOW\UH GHIOHFWLRQ DQGSDWKIUHTXHQF\ IRUWKHEDUHVWULQJPRGHO
DFFRUGLQJWRWKHDGDSWHGEDUHVWULQJPRGHOIRUWKHLQYHVWLJDWLRQRIWKHPRGHOZLWK WUHDGHOHPHQWV )RUDVHULHVRIDPSOLWXGHV Ö DQGZDYHOHQJWKV WKHYDULDWLRQRIWKHFRUQHULQJ IRUFHRUUDWKHURIWKHFRUQHULQJVWLIIQHVVKDVEHHQFDOFXODWHG6LQFHGXULQJORVV RIFRQWDFWWKHYHUWLFDOORDGUHPDLQV]HURWKHSHULRGRIWKHWRWDOORDGLQJF\FOH PXVWEHFRPHORQJHULQRUGHUWRNHHSWKHDYHUDJHYHUWLFDOORDGXQFKDQJHG7KLV FKDQJHLQSHULRGKDVEHHQWDNHQLQWRDFFRXQWLQWKHFDOFXODWLRQRIWKHDYHUDJH VLGHIRUFHRUFRUQHULQJVWLIIQHVV&) )LJXUHV DQG VKRZ WKH ILQDO UHVXOW RI WKLV LQYHVWLJDWLRQ YL] WKH
average cornering stiffness Do 1 = λ 2
_ 1
tyre vertical deflection
1 4
0
&)α &)αo
1 8
ρ
1 16 0
ρo ρ8sin(2π Vλ)
σ = 3D o FF=15 SF
0.8
0
0.5
ρ8 ρo 0
)LJ
1 16
1 3.15
ρ8
periodic lift off
ρ8
1 2 periodic lift off
1 4
1 8
Do λ
&RPSXWHGYDULDWLRQRIDYHUDJHFRUQHULQJVWLIIQHVVZLWKDPSOLWXGHRIWKHYHUWLFDOW\UH GHIOHFWLRQ DQG SDWK IUHTXHQF\ XVLQJ WKH DGDSWHG EDUH VWULQJ PRGHO WKDW DSSUR[LPDWHVWKHVWULQJPRGHOZLWKWUHDGHOHPHQWV
12167($'<67$7(675,1*%$6('7<5(02'(/6
FRUQHULQJVWLIIQHVVDYHUDJHGRYHURQHFRPSOHWHORDGLQJF\FOHDVDIXQFWLRQRIWKH YHUWLFDOGHIOHFWLRQDPSOLWXGH Ö DQGWKHSDWKZDYHOHQJWK
IRUERWKPRGHOVZLWK
DQGZLWKRXWWUHDGHOHPHQWVWKDWLV ZLWKWKHDGDSWHGEDUHVWULQJPRGHODQGZLWK WKHRULJLQDOEDUHVWULQJPRGHO:LWKWKHSDUDPHWHUYDOXHV
DRDQGFS
FF
WKHUHOD[DWLRQOHQJWKRIWKHPRUHDGYDQFHGPRGHOZLWKWUHDGHOHPHQWVDWWKHVWDWLF ORDG EHFRPHV
(
R
DR )RU WKH PRGHO ZLWKRXW WUHDG UXEEHU WKH UHOD[DWLRQ
OHQJWKKDVEHHQWDNHQHTXDOWR
DR
7KH ILJXUHV FOHDUO\ LOOXVWUDWH WKH XQIDYRXUDEOH HIIHFW RI LQFUHDVLQJ WKH DPSOLWXGH Ö DQGWKHSDWKIUHTXHQF\ GHFUHDVLQJWKHZDYHOHQJWK 7KHFXUYHDW ]HURSDWKIUHTXHQF\LVSXUHO\GXHWRWKHQRQOLQHDU YDULDWLRQRIWKHFRUQHULQJ VWLIIQHVV&) YHUVXVYHUWLFDOORDG)]VKRZQLQ)LJ FI)LJDQGGLVFXVVLRQ RQWKHHIIHFWRIORDGWUDQVIHULQ&KDSWHU)LJ DQGUHIOHFWVWKHµVWDWLF¶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µG\QDPLF¶GURSLVUHSUHVHQWHGE\WKHGRWWHG FXUYHV DORQJ ZKLFK WKH SDWK IUHTXHQF\ FKDQJHV DQG WKH GHIOHFWLRQ DPSOLWXGH UHPDLQVWKHVDPH7KHYHU\ORZOHYHOWRZKLFKWKHDYHUDJHVWLIIQHVVLVUHGXFHG RQFH ORVV RI FRQWDFW RFFXUV LV RI WKH VDPH RUGHU RI PDJQLWXGH IRU ERWK W\UH PRGHOV
&KDSWHU 7+(25<2)7+(:+((/6+,00< 3+(120(121
,QWURGXFWLRQ
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µJ\URVFRSLF¶ VKLPP\ IRU V\VWHPV ZLWK OLYH D[OHV 7KH SKHQRPHQRQZDVH[SHULPHQWDOO\H[DPLQHGE\2OOH\ $QRWKHU NLQG RI VKLPP\ LV FORVHO\ UHODWHG WR W\UH DQG VXVSHQVLRQ ODWHUDO FRPSOLDQFH DQG KDV EHHQ REVHUYHG WR RFFXU ZLWK DLUFUDIW ODQGLQJ JHDUV DQG DXWRPRELOHVHTXLSSHGZLWKLQGHSHQGHQWIURQWZKHHOVXVSHQVLRQV0DQ\RIWKH DXWKRUVPHQWLRQHGLQFRQQHFWLRQZLWKWKHGHYHORSPHQWRIQRQVWHDG\VWDWHRXW
7+(:+((/6+,00<3+(120(121
RISODQHW\UHPRGHOVLQ&KDSWHUKDYHXVHGWKHLUPRGHOLQFRQQHFWLRQZLWKWKH DQDO\VLVRIWKHVKLPP\SUREOHP
7KH6LPSOH7UDLOLQJ:KHHO6\VWHP ZLWK
)LUVW WKH VLPSOHVW V\VWHP WKDW FDQ JHQHUDWH VKLPP\ ZLOO EH GLVFXVVHG &RQVLGHUWKHWUDLOLQJZKHHOV\VWHPGHSLFWHGLQ)LJ7KHYHUWLFDOVZLYHOD[LV
2
PRYHVDORQJWKH[D[LVZLWKVSHHG97KHPRWLRQYDULDEOHLVWKH\DZDQJOH
RI
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
.
H
ψ
N
\
0]
\
)LJ
ψ
Fψ
9
)\
9
[
H
6LPSOH WUDLOLQJ ZKHHO V\VWHP WKDW LV FDSDEOH RI VKRZLQJ VKLPP\ 7ZR ZD\VRI
H
UHDOLVLQJ WKH PHFKDQLFDO FDVWHU OHQJWK RI WKH FRQWDFW FHQWUH ZLWK UHVSHFW WR WKH VWHHULQJD[LV NLQJSLQ KDYHEHHQLQGLFDWHG
7KHW\UHZLOOEHFRQVLGHUHGPDVVOHVVDQGWKHFRQWDFWOLQHLVDSSUR[LPDWHGE\ D VWUDLJKW OLQH WDQJHQW WR WKH DFWXDO FRQWDFW OLQH DW WKH OHDGLQJ HGJH VWUDLJKW WDQJHQWDSSUR[LPDWLRQ IRUZKLFKWKH(TV KROGRUE\DVLQJOHSRLQW (TV ZLWKDQGZLWKRXWWKHHIIHFWRIWUHDGZLGWK7KHUHVXOWVZLOOEH FRPSDUHGZLWKWKRVHREWDLQHGZKHQXVLQJWKHDOPRVWH[DFW9RQ6FKOLSSHW\UH PRGHO:HKDYHWKHHTXDWLRQV
7+(:+((/6+,00<3+(120(121
,
%
)
\'
%
'&
F
&
]
(
('& (G
]
GY
Y
%
GV
GV
Y
'
V
0
] % 0]
0
] ' & &0
\
0
0
%
) H
)
]'
0
0
N
' O' [
'
&
D
G GV
&
G\O
GV
9W
O ' &H
\
ZKHUHD
ZKHQWKHVLQJOHFRQWDFWSRLQWW\UHPRGHOLVFRQVLGHUHG7RUHGXFHWKH
QXPEHU RI JRYHUQLQJ V\VWHP SDUDPHWHUV ZH ZLOO LQWURGXFH WKH IROORZLQJ QRQGLPHQVLRQDOTXDQWLWLHVZLWKWKHUHIHUHQFHOHQJWKDRUHSUHVHQWLQJWKHDFWXDO RUWKHQRPLQDOKDOIFRQWDFWOHQJWK
D
'
9
'
D D
V
R
'
V D
&
'
H D
R
'
W
R
(
('
D
)
H
R
,
9
&
)
D
R
&
0
D
'
'
&
R )
(W &
0
D
N
R
D
Y
R
'
'
Y
D
N ,&
)
D
R
V
R
F
'
'
VDR
F &
)
D
R
ZKLFKLQFOXGHVWKHQRQGLPHQVLRQDOSQHXPDWLFWUDLOW WDR$IWHUHOLPLQDWLRQRI WKH WLPH DQG DOO WKH YDULDEOHV H[FHSW
DQG Y LQ (TV DQG XVLQJ WKH
FRQYHUVLRQV ZHREWDLQWKHQRQGLPHQVLRQDOGLIIHUHQWLDOHTXDWLRQV
9
G
N9
GV GY
GV
%
%
Y
%
( G
GV
%
F
% % H
Y W
'
'
H
&
D
G GV
%
7+(:+((/6+,00<3+(120(121
7KHV\VWHPLVOLQHDUDQGRIWKHWKLUGRUGHU,WLVDVVXPHGWKDWWKHW\UHGHIOHFWLRQ UHOD[DWLRQOHQJWK
DRUQRQGLPHQVLRQDOO\
,QFDVHRIWKHVLQJOHFRQWDFW
SRLQWW\UHPRGHODPXVWEHWDNHQHTXDOWR]HURDQG D
PXVWEHJLYHQWKHYDOXHRI
D,QWKHQRQGLPHQVLRQDOQRWDWLRQ WDNHVWKHYDOXHRI
)RUWKHSQHXPDWLFWUDLOZHDVVXPHW
DRUW
D
7KHFKDUDFWHULVWLFHTXDWLRQRIV\VWHP EHFRPHV
9
S
%
N9
( S % F
%
%
H
W
/
/
0 0
H
0 0 0
&
%
D S
&
S
%
0 0
'
0 0 0
RU
9
S
%^
9
%^
%
N9
%
N9
(%
,QJHQHUDOZHPD\ZULWH D S
%
D
%
S
D
S
F
%
% ( ` % % H
D
'
% &
S
W
H
D ` S
% % % H
W
F
'
$FFRUGLQJWRWKHFULWHULXPRI+XUZLW]WKHFRQGLWLRQVIRUVWDELOLW\RIWKHPRWLRQ RIWKHWKLUGRUGHUV\VWHPUHDG
DOOFRHIILFLHQWVDLRIWKHFKDUDFWHULVWLFHTXDWLRQPXVWEHSRVLWLYH
!
D
D
!
!
D
!
WKH+XUZLW]GHWHUPLQDQWV+Q+QHWFPXVWEHSRVLWLYHZKLFK\LHOGVIRUWKH V\VWHP
WKLUGRUGHU Q
D
+
'
D
D
D
D
/
/
0 0 0 0
0
!
0
0 0
7KHILUVWWZRFRHIILFLHQWVRI(T DUHDOZD\VSRVLWLYH)RUWKH UHPDLQLQJ FRHIILFLHQWVWKHFRQGLWLRQVIRUVWDELOLW\EHFRPH
D
D
'
N9
%
' % % H
W
(%
F
F
% %
!
ZKLOHDFFRUGLQJWR
H
W H
&
D
!
7+(:+((/6+,00<3+(120(121
+
'
D
^9
%
N 9
%
( ` &
9
H
% % W
F
!
,WFDQEHHDVLO\VHHQWKDWLIWKHODVWWZRFRQGLWLRQVDUHVDWLVILHGWKHILUVWRQHLV VDWLVILHGDXWRPDWLFDOO\,IWKHFRQGLWLRQDQ
D!LVWKHILUVWWREHYLRODWHGWKHQ
WKHPRWLRQWXUQVLQWRDPRQRWRQRXVXQVWDEOHPRWLRQ GLYHUJHQWLQVWDELOLW\WKDW LV ZLWKRXWRVFLOODWLRQV &RQVHTXHQWO\LIH WF PHDQLQJWKDWWKHVWHHULQJ D[LVOLHVDGLVWDQFHHEHKLQGWKHFRQWDFWFHQWUHWKDWLVODUJHUWKDQWF &) WKH +LV
ZKHHOVZLQJVDURXQGRYHUGHJUHHVWRWKHQHZVWDEOHVLWXDWLRQ,I+Q
WKH ILUVW WR EHFRPH QHJDWLYH WKH PRWLRQ EHFRPHV RVFLOODWRULO\ XQVWDEOH 7KH ERXQGDULHVRIWKHWZRXQVWDEOHDUHDVDUH D ,QWKHFDVHRIYDQLVKLQJGDPSLQJ N
H
%
W H
& & D
DQG+
FRQGLWLRQ UHGXFHVWR
!
$SSDUHQWO\LQWKH H9 SDUDPHWHUSODQHWKHERXQGDULHVRI UHGXFHWRWZR SDUDOOHOOLQHVDWH
DDQGH
W:KHQWKHFDVWHUOLHVLQEHWZHHQWKHVHWZR
YDOXHV WKH \DZ DQJOH SHUIRUPV DQ RVFLOODWLRQ ZLWK H[SRQHQWLDOO\ LQFUHDVLQJ DPSOLWXGH DW DQ\ VSHHG RI WUDYHO $SSDUHQWO\ ZKHQ WKH GDPSLQJ LV ]HUR WKH VSHHGDQGWKHWRUVLRQDOVWLIIQHVVGRQRWLQIOXHQFHWKHH[WHQWRIWKHXQVWDEOHDUHD 7KH\ZLOOKRZHYHUFKDQJHWKHGHJUHHRILQVWDELOLW\DQGWKHQDWXUDOIUHTXHQF\ HLJHQYDOXHV ,IZHKDYHGDPSLQJWKHOLPLWVLWXDWLRQPD\EHFRQVLGHUHGZKHUHWKHVSHHG 9WHQGVWR]HUR7KHFRQGLWLRQIRUVWDELOLW\WKHQEHFRPHV
(%
F
% % H
W H
&
D
!
ZKLFKVKRZVWKDWWKHYDOXHVIRUHEHFRPHFRPSOH[LI
F
!ó W D
W D
7KLVLPSOLHVWKDWLQWKDWFDVHWKHXQVWDEOHDUHDEHFRPHVGHWDFKHGIURPWKHHD[LV ,Q )LJ WKH XQVWDEOH DUHDV KDYH EHHQ VKRZQ IRU GLIIHUHQW YDOXHV RI WKH YLVFRXVVWHHUGDPSLQJNDQGZLWKVWHHUVWLIIQHVVF
'DPSLQJUHGXFHVWKHVL]H
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
V
LVUHODWLYHO\KLJKDQGWKXVWKHZDYHOHQJWK UHODWLYHO\VKRUW
7KLVLVLQDFFRUGDQFHZLWKILQGLQJVLQ)LJZKHUHWKHIUHTXHQF\UHVSRQVH
7+(:+((/6+,00<3+(120(121
ω BV 4 ω BV
σB + BD = 4
ω BV
1
ω BV
0.47
Fψ = 0
0.2
Von Schlippe B* 0 straight tangent κ single point straight tangent κ B * 0.3
NB= 0
HB
NB= 0.25
NB= 0.5
ω BV
oscillatory instability
0.6
1
0.1
0 -BW = -0.5 divergent instability
0 )LJ
5
10
9B
15
$UHDVRILQVWDELOLW\IRUWKHWUDLOLQJZKHHOV\VWHPRI)LJZLWKRXWVWHHUVWLIIQHVVDW GLIIHUHQW OHYHOV RI YLVFRXV VWHHU GDPSLQJ &RQVHTXHQFHV RI XVLQJ GLIIHUHQW W\UH PRGHOVFDQEHREVHUYHG
IXQFWLRQVDFFRUGLQJWRYDULRXVDSSUR[LPDWLRQVKDYHEHHQFRPSDUHG7KHVWUDLJKW WDQJHQWDSSUR[LPDWLRQVKRZVDWHQGHQF\WRSUHGLFWDVKLPP\LQVWDELOLW\WKDWLV VRPHZKDWWRRVWURQJ2QWKHRWKHUKDQGWKHVLQJOHSRLQWFRQWDFWPRGHOWXUQVRXW WR EH WRR VWDEOH IRU N
WKH XQVWDEOH DUHD YDQLVKHV DOO WRJHWKHU :KHQ
HPSOR\LQJWKHVWUDLJKWWDQJHQWDSSUR[LPDWLRQZHDWOHDVWDSSHDUWREHRQWKHVDIH VLGH)RUWKHFDVHN
DFXUYHKDVEHHQDGGHGLQ)LJWRVKRZWKHHIIHFWRI
DUWLILFLDOO\LQWURGXFLQJDSURSHUYDOXHRI VWUDLJKWWDQJHQWPRGHO7KHYDOXH
(
(
&0 n
WRLPSURYHWKHSHUIRUPDQFHRIWKH D&0 WKDWFRUUHVSRQGVWR
(
LQGHHGSURGXFHVDEHWWHUUHVXOW7KLVLVLQDFFRUGDQFHZLWKWKHFXUYHDGGHG LQWKH1\TXLVWGLDJUDPRI)LJIRU&0n
D&0 WKDWVKRZVDQLPSURYHG
DOLJQLQJWRUTXHIUHTXHQF\UHVSRQVH 7KH QRQGLPHQVLRQDO SDWK IUHTXHQFLHV VKRZQ LQ WKH ILJXUH RFFXU RQ WKH ERXQGDULHVFRPSXWHGIRUWKHV\VWHPZLWKVWUDLJKWWDQJHQWDSSUR[LPDWLRQ+HUH WKHPRWLRQVKRZVDQXQGDPSHGRVFLOODWLRQ7KHQWKHVROXWLRQRI FRQWDLQV D SDLU RI SXUHO\ LPDJLQDU\ URRWV %\ UHSODFLQJ S E\ L
V
LQ WKH FKDUDFWHULVWLF
HTXDWLRQDQGWKHQVHSDUDWLQJWKHLPDJLQDU\DQGWKHUHDOWHUPVZHJHWWZRHTXD WLRQVZKLFKFDQEHUHGXFHGWRRQHE\HOLPLQDWLQJWKHWRWDOGDPSLQJFRHIILFLHQW N 9
(
7KH UHVXOWLQJ UHODWLRQVKLS EHWZHHQ 9 DQG H KDV EHHQ VKRZQ IRU D
7+(:+((/6+,00<3+(120(121
ω BV 4 ω BV
1 ω B V 0.47
ω BV
Fψ = 0
0.2
straight tangent
σB + DB = 4
0 0.3 0.5
HB
NB = 0
1
ω BV 0
κB *
0.5
0.9
1
0.3 oscillatory instability
κB *
1 0.5
0.1
0.3 0
NB = 0.25
NB = 0.5
0 -BW = -0.5
divergent instability
0 )LJ
9B
10
20
$UHDVRILQVWDELOLW\IRUWKHWUDLOLQJZKHHOV\VWHPRI)LJZLWKRXWVWHHUVWLIIQHVV ,QIOXHQFHRIWUHDGZLGWKGDPSLQJDWGLIIHUHQWOHYHOVRIYLVFRXVVWHHUGDPSLQJ
QXPEHURIYDOXHVRI
$WSDWKIUHTXHQF\
V
V
%^
W `
DVSHFLDOFDVH
DULVHVZKHUHWKHIUHTXHQF\FXUYHUHGXFHVWRDVWUDLJKWOLQHRULJLQDWLQJIURP W %\XVLQJRQHRIWKHWZRHTXDWLRQVDQH[SUHVVLRQIRUWKHQRQGLPHQVLRQDOSDWK IUHTXHQF\PD\EHREWDLQHG
V
'
H%W%F D ' D 9 % N9 %
(
)LJXUHSUHVHQWVWKHGLDJUDPLQGLFDWLQJWKHLQIOXHQFHRIGDPSLQJGXHWR WUHDGZLGWK
(
7KLVW\SHRIGDPSLQJLVHVSHFLDOO\HIIHFWLYHDWORZVSHHG7KLVLV
XQGHUVWDQGDEOH EHFDXVH ZH KDYH VHHQ WKDW WKH HTXLYDOHQW YLVFRXV GDPSLQJ FRHIILFLHQW GHFUHDVHV LQYHUVHO\ SURSRUWLRQDOO\ ZLWK WKH IRUZDUG YHORFLW\ 7KH ERXQGDULHV QRZ KDYH D FKDQFH WR EHFRPH FORVHG DW WKH OHIWKDQG VLGH FI (T $V)LJGHSLFWVWKHVWHHUWRUVLRQDOVWLIIQHVVUHGXFHVWKHXQVWDEOHVKLPP\ DUHD LQ VL]H HVSHFLDOO\ ZKHQ VXIILFLHQW GDPSLQJ LV SURYLGHG $W YDQLVKLQJ GDPSLQJ WKH DUHD RI RVFLOODWRU\ LQVWDELOLW\ DSSHDUV WR UHPDLQ OLPLWHG E\ WKH KRUL]RQWDOOLQHVREWDLQHGIURP 7KHDUHDRIGLYHUJHQWLQVWDELOLW\LVUHGXFHG WKURXJKWKHUHVWRULQJDFWLRQRIWKHWRUVLRQDOVSULQJDERXWWKHVWHHULQJD[LV,WV
7+(:+((/6+,00<3+(120(121
straight tangent
σB + DB = 4 ω B V
HB
1
NB = 0
FBψ = 0
.5 .4 1 .7
FBψ = 0
.1
1.5
.2
0.5 1.0
ωV B
NB = 0.5
.2
1.5
1 .3
oscillatory instability
.2
0
FBψ = 0
-BW = -0.5 -1
WBBFψ
0.5
divergent instability
1.0 1.5
-2
10
0 )LJ
NB = 0.25
.3
,QIOXHQFHRIVWHHUVWLIIQHVV
F
9B
20
RQVKLPP\LQVWDELOLW\DQGRQGLYHUJHQWLQVWDELOLW\DW
GLIIHUHQWOHYHOVRIVWHHUGDPSLQJZLWKRXWWUHDGZLGWKGDPSLQJ
XSSHUERXQGDU\VKLIWVGRZQZDUGVDFFRUGLQJWRWKHQHJDWLYHWUDLOH
W F
7KHLQIOXHQFHRIW\UHLQHUWLDKDVQRWEHHQLQGLFDWHGLQWKHGLDJUDPV$VPD\ EHH[SHFWHGIURPWKHGLDJUDPRI)LJWKDWVKRZVWKDWWKHSKDVHODJRIWKH PRPHQWUHVSRQVHLVUHGXFHGWKURXJKWKHDFWLRQRIWKHJ\URVFRSLFFRXSOH0]J\U HVSHFLDOO\DWKLJKHUVSHHGVWKHDUHDEHFRPHVOLPLWHGDWWKHULJKWKDQGVLGHHYHQ LQWKHDEVHQFHRIGDPSLQJ([HUFLVHJLYHQEHORZDGGUHVVHVWKLVSUREOHP ,IWKHPRPHQWRILQHUWLDDERXWWKHVWHHULQJD[LVFDQQRWEHFRQVLGHUHGDVD FRQVWDQWEXWUDWKHUDVDIXQFWLRQRIWKHWUDLOHHJUHSODFH,E\, ] PH WKHQ
FRQVLGHULQJ WKH QRQGLPHQVLRQDOLVDWLRQ GHILQHG E\ WKH FXUYHV RI WKH YDULRXV GLDJUDPV PXVW EH UHLQWHUSUHWHG DW D FRQVWDQW N WKH DFWXDO GDPSLQJ FRHIILFLHQWNLVQRWDFRQVWDQWDORQJWKHFXUYHEXWLQFUHDVHVZLWKLQFUHDVLQJH ZKLOHDWDJLYHQ9WKHDFWXDOVSHHG9GHPLQLVKHVDWLQFUHDVLQJH
7+(:+((/6+,00<3+(120(121
([DPSOH &RQVLGHUWKHV\VWHPZLWKSDUDPHWHUV D
DR
PW
NJP &)
,
H
(
1UDG
W
6SHHGRIWUDYHO
,
9' 9
&) D
6
9
9>PV@
9>NPK@
6KLPP\ZDYHOHQJWK D
V
V
VRWKDWDW V
V
6
6
DQGDW V
6
V
V
>UDGP@
>P@ V
P
D
P
D
6KLPP\IUHTXHQF\
Q DW 9
9
V
9>UDGV@
>+]@
ZH KDYH 9
PV DQG DFFRUGLQJ WR WKH VWDELOLW\ JUDSK RI )LJ
ZHKDYHDWWKLVVSHHGDQGFDVWHUH
F
DQRQGLPHQVLRQDOSDWKIUHTXHQF\
.VRWKDW . UDGP . P . UDGVDQGQ . +]7KLV ( $W ZRXOGEHWKHFDVHDWWKHERXQGDU\RIVWDELOLW\ZKHUHN . ZLWK V
V
DERXW WKH VDPH SRLQW LQ WKH GLDJUDP RI )LJ ZH ZRXOG KDYH ZLWK D QRQ GLPHQVLRQDO VWHHU VWLIIQHVV F
DQG GDPSLQJ FRHIILFLHQW N
. PDQGDIUHTXHQF\Q . +]
D QRQ
.ZKLFK\LHOGV . UDGPDZDYHOHQJWK
V
GLPHQVLRQDOSDWKIUHTXHQF\
V
([HUFLVH,QIOXHQFHRIWKHW\UHPDVVRQWKHVWDELOLW\ERXQGDU\ &RQVLGHUWKHZKHHOV\VWHPRI)LJ'HULYHWKHVWDELOLW\FRQGLWLRQVIRUWKLVV\VWHP ZLWK WKH LQFOXVLRQ RI WKH HIIHFW RI W\UH LQHUWLD DSSUR[LPDWHG E\ WKH LQWURGXFWLRQ RI
0
]J\U
,QQRQGLPHQVLRQDOIRUPWKHJ\URVFRSLFFRXSOHEHFRPHV
0 ] J\U '
ZLWK
0] J\U ' & & J\U 9 &) D
)\ GV
G
7+(:+((/6+,00<3+(120(121
&
J\U '
)
D& &
J\U
,
P
'
J\U
F
W
D&
)\
&
)
'
,
P
J\U
F
W
D
,
%
D
FI(TV DQG
F
'HWHUPLQHKRZWKHVWDELOLW\ERXQGDULHVIRUWKHV\VWHPZLWKRXWGDPSLQJ
N
DUH FKDQJHG GXH WR WKH LQFOXVLRQ RI WKH J\URVFRSLF FRXSOH &RQVLGHU WKH
IROORZLQJSDUDPHWHUYDOXHV
W
'
'
H 9
'UDZLQWKH
&
J\U '
GLDJUDPWKHERXQGDULHVIRU
&
J\U
DQGVNHWFKWKHQHZVWDELOLW\
9 H WH
ERXQGDU\E\FDOFXODWLQJDWOHDVWWKHERXQGDU\YDOXHVIRU
H
DW
H
DQG
DQG
PRUHRYHUWKHERXQGDU\RQWKH D[LV,QGLFDWHZKHUHWKHV\VWHPLVVWDEOH
6\VWHPVZLWK
7KH DGGLWLRQDO GHJUHH RI IUHHGRP WKDW DOORZV WKH NLQJSLQ WR PRYH VLGHZD\V OHDGVWRDV\VWHPZKLFKLVRIWKHILIWKRUGHU)LJXUHGHSLFWVWKHV\VWHPZLWK ODWHUDOFRPSOLDQFHRIWKHZKHHOVXVSHQVLRQSRVVLEO\DVVRFLDWHGZLWKDWRUVLRQDO FDPEHU FRPSOLDQFHZLWK YLUWXDO D[LVRIURWDWLRQDVVXPHGWREHORFDWHGDWD KHLJKWKDERYHWKHJURXQG ULJKWKDQGGLDJUDP 7KHLQHUWLDRIWKHV\VWHP P,] PD\EHUHSUHVHQWHGE\WZRSRLQWPDVVHVP$DQGP%DQGDUHVLGXDOPRPHQWRI
LQHUWLD , DERXW WKH YHUWLFDO D[LV 7KH WZR PDVV SRLQWV DUH FRQQHFWHG E\ D PDVVOHVVURGZLWKOHQJWKHDQGDUHORFDWHGRQWKHZKHHOD[LVDQGWKHVWHHUD[LV UHVSHFWLYHO\/DWHUDOGDPSLQJZLOOQRWEHFRQVLGHUHG 7KHIROORZLQJVHWRIHTXDWLRQVKROGIRUWKHG\QDPLFVRIWKHV\VWHPRI)LJ DVVXPLQJDYHUWLFDOVWHHULQJD[LVDQGDGRSWLQJWKHVWUDLJKWWDQJHQWW\UHPRGHO (T
\
]
P
9
,
ψ
0] P$
)\
H
F\ Fψ
\
\
9
K
I
N
F\
,
ψ
P%
U
)LJ/DWHUDOFRPSOLDQFHLQWURGXFHGLQWKHV\VWHP
Fψ
7+(:+((/6+,00<3+(120(121
&
$ H & \ % P% \ % F \ \ ' )\ % ,( % N 0 % F P H H & \ ' & )\ H % 0 ] $ P
)
\'
)
&
]'
0
( ] % 0]
0
] ' & &0
( (G ] '&
0
GY
GV
GV
Y
'
V
0
%
Y
' O' [
' &
D
&
GV
9W
ZKHUHWKHVOLSDQJOH
'
G
0 \
9
&
LVREWDLQHGIURP
H
0
%HIRUHIXUWKHUVWXG\LQJWKHILIWKRUGHUV\VWHPOHWXVWU\WRUHGXFHWKHV\VWHPWR DORZHURUGHUZKLOHUHWDLQLQJWKHODWHUDOFRPSOLDQFH
'
&
0\
9
&
H
0
'
,QDGGLWLRQZHDVVXPHWKDWIRUWKHULJLGGLVFWKHDOLJQLQJWRUTXHDQGWKHFRQWDFW OHQJWKYDQLVK 0
]'
D
'
7+(:+((/6+,00<3+(120(121
$IWHUHOLPLQDWLRQRI)\IURPWKH(TV DQGH[SUHVVLQJ \ 0 LQWHUPVRI DQGLWVGHULYDWLYHZHILQGDWKLUGRUGHUGLIIHUHQWLDOHTXDWLRQ,WVFKDUDFWHULVWLF HTXDWLRQZLWKYDULDEOHVLQ>UDGV@UHDGV
(% P H V %
,
% % N
% H 9 V
P
% % F
\
F H
V
%
F
\H9 '
7KHWULYLDOFRQGLWLRQDJDLQVW GLYHUJHQW LQVWDELOLW\WKDWUHVXOWVIURPWKHODVWWHUP RI ZKLFKPXVWUHPDLQSRVLWLYHEHFRPHV H
!
7KHFRQGLWLRQIRUVWDELOLW\+!FI UHDGV
%
N
% H 9 F
P
%
F
( \ H ! , % P% H F \ H 9
)RUWKHXQGDPSHGFDVH N
WKLVUHGXFHVWRWKHVLPSOHFRQGLWLRQIRUVWDELOLW\
LIH!
%
P F
!
(
, F
\
$SSDUHQWO\DQHJDWLYHUHVLGXDOPRPHQWRILQHUWLD , ZLOOHQVXUHVWDELOLW\ :KHQIGHILQHVWKHORFDWLRQRIWKHFHQWUHRIJUDYLW\EHKLQGWKHVWHHUD[LVDQGL GHQRWHV WKH UDGLXV RI LQHUWLD RI WKH FRPELQDWLRQ P$ P% ,
(
RU RQO\ RI WKH
VWHHUDEOHSDUWDQHJDWLYHUHVLGXDOPRPHQWRILQHUWLDZRXOGRFFXULIL
(
I HI
)RUVXFKDV\VWHPDQLQFUHDVHRIWKHODWHUDOVWLIIQHVVVWUHQJWKHQVWKHVWDELOLW\ ,I RQ WKH RWKHU KDQG ,
! D VXIILFLHQWO\ ODUJH VWHHU VWLIIQHVV LV QHHGHG WR
VWDELOLVH WKH V\VWHP ,W LV VXUSULVLQJ LV WKDW D ODUJHU ODWHUDO VWLIIQHVV PD\ WKHQ FDXVHWKHV\VWHPWREHFRPHXQVWDEOHDJDLQ 7KHFRQFOXVLRQWKDWWKHIUHHO\VZLYHOOLQJZKHHOV\VWHP WKDWLV F
N
LV
VWDEOHLIWKHUHVLGXDOPRPHQWRILQHUWLD, HQWDLOVWKDWZKHQFDVWHULVUHDOLVHG E\LQFOLQLQJWKHVWHHULQJD[LVEDFNZDUGV )LJULJKWKDQGGLDJUDP ZHKDYH WKHVLWXDWLRQWKDWP$
VRWKDW,
,ZKLFKLVDOZD\VSRVLWLYH&RQVHTXHQWO\IRU
VXFK D IUHHO\ VZLYHOOLQJ V\VWHP HTXLSSHG ZLWK D ULJLG WKLQ W\UH WKH PRWLRQ LV DOZD\VRVFLOODWRULO\XQVWDEOH
7KHILIWKRUGHUV\VWHP 7KHJ\URVFRSLFFRXSOLQJWHUPVWKDWDULVHGXHWRWKHDQJXODUFDPEHUYHORFLW\RI WKH ZKHHO V\VWHP DERXW WKH ORQJLWXGLQDO D[LV ORFDWHG DW D KHLJKW K DERYH WKH JURXQG ZLOO EH DGGHG WR WKH (TV )RU WKLV WKH FRHIILFLHQW
J\U
LV
LQWURGXFHG
J\U '
,
S
PUK
ZKHUH,SGHQRWHVWKHSRODUPRPHQWRILQHUWLDRIWKHZKHHO)XUWKHUPRUHWKHUDWLR
7+(:+((/6+,00<3+(120(121
RIGLVWDQFHVZLWKUHVSHFWWRWKHFDPEHUWRUVLRQFHQWUHLVLQWURGXFHG
K'
K K
&
U
0RUHRYHUWRIXOO\DFFRXQWIRUWKHHIIHFWVRIWKHDQJXODUGLVSODFHPHQWDERXWWKH ORQJLWXGLQDOD[LVWKDWDOVRDULVHVZKHQVWHHULQJDURXQGDQLQFOLQHGNLQJSLQ WR SURYLGH FDVWHU RQH PD\ LQFOXGH WKH FDPEHU IRUFH LQ (T DQG WKH VPDOO QHJDWLYH VWLIIQHVVHIIHFWVUHVXOWLQJIURPWKHODWHUDOVKLIWRIWKHQRUPDOIRUFH)] %HFDXVHRIWKHIDFWWKDWWKHVHHIIHFWVDUHUHODWLYHO\VPDOODQGDSSHDUWRSDUWO\ FRPSHQVDWHHDFKRWKHUZHZLOOQHJOHFWWKHH[WUDWHUPV 7KHGLIIHUHQWLDOHTXDWLRQVWKDWUHVXOWZKHQUHWXUQLQJWRWKHVHW,]DQGPWKDW
UHSODFHVWKHHTXLYDOHQWVHWP$P%DQG, DQGHOLPLQDWLQJDOOYDULDEOHVH[FHSW\ DQG
NUHDG
&
P \ ,
I
%
9
N \ 0
\
%
\
F \
(9 0 % F
] % N% 0
%
'&
& %
J\U 9 0 '
P
K \0 % H &
K&)
J\U 9 \0 % I N\ \0 % I F\ \ '
P
D 0
%
&
) H%W&I
&
9
,QWKHHTXDWLRQV WKHODWHUDOVXVSHQVLRQGDPSLQJN\KDVEHHQLQFOXGHG,QWKH IXUWKHUDQDO\VLVWKLVTXDQWLW\ZLOOEHOHIWRXWRIFRQVLGHUDWLRQ7RVWXG\WKHSXUH J\URVFRSLFFRXSOLQJHIIHFWWKHUDWLR
K
ZLOOEHWDNHQHTXDOWRXQLW\ZKLFKZRXOG
UHSUHVHQWWKHFDVHWKDWWKHFHQWUHRIJUDYLW\DQGWKHODWHUDOVSULQJDUHORFDWHGDW JURXQGOHYHO)RUWKHVSHFLDOFDVHWKDWWKHFHQWUHRIJUDYLW\OLHVRQWKH LQFOLQHG WKHV\VWHPGHVFULSWLRQLVFRQVLGHUDEO\VLPSOLILHG2XU
VWHHUD[LVGLVWDQFHI
DQDO\VLVZLOOPDLQO\EHOLPLWHGWRWKLVFRQILJXUDWLRQ%HVVHOLQN KDVFDUULHG RXW DQ HODERUDWH VWXG\ RQ WKH V\VWHP RI )LJ ZLWK WKH PDVV FHQWUH ORFDWHG EHKLQGWKH YHUWLFDO VWHHULQJD[LV I! HYHQEHKLQGWKHZKHHOFHQWUH I! H )RUWKHFRPSOHWHUHVXOWVRIWKLVIRUDLUFUDIWVKLPP\LPSRUWDQWDQDO\VLVZHUHIHU WR WKH RULJLQDO SXEOLFDWLRQ 2QH W\SLFDO UHVXOW KRZHYHU ZKHUH I
H ZLOO EH
GLVFXVVHGKHUH 7KH FRPSOHWH VHW RI QRQGLPHQVLRQDO TXDQWLWLHV H[WHQGHG DQG VOLJKWO\ FKDQJHGZLWKUHVSHFWWRWKHVHWGHILQHGE\(TV UHDGV
D
'
D D
'
H
R (
(' )
&
D
R
H D
'
'
I
R
I D
(W 0
&
D
R
R
W
N
'
'
0
& D
'
R ) &
N
] )
, &
D
R
D
F
R
'
L
]'
L
]
D
R
5\ 5
'
F
F
)
&
D
R
\ D
V'
R
\'
F
\ DR )
&
VDR
7+(:+((/6+,00<3+(120(121
'
5P 5
PD ,
R
'
]
L
9
'
, 9
]
)
&
]
55
D
,
'
]
)
&
R
D
V
R
'
, V
)
&
] D
R
+HUH V GHQRWHV WKH QRQGLPHQVLRQDO /DSODFH YDULDEOH 7KH QRQGLPHQVLRQDO GLIIHUHQWLDOHTXDWLRQVIROORZHDVLO\IURPWKHRULJLQDOHTXDWLRQV )RUWKH VWDELOLW\ DQDO\VLV ZH QHHG WKH FKDUDFWHULVWLF HTXDWLRQ ZKLFK UHDGV LQ QRQ GLPHQVLRQDOIRUP
55 5
D V
% 55
D
5V
% 55
D
5V
% 55
D
% 55 5 % 55 '
5V
D
7KHFRHIILFLHQWVRI ZLWKI
55
D
55
D
55
D
55
D
' ' ' '
55
N\
DUH
% ( % \ % K % % & % J\U % % % \ % % % K% \ % ( % & % K & J\U J\U ' \% K % % & \ % J\U % \ % ( ' \ % % %
P9 P
N
P
F
PF
F
9
F
9
F
W
P
9
9
F
F
P
P
F
55
D
D
P
D
D
V
H
H
W
H
H
W
H
D
W 9
F
P
9
H
D F
W
P
9
N
P
9
P
N
(9 9
9
F
N
9
9
9
7KH LQIOXHQFH RI D QXPEHU RI QRQGLPHQVLRQDO SDUDPHWHUV ZLOO EH LQYHVWLJDWHGE\FKDQJLQJWKHLUYDOXHV7KHRWKHUSDUDPHWHUVZLOOEHNHSWIL[HG 7KHYDOXHVRIWKHIL[HGVHWRISDUDPHWHUVDUH D
' ' ' W
'
P
K'
)RUWKHILIWKRUGHUV\VWHPWKH+XUZLW]FRQGLWLRQVIRUVWDELOLW\UHDG
DOOFRHIILFLHQWVDLRIWKHFKDUDFWHULVWLFHTXDWLRQPXVWEHSRVLWLYH
D
WKH+XUZLW]GHWHUPLQDQWV+QDQG+QPXVWEHSRVLWLYHZKLFK\LHOGV
!
D
!
D
!
D
!
D
D
!
D
D
!
/
/
0
+ 55
'
D
D
0 0
/
/
0
0
0
0
0 0 0
D
D
! 55 +
'
0
0
D
D
D
D
0 0
0
0
0
0
0
0
0
0
0
0
D
D
D
D
0
0 0
0
0
0
0
0
0
0
0
0 0
0
0
0
D
D
0
0 0 0
!
7+(:+((/6+,00<3+(120(121
,WWXUQVRXWWKDWIRUWKHFDVHZLWKRXWGDPSLQJDQGJ\URVFRSLFFRXSOLQJWHUPV
N
DUHODWLYHO\VLPSOHVHWRIDQDO\WLFDOH[SUHVVLRQVFDQEHGHULYHG
J\U
IRUWKHFRQGLWLRQVRIVWDELOLW\,WFDQEHSURYHQWKDWZKHQD!+!+!WKH RWKHUFRQGLWLRQVDUHVDWLVILHGDVZHOO&RQVHTXHQWO\WKHJRYHUQLQJFRQGLWLRQVIRU VWDELOLW\RIWKHV\VWHPZLWKRXWGDPSLQJDQGJ\URVFRSLFFRXSOLQJUHDG
55
D
' % % F
\
F 9
+ 55
P
'
9
+ 55
P
9
H
H
%
W
W H
' % H
W
!
& & % D
&
F
F
\
!
P
&
F
P
F
\
H
P
& & % D
+ 55 P
!
64DQGD 6WKHFRQGLWLRQ FRUUHVSRQGVZLWK DQGWKHFRQGLWLRQ
:HPD\DVFHUWDLQWKDWIRUWKHVSHFLDOFDVHRIDULJLGZKHHOZKHUH&) W
9
F
&
F
\
P
!
WKDWUHVXOWVIURP FRUUHVSRQGVZLWKFRQGLWLRQ LIZHUHDOLVHWKDWLQWKH FRQILJXUDWLRQVWXGLHGKHUHZHKDYH P$
DQG,
,]
)RUWKHFDVHZLWKDQHODVWLFW\UHZLWKVWDELOLW\FRQGLWLRQV ZHZLOO SUHVHQWVWDELOLW\GLDJUDPVDQGVKRZWKHHIIHFWVRIFKDQJLQJWKHVWLIIQHVVHV,Q DGGLWLRQEXWWKHQQHFHVVDULO\VWDUWLQJRXWIURP(TV WKHLQIOXHQFH RIWKHVWHHUDQGWUHDGGDPSLQJDQGWKDWRIWKHJ\URVFRSLFFRXSOLQJWHUPVZLOOEH DVVHVVHG )LUVW OHW XV FRQVLGHU WKH V\VWHP ZLWK ODWHUDO VXVSHQVLRQ FRPSOLDQFH EXW ZLWKRXWVWHHUWRUVLRQDOVWLIIQHVVF
WKDWLV DIUHHO\VZLYHOOLQJZKHHOSRVVLEO\
GDPSHG JRYHUQHG E\ WKH FRPELQHG FRHIILFLHQW N
9 DQG VXEMHFWHG WR WKH
DFWLRQ RI WKH J\URVFRSLF FRXSOH WKDW DULVHV ZKHQ WKH ODWHUDO GHIOHFWLRQ LV FRQQHFWHG ZLWK FDPEHU GLVWRUWLRQ DV JRYHUQHG E\ D QRQ]HUR FRHIILFLHQW
J\U
)URPWKHDERYHFRQGLWLRQVLWFDQEHVKRZQWKDWIRUWKHVLPSOHV\VWHPZLWKRXW GDPSLQJDQGFDPEHUFRPSOLDQFHVWDELOLW\PD\EHDFKLHYHGRQO\DWWKHDFDGHPLF FDVHRIODUJHFDVWHUH ! D ZKHQWKHODWHUDOVWLIIQHVVRIWKHZKHHOVXVSHQVLRQ LVVXIILFLHQWO\ODUJH
F
\!
P H
% %
W
H
& & D
ZKLOH H
!
D
%
7KHPLQLPXPVWLIIQHVVZKHUHVWDELOLW\PD\RFFXULV
7+(:+((/6+,00<3+(120(121
F
\ FULW '
% % % ]
P D
W
L
ZLWKL]UHSUHVHQWLQJWKHUDGLXVRILQHUWLD
L
]
'
P
,
'
]
PD
7KHYDOXHRIWKHFDVWHUOHQJWKHFZKHUHVWDELOLW\FRPPHQFHV DWF\FULW LV H
F'
D
% %
L
]
&RQVHTXHQWO\LWPD\EHFRQFOXGHGWKDWZLWKUHVSHFWWRWKHWKLUGRUGHUV\VWHPRI )LJ WKH LQWURGXFWLRQ RI ODWHUDO VXVSHQVLRQ FRPSOLDQFH VKLIWV WKH XSSHU ERXQGDU\RIWKHDUHDRILQVWDELOLW\WRYDOXHVRIFDVWHUOHQJWKHODUJHUWKDQD $WWKHVDPHWLPHDQHZDUHDRILQVWDELOLW\HPHUJHVIURPDERYH:KHQWKHODWHUDO VWLIIQHVVEHFRPHVORZHUWKDQWKHFULWLFDOYDOXH WKHVWDEOHDUHDYDQLVKHV HF
DOWRJHWKHUDWH
7KHFDVHZLWKVWHHUDQGWUHDGGDPSLQJVKRZVUDQJHVRIVWDELOLW\WKDWYDU\ ZLWK VSHHG RI WUDYHO 9 ,Q )LJ DUHDV RI LQVWDELOLW\ KDYH EHHQ GHSLFWHG IRU GLIIHUHQWYDOXHVRIWKHODWHUDOVWLIIQHVV7KHXSSHUOHIWLQVHWLOOXVWUDWHVWKHFDVH ZLWKRXW GDPSLQJ $W YDQLVKLQJ ODWHUDO VWLIIQHVV D QDUURZ UDQJH RI VWDELOLW\ DSSHDUVWRUHPDLQLQWKHQHJDWLYHWUDLOUDQJHZLWKDPDJQLWXGHVPDOOHUWKDQWKH SQHXPDWLFWUDLOW :KHQ D ILQLWH KHLJKW K RI WKH ORQJLWXGLQDO WRUVLRQ D[LV LV FRQVLGHUHG SDUDPHWHU
J\U
GHILQHGE\ REWDLQVDYDOXHODUJHUWKDQ]HUR)RU
J\U
WKHGLDJUDPRI)LJDULVHV$QLPSRUWDQWSKHQRPHQRQDSSHDUVWRRFFXUWKDW LVHVVHQWLDOIRUWKLVWZRGHJUHHRIIUHHGRPV\VWHP QRWFRXQWLQJWKHµKDOI¶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
WKH IRUPXOD
9
DLVREWDLQHGE\XVLQJ
,Q 7DEOH WKH QRQGLPHQVLRQDO IUHTXHQFLHV
ZDYHOHQJWKV DQGSDWKIUHTXHQFLHV
D KDYHEHHQSUHVHQWHGDVFRPSXWHG
IRUWKHIRXUGLIIHUHQWYDOXHVRIQRQGLPHQVLRQDOYHORFLWLHVZKHUHWKHWUDQVLWLRQ IURPVWDEOHWRXQVWDEOHRUYLVDYHUVDRFFXUV$OVRWKHPRGHRIYLEUDWLRQLVTXLWH GLIIHUHQWIRUWKLVµJ\URVFRSLF¶VKLPP\H[KLELWLQJDFRQVLGHUDEO\ODUJHUODWHUDO PRWLRQ DPSOLWXGH RI WKH ZKHHO FRQWDFW FHQWUH ZLWK UHVSHFW WR WKH \DZ DQJOH
7+(:+((/6+,00<3+(120(121
DPSOLWXGH WKDQ LQ WKH RULJLQDO µODWHUDO W\UH FRPSOLDQFH¶ VKLPP\ PRGH 7R LOOXVWUDWHWKLVWKHWDEOHVKRZVWKHUDWLRRIDPSOLWXGHVRIWKHODWHUDOGLVSODFHPHQW \ RIWKHFRQWDFWFHQWUH FI)LJ DQGWKH\DZDQJOHPXOWLSOLHGZLWKKDOIWKH FRQWDFWOHQJWK D
)RUWKHWZRORZHUYDOXHVRIVSHHGWKHODWHUDOGLVSODFHPHQW
E
DSSHDUVWRODJEHKLQGWKH\DZDQJOHZLWKDERXW ZKLOHIRUWKHWZRKLJKHU
E
VSHHGVWKHSKDVHODJDPRXQWVWRDERXW 7DEOH 1RQGLPHQVLRQDO IUHTXHQFLHV ZDYHOHQJWKV DQG UDWLR RIDPSOLWXGHVRIODWHUDO GLVSODFHPHQW\ DWFRQWDFWFHQWUHDQG\DZDQJOH
9
D
D
WLPHVKDOIFRQWDFWOHQJWK IRU
H
VSHHGVDWVWDELOLW\ERXQGDULHVDVLQGLFDWHGLQ)LJDWWUDLO
_\ D
_
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
F\ P :H PD\ LQWURGXFH DQ HIIHFWLYH \DZ VWLIIQHVV DQG DQ
HIIHFWLYHODWHUDOVWLIIQHVV
(
F
'
F
&
F
\
P
DQG F
( \
'
F
\ & PF
DQGHVWDEOLVKDVLQJOHGLDJUDPIRUWKHUDQJHVRIVWDELOLW\IRUWKHWZRHIIHFWLYH VWLIIQHVVHVDVSUHVHQWHGLQ)LJ7KHZD\LQZKLFKWKHERXQGDULHVDUHIRUPHG PD\EHFODULILHGE\WKHGLDJUDPRI)LJ 7KHSDUDERODUHSUHVHQWVWKHYDULDWLRQ
RI WKH VHFRQG +XUZLW] GHWHUPLQDQW + GLYLGHG E\ P 9 FI (T DV D IXQFWLRQRIWKHFDVWHUOHQJWKH7KLVVDPHWHUPDOVRDSSHDUVLQWKHH[SUHVVLRQIRU
WKHIRXUWK+XUZLW]GHWHUPLQDQW+DFFRUGLQJWR )RUERWKFDVHVF !DQG
F \!WKHVWUDLJKWOLQHRULJLQDWLQJIURPWKHSRLQW D DWDVORSHF DQGF \P UHVSHFWLYHO\ KDYH EHHQ GHSLFWHG 7KH SRLQWV RI LQWHUVHFWLRQ ZLWK WKH SDUDEROD
7+(:+((/6+,00<3+(120(121
FRUUHVSRQGWRYDOXHVRIHZKHUH+
'HSHQGLQJRQWKHVLJQVRIIDFWRUHWDQG
RI+DQGDWKHUDQJHVRIVWDELOLW\FDQEHIRXQG)LYHW\SLFDOFDVHV$(KDYH EHHQLQGLFDWHG7KH\FRUUHVSRQGWRVLPLODUFDVHVLQGLFDWHGLQ)LJ&DVH'
DWVPDOOQHJDWLYHHOLHVLQVLGHWKHVPDOOVWDEOHWULDQJOHRI)LJ+HUHF !
HW!DQG+P 9!F DH &DVH&VKRZVDUDQJHRIHZKHUH+!EXW DWWKHVDPHWLPH+ VRWKDWLQVWDELOLW\SUHYDLOV)RUODUJHUQHJDWLYHYDOXHVRI WKHFDVWHUVWDELOLW\DULVHVEHFDXVHH WFKDQJHVVLJQZKLOHDUHPDLQVSRVLWLYH
$OVRIRUODUJHUYDOXHVRIF DVWDEOHDUHDDSSHDUVWRVKRZXSIRUH W)RUFDVHV OLNH'DQG(VWDELOLW\RFFXUVIRUODUJHYDOXHVRIWKHPHFKDQLFDOWUDLOH QRWYHU\ UHDOLVWLFIRURXUSUHVHQW DVVXPSWLRQWKDWI
ZKLFKPHDQV FJRQ LQFOLQHG
VWHHULQJD[LV ,QDVLPLODUODUJHUDQJHRIHVWDELOLW\RFFXUVIRUODUJHHIIHFWLYH ODWHUDOVWLIIQHVVHJFDVH$7KLVFDVHKDVDOUHDG\EHHQUHIHUUHGWRSUHYLRXVO\ ZKHQGLVFXVVLQJWKHVLWXDWLRQZLWKRXWVWHHUVWLIIQHVVFI)LJXSSHUOHIWLQVHW (T &DVH%VKRZVLQVWDELOLW\IRUDOOYDOXHVRIH$VKDVEHHQLQGLFDWHGLQ )LJZHFDQHVWDEOLVKWKHDEVFLVVDF E\VKLIWLQJWKHYHUWLFDOD[LVWRWKHOHIW RYHUDGLVWDQFHF\ P$WWKHOHIWRIWKHVKLIWHGD[LVWKHVWHHUVWLIIQHVVPXVWEH
FRQVLGHUHGHTXDOWR]HURVRWKDWF \ OHIWWRJHWKHUZLWKWKHF
F\7KHVWDELOLW\ERXQGDU\D
VKLIWVWRWKH
D[LV
,Q)LJWKHGLDJUDPRI)LJLVUHSHDWHGEXWRQO\IRUWKHYHUVLRQZLWK F DVDEVFLVVDDQGIRURQHIL[HGYDOXHRIF\)LJXUHVDQGFOHDUO\VKRZWKH LQIOXHQFHRIGDPSLQJ7KHVWDEOHDUHDVLQFUHDVHLQVL]HZKLOHWKHXQVWDEOHDUHD
-F
WHQGVWRVSOLWLWVHOILQWRWZRSDUWVWKHKLJKHUVWLIIQHVVSDUWRIZKLFK F !
P
\
DSSHDUVWRYDQLVKDWVXIILFLHQWGDPSLQJ$OVRLQWKHORZ\DZVWLIIQHVVUDQJHWKH LQVWDELOLW\ LV VXSSUHVVHG HVSHFLDOO\ LQ WKH ORZHU UDQJH RI VSHHG )RU D FDVWHU OHQJWKFORVHWR]HURLWDSSHDUVSRVVLEOHWRDFKLHYHVWDELOLW\WKURXJKDGDSWLQJWKH VWHHUVWLIIQHVV PDNLQJLWDELWODUJHUWKDQF\PFI&DVH'RI)LJ DQGRU E\VXSSO\LQJVXIILFLHQWGDPSLQJ 7KHPRUHFRPSOH[VLWXDWLRQZKHUHJ\URVFRSLFFRXSOLQJLVLQFOXGHGKDVEHHQ FRQVLGHUHGLQ)LJV)RUYDQLVKLQJVWHHUVWLIIQHVVF
6WKHVLWXDWLRQRI
)LJVLVUHDFKHGDJDLQ,Q)LJZKHUHWKHGDPSLQJLV]HURWKHFXUYHDW ORZVSHHGUHVHPEOHVWKHJUDSKRI)LJ$VZDVREVHUYHGWRRFFXULQWKH]HUR \DZVWLIIQHVVFDVHZHVHHWKDWLQWKHORZHU\DZVWLIIQHVVUDQJH F
-F
\
P DQ
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
HDQGWKHSHFXOLDU
7+(:+((/6+,00<3+(120(121
βJ\U = 0, _Fψ =0
no damping
6
H_
00 11
2
4
2
2.7 2.7
4
VWDEOH
NB = 0.25,κB * 1
33
_ F\= 3.7
<3.66
D σ _ +_
F_\\= 22
F_\\=333
)LJ ,QIOXHQFH RI ODWHUDO
VXVSHQVLRQ
FRPSOLDQFH
ZLWK
QRQ
GLPHQVLRQDO VWLIIQHVV
F\
RQ VWDELOLW\ *\URVFRSLF
RVFLOODWRULO\ RVFLOODWRULO\
1
XQVWDEOH XQVWDEOH
FRXSOHDQGVWHHUVWLIIQHVV QRWUHJDUGHG
00
0
0 -_W
-_W GLYHUJHQWO\
0
XQVWDEOH
10
9_
20
30
NB = 0.25, κB * 1
H_
XQVWDEOH XQVWDEOH
0.5 0.5 0.5
0
P
W
FDPEHU
VXV
D σ _ +_
VWDEOH
J\URVFRSLF
FRXSOLQJ
6WHHUVWLIIQHVVLVHTXDOWR ]HUR $WLQFUHDVLQJFRPSOLDQFH
4
22 4 4 55
F
\DQHZDUHDRIRVFLO
55
44
2
ω _ = 4.5
2.4
0.84
2
1
0.1 0.1 0.5 0.5
0
)LJ ,QIOXHQFH RI ODWHUDO
F_F\\==11 2
SHQVLRQFRPSOLDQFHZLWK
RVFLOODWRULO\ RVFLOODWRULO\
4
I
HWHUV
40
βJ\U = 0.2, _Fψ = 0
6
1RQGLPHQVLRQDOSDUDP
GLYHUJHQWO\ XQVWDEOH
ODWRU\ LQVWDELOLW\ VKRZV
1
XS ZKLOH WKH RULJLQDO
0.5 F_\= 0.1
0.5 0
-_W 10
20
9_
30
DUHDVKULQNV )UHTXHQFLHV KDYH EHHQ LQGLFDWHG IRU WKH FDVH
F\
40
H
DW
βJ\U = 0.2, _Fψ =0
6
H_
no damping
VWDEOH
σ D _ +_
4 1
ODWHUDO
FF\\== 44 _
20 20
J\URVFRSLF
A A
GDPSLQJHTXDOWR]HUR
0.5 0
B
-_W 0
GLYHUJHQWO\ XQVWDEOH
10
20
FRXSOLQJ
]HUR 6WHHU DQG WUHDG
20
44
VXV
6WHHUVWLIIQHVVLVHTXDOWR
XQVWDEOH XQVWDEOH
1
FDPEHU
SHQVLRQFRPSOLDQFHZLWK
RVFLOODWRULO\ RVFLOODWRULO\
2
0
)LJ ,QIOXHQFH RI
9_
30
40
(LJHQYDOXHV
DQG
HLJHQYHFWRUVRISRLQWV$ DQG % DUH DQDO\VHG LQ )LJV
7+(:+((/6+,00<3+(120(121
F_ψ* = F_ψ - F_ \ , P_
F\* _
F_ \ - F_
P_ P_ =
stable ranges
ψ
+4 >0
+4>0 +2>0 D5> 0
C
+4>0
E D A
_ P9 _ 2_ +2
+2< 0
* _ψ E F (>0)
H_ W_ >0
F_ψ
)LJ
F*\ P _ _ (>0) F* >F_*\FULW) (_
P1_
(>0) D F * _ψ
D5> 0
F*\ < F_*\FULW B _
\
W _
+4 >0
0
+2< 0 +4 >0
+4<0
H_
σ _ +D _
+4<0
C
(VWDEOLVKLQJWKHUDQJHVRIVWDELOLW\XVLQJWKHFRQGLWLRQV ZKLFKKROGIRUWKH ILIWKRUGHU V\VWHP ZLWKRXW GDPSLQJ DQG J\URVFRSLF FRXSOLQJ 7KH FDVHV $( FRUUHVSRQGWRWKRVHLQGLFDWHGLQWKHVWDELOLW\GLDJUDPRI)LJ
_ F P \ _
8
A
H_
B
C
D
+4=0
βJ\U= 0 no damping
E
F_ψ* = F_ψ - _F\ P_ F\* F_\ _ = -F P_ P_ _ψ
VWDEOH
VWDEOH
+4 =0
RVFLOODWRULO\
4
_ +_ σ D
XQVWDEOH
+2=0 0
D5=0
GLYHUJHQWO\ XQVWDEOH
-2
9 _ F\ P _ 8
4
F_ψ =0
RVFLOO XQVWDEOH
2 _ F*\ P _ 1
0
1
2
0
1
F_ψ
3
+4=0 F_ψ*
-_W
2
3
4
5
)LJ %DVLFVWDELOLW\GLDJUDPDFFRUGLQJWRFRQGLWLRQV IRUWKHILIWKRUGHUV\VWHP ZLWKRXW GDPSLQJ DQG J\URVFRSLF FRXSOLQJ 7KH FDVHV $( FRUUHVSRQG WR WKRVH LQGLFDWHGLQWKHVWDELOLW\GLDJUDPRI)LJ 7KHGLDJUDPKROGLQJHQHUDOWKHFDVH
F P
ZLWK \
P
LVDQH[DPSOHQRWH
7+(:+((/6+,00<3+(120(121
F\ P _ _
βJ\U = 0, _F\=1
5
no damping
+4 =0
H_ +2 =0
_ +D σ _
VWDEOH
RVFLOODWRULO\ RVFLOODWRULO\
)LJ6WDELOLW\
XQVWDEOH XQVWDEOH
GLDJUDP DFFRUGLQJ WR )LJ
0
0
D5=0 -2
0 GLYHUJ
1
XQVWDEOH XQVWDEOH
F_ψ
XQVWDEOH
5 55
9__ ==2.5 2.5 H_
VWDEOH
YDOXH RI WKH ODWHUDO VWLIIQHVV
4
5
J\URV
FRSLFFRXSOLQJ
NB= 0.125, κ_ * 0.5
RVFLOODWRULO\ RVFLOODWRULO\
RVFLOODWRULO\
XQVWDEOH XQVWDEOH
XQVWDEOH
20 20
2.5 2.5
)LJ 6DPH DV )LJ
1
2
9_ ==55
ZLWK
$UHDVRILQVWDELOLW\DW
VWDEOH
GLIIHUHQW
10 10 XQVWDEOH
EXW
VRPHGDPSLQJ
5
0 GLYHUJ
Z LWKRXW RU
9_ =10
20
5
DEVFLVVD DW D IL[HG
GDPSLQJ
3
VWHHU
DORQJ
βJ\U= 0, _F\=1
10 10 20 20
0
-2
+4 =0
RVFLOO RVFLOO
2
-_W
ZLWK
VWLIIQHVV
F_ψ
OHYHOV
RI
VSHHG
3
4
RVFLOO
5
XQVWDEOH
βJ\U= 0, _F\=1
NB= 0.25, κB* 1
10 20 10 20
RVFLOODWRULO\ RVFLOODWRULO\
H_
XQVWDEOH XQVWDEOH
2.5 2.5
VWDEOH
0 )LJ 6DPH DV
20
-2
)LJ EXW ZLWK PRUHGDPSLQJ
0 GLYHUJ XQVWDEOH
1
2
F_ψ
3
4
5
7+(:+((/6+,00<3+(120(121
5
βJ\U = 0.2, F_\=1
2.5
VWDEOH
no damping
10
H_
9= 5
RVFLOODWRULO\ RVFLOODWRULO\ XQVWDEOH XQVWDEOH
0
)LJ'LDJUDPIRUWKH FDVHRI)LJEXW ZLWK
10
-2
0
9 = 2.5 2.5
1
GLYHUJ
2
XQVWDEOH
F_ψ
3
J \ U R V FRSLF
FRXSO L Q J
$UHDV RI LQVWDELOLW\ DW
4
5
GLIIHUHQWOHYHOVRIVSHHG
βJ\U = 0.2, F_\=1
5
NB= 0.125,κB*= 0.5
VWDEOH
2.5
10
H_
RVFLOODWRULO\XQVWDEOH RVFLOODWRULO\XQVWDEOH
9 == 55
0 VWDEOH
2.5 -2
)LJ 6DPH DV )LJ
0 GLYHUJ
1
2
XQVWDEOH
F_ψ
3
4
5
EXW
ZLWK
VRPH
GDPSLQJ
βJ\U= 0.2, _F\=1
5
NB = 0.25, κB * 1
RVFLOODWRULO\
XQVWDEOH
VWDEOH
H_ 2.5 2.5
9 == 10 10
5
0 VWDEOH
-2
)LJ 6DPH DV )LJ
0 GLYHUJ XQVWDEOH
1
2
F_ψ
3
4
5
EXW
GDPSLQJ
ZLWK
PRUH
7+(:+((/6+,00<3+(120(121
I
FDVH ZLWK
H
!
ZKLFK PD\ EH EHQLILFLDO WR VXSSUHVV VKLPP\ )RU WKH
FRQILJXUDWLRQWKDW
I HDQGGDPSLQJDQGJ\URVFRSLFFRXSOLQJDUHGLVUHJDUGHG
WKHIROORZLQJFRQGLWLRQVIRUVWDELOLW\KROG
D
+
!
+
!
F
!
F ' & H % W
!
L ] ' P ! % D W PLQ F F F PD[ F F
H
ZLWKWKHWZRSDUDEROLFIXQFWLRQVRI
& HF\ F \ % % F H % D & F \ H & W P % D P \ F\ H % W & HPW
F ' F '
PW
)LJXUHGHSLFWVWKHERXQGDULHVRIVWDELOLW\DFFRUGLQJWRWKHDERYHIXQFWLRQV 7KHRUGLQDWHVRIWZRFKDUDFWHULVWLFSRLQWVKDYHDOUHDG\EHHQJLYHQLQWKHILJXUH
H
)RUWKHWKUHHRWKHUSRLQWVZHKDYHWKH YDOXHV
L]
H '&
%D
H '
W
F\ % D % W
L H ' ] W
7KHDSSHDUDQFHRIWKHVWDELOLW\GLDJUDPLVTXLWHGLIIHUHQWZLWKUHVSHFWWRWKDW RI)LJZKHUH
W H
RIWKHWUDLO
I
6WLOODQDUHDRILQVWDELOLW\H[LVWVLQEHWZHHQWKHOHYHOV
D/LNHLQWKHI
FDVH FOHDUO\YLVLEOHLQ)LJ ZH
KDYHWZRGLIIHUHQWUDQJHVRILQVWDELOLW\ DUHDDQGDUHDZKHUHGLIIHUHQWPRGHV
W H
RIXQVWDEOHRVFLOODWLRQVRFFXU,QWKHUDQJHRIWKHFDVWHUOHQJWK
D
WKHVH
DUHDVFRUUHVSRQGWRWKHORZHUDQGWKHKLJKHU\DZVWLIIQHVVUHJLRQV )RU WKH SRLQWV $ DQG % ZKLFK OLH RQ WKH ERXQGDULHV RI WKH ORZHU DQG WKH KLJKHUVWLIIQHVVLQVWDELOLW\DUHDVUHVSHFWLYHO\WKHPRGHVRIYLEUDWLRQKDYHEHHQ DQDO\VHG7KHFRQGLWLRQVIRUVWDELOLW\ RIWKHXQGDPSHGV\VWHPZLWKRXW J\UR¶V DUH LQGHSHQGHQW RI WKH VSHHG :KHQ FKDQJLQJ WKH VSHHG RI WUDYHO WKH
7+(:+((/6+,00<3+(120(121
GDPSLQJRIWKHPRGHWKDWLVMXVWRQDERXQGDU\RILQVWDELOLW\UHPDLQVHTXDOWR ]HUR7KHRWKHUPRGHVKRZVDSRVLWLYHGDPSLQJZKLFKFKDQJHVZLWKVSHHGDV GRHV WKH IUHTXHQF\ RI WKLV VWDEOH PRGH ,Q WKH )LJXUHV DQG IURP %HVVHOLQN WKHGDPSLQJIUHTXHQF\DPSOLWXGHUDWLRDQGSKDVHOHDGRIWKH ODWHUDOGLVSODFHPHQWRIWKHFRQWDFWFHQWUH\ ZLWKUHVSHFWWRWKH\DZDQJOH
KDYH
EHHQ SUHVHQWHG IRU WKH WZR PRGHV RI YLEUDWLRQ LQ )LJ IRU FDVH $ DQG LQ )LJ IRUFDVH%RI)LJ8VLQJDUHIHUHQFHW\UHUDGLXVUUHIWKHTXDQWLWLHV
\ DQG9 KDYHEHHQGHILQHGDVIROORZV \
\UUHI>@9
9UUHI>VHF@)URPWKH
JUDSKVFDQEHREVHUYHGWKDWWKHORZHU\DZVWLIIQHVVFDVH$VKRZVDPRGHRQWKH YHUJHRILQVWDELOLW\ PRGH WKDWKDVDORZHUIUHTXHQF\DQGDODUJHUDPSOLWXGH
UDWLR_\
_WKDQPRGHLQFDVH%(YLGHQWO\FDVH%H[KLELWVDPRUHSURQRXQFHG
\DZRVFLOODWLRQZKHUHDVWKHPRWLRQSHUIRUPHGE\PRGHRIFDVH$FRPHVFORVHU WRDODWHUDOWUDQVODWLRQDORVFLOODWLRQ7KHQH[WVHFWLRQGLVFXVVHVWKHPRGHVKDSHV RI WKHVH SHULRGLF PRWLRQV LQ UHODWLRQ ZLWK WKH HQHUJ\ IORZ LQWR WKH XQVWDEOH V\VWHP
βJ\U = 0, I = H
H_
H_3
RVFLOOXQVWDEOH RVFLOOXQVWDEOH
+4 =0 D_ +σ_ + F_
1 \
VWDEOH
A
VWDEOH
H_1 GLYHUJHQW XQVWDEOH 0
B
1
XQVWDEOH XQVWDEOH
0
_W
F_ψ = F_ψ3 (H_ )
RVFLOODWRULO\ RVFLOODWRULO\
H_ 2
no damping
2
_ Fψ = _Fψ1(H_)
F_ψ = F_ψ2(H_ )
F_ψ I
H
H
)LJ %DVLFVWDELOLW\GLDJUDPIRUWKHFDVH FJGLVWDQFH EHKLQGVWHHUD[LV 6WHHU DQG WUHDG GDPSLQJ DUH HTXDO WR ]HUR *\URVFRSLF FRXSOLQJ LV GLVUHJDUGHG 7KH
F
IXQFWLRQV
L
H
DUHGHILQHGE\(TV (LJHQYDOXHVDQGHLJHQYHFWRUVRI
SRLQWV$DQG%DUHDQDO\VHGLQ)LJ )URP %HVVHOLQN
7+(:+((/6+,00<3+(120(121
frequency [Hz] 25
_ *ψ amplitude ratio \ 3
20
2.5
mode 1 mode 2
case A
2
15
1.5 10
1
5 0 50
0.5 100
150
200
damping [%] 5
0 50
100 _ *ψ phase lead \ 180
4
150
3
120
2
90
1
60
0
30
-1 50
100 150 velocity 9 *
200
0 50
150
200
100 150 velocity 9 *
200
)LJ1DWXUDOIUHTXHQFLHVGDPSLQJDQGPRGHVKDSHVIRU&DVH$RI)LJ
frequency [Hz] 25
_ *ψ amplitude ratio \ 3
20
2.5
mode 1 mode 2
case B
2
15
1.5 10
1
5 0 50
0.5 100
150
200
damping [%] 5
0 50
100 _ * phase lead \ψ 180
4
150
3
120
2
90
1
60
0
30
-1 50
100 150 velocity 9 *
200
0 50
150
200
100 150 velocity 9 *
200
)LJ 1DWXUDOIUHTXHQFLHVGDPSLQJDQGPRGHVKDSHVIRU&DVH%RI)LJ
7+(:+((/6+,00<3+(120(121
%HVVHOLQNKDVDQDO\VHGWKHHIIHFWVRIFKDQJLQJDQXPEHURISDUDPHWHUVRID UHDOLVWLFWZRZKHHOVLQJOHD[OHPDLQODQGLQJJHDUPRGHOFRQILJXUDWLRQ RIFLYLO DLUFUDIWDOVRVHH9DQGHU9DON 'HJUHHVRIIUHHGRPUHSUHVHQW D[OH\DZ DQJOHVWUXWWODWHUDOGHIOHFWLRQD[OHUROODQJOHDQGW\UHODWHUDOGHIOHFWLRQ VWUDLJKW WDQJHQW 7KLV PDNHV WKH WRWDO RUGHU RI WKH V\VWHP HTXDO WR VHYHQ ,PSRUWDQW FRQFOXVLRQV UHJDUGLQJ WKH LPSURYHPHQW RI DQ H[LVWLQJ V\VWHP DLPLQJ DW DYRLGDQFHRIVKLPP\KDYHEHHQGUDZQ7ZRPXWXDOO\GLIIHUHQWFRQILJXUDWLRQV KDYHEHHQVXJJHVWHGDVSRVVLEOHVROXWLRQVWRWKHSUREOHPZLWKWKHFJSRVLWLRQ IDVVXPHGWRUHPDLQHTXDOWRWKHWUDLOH
ODUJHSRVLWLYHWUDLO H! ³$QLQFUHDVHRIWKHPHFKDQLFDOWUDLODQGDUHGXFWLRQRIWKHODWHUDOVWLIIQHVVDUH
UHTXLUHG WR LPSURYH VWDELOLW\ DW KLJK IRUZDUG YHORFLW\ ,QFUHDVLQJ WKH \DZ VWLIIQHVVZLOOJHQHUDOO\LPSURYHWKHVKLPP\VWDELOLW\RIDJHDUZLWKDSRVLWLYH PHFKDQLFDO WUDLO SURYLGHG WKDW VRPH VWUXFWXUDO GDPSLQJ LV SUHVHQW $ OLPLWHG UHGXFWLRQRIWKHUROOVWLIIQHVVLPSURYHVWKHVWDELOLW\DWKLJKIRUZDUGYHORFLWLHV 7KH\DZPRPHQWRILQHUWLDVKRXOGEHNHSWDVVPDOODVSRVVLEOHLQFDVHRIDODUJH SRVLWLYHPHFKDQLFDOWUDLO´
VPDOOQHJDWLYHWUDLO H ³)RUDODQGLQJJHDUZLWKDQHJDWLYHWUDLODQXSSHUERXQGDU\H[LVWVIRUWKH\DZ
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´ ,Q FRPSDULVRQ ZLWK WKH EDVHOLQH FRQILJXUDWLRQ WKH SURSRVHG PRGLILFDWLRQV UHVXOW LQ D PDMRU JDLQ LQ VWDELOLW\ 7KLV LOOXVWUDWHV WKDW LW PD\ EH SRVVLEOH WR GHVLJQ D VWDEOH FRQYHQWLRQDO ODQGLQJ JHDU SURYLGHG WKDW WKH FRPELQDWLRQ RI SDUDPHWHUYDOXHVLVVHOHFWHGFRUUHFWO\ )RUGHWDLOHGLQIRUPDWLRQDOVRRQWKHUHVXOWVRIVLPXODWLRQVZLWKDPXOWLERG\ FRPSOH[PRGHOLQFRPSDULVRQZLWKIXOOVFDOHH[SHULPHQWVRQDODQGLQJDLUFUDIW H[KLELWLQJVKLPP\ZHUHIHUWRWKHRULJLQDOZRUNRI%HVVHOLQN
7+(:+((/6+,00<3+(120(121
6KLPP\DQG(QHUJ\)ORZ
7RVXVWDLQWKHXQVWDEOHVKLPP\RVFLOODWLRQHQHUJ\PXVWEHWUDQVPLWWHGµIURPWKH URDGWRWKHZKHHO¶:HUHDOLVHWKDWXOWLPDWHO\WKHSRZHUFDQRQO\RULJLQDWHIURP WKHYHKLFOH¶VSURSXOVLRQV\VWHP7KHUHODWLRQEHWZHHQXQVWDEOHPRGHVDQGWKH VHOIH[FLWDWLRQHQHUJ\JHQHUDWHGWKURXJKWKHURDGWRW\UHVLGHIRUFHDQGDOLJQLQJ PRPHQW LV GLVFXVVHG LQ GHWDLO LQ VXEVHFWLRQ 7KH WUDQVLWLRQ RI GULYLQJ HQHUJ\ RUWKHYHKLFOHIRUZDUGPRWLRQNLQHWLFHQHUJ\ WRVHOIH[FLWDWLRQHQHUJ\ LV DQDO\VHG LQ VXEVHFWLRQ 2EYLRXVO\ EHVLGHV IRU WKH VXSSO\ RI VHOI H[FLWDWLRQHQHUJ\WKHGULYLQJHQHUJ\LVQHHGHGWRFRPSHQVDWHIRUWKHHQHUJ\WKDW LVGLVVLSDWHGWKURXJKSDUWLDOVOLGLQJLQWKHFRQWDFWSDWFK
8QVWDEOH0RGHVDQGWKH(QHUJ\&LUFOH
7RVWDUWRXWZLWKWKHSUREOHPPDWWHUOHWXVFRQVLGHUWKHVLPSOHWUDLOLQJZKHHO V\VWHPRI)LJZLWKWKH\DZDQJOH
UHSUHVHQWLQJWKHRQO\GHJUHHRIIUHHGRP
RIWKHZKHHOSODQH,IWKHPHFKDQLFDOWUDLOLVHTXDOWR]HURLWLVRQO\WKHDOLJQLQJ WRUTXHWKDWH[HUWVZRUNZKHQWKHZKHHOSODQHLVURWDWHGDERXWWKHVWHHULQJD[LV ,IWKH\DZDQJOHYDULHVKDUPRQLFDOO\
V
'
R
VLQ
V
V
WKHPRPHQWUHVSRQVHEHFRPHVIRUVPDOOSDWKIUHTXHQF\
'&
0 V ]
&
0
R
VLQ
V
V
V
DSSUR[LPDWHO\
% k
ZKHUHWKHSKDVHDQJOHIROORZVIRULQVWDQFHIURPWUDQVIHUIXQFWLRQV RU RU7DEOH EHORZ)LJ
k ' k&
0
'&
0
V
' & %
D
V
WKHQHJDWLYHYDOXHRIZKLFKFRUUHVSRQGVWRWKHIDFWWKDWWKHPRPHQWODJVEHKLQG WKH\DZDQJOH7KHHQHUJ\WUDQVPLWWHGGXULQJRQHF\FOHFDQEHIRXQGDVIROORZV
V
V
'm
:
'
V
G ]
0
'
&
0
R
m
FRV
%
VLQ
G
'&
&
0
R
k
VLQ
!
$SSDUHQWO\EHFDXVHRIWKHQHJDWLYHSKDVHDQJOHWKHZRUNGRQHE\WKHDOLJQLQJ WRUTXHLVSRVLWLYHZKLFKPHDQVWKDWHQHUJ\LVIHGLQWRWKHV\VWHP$VDUHVXOWWKH XQGDPSHGZKHHOV\VWHPZLOOVKRZXQVWDEOHRVFLOODWLRQVZKLFKLVLQDJUHHPHQW ZLWKRXUHDUOLHUILQGLQJV,WLVRI LQWHUHVW WR QRWHWKDWLQFRQWUDVWWRWKHDERYH
7+(:+((/6+,00<3+(120(121
UHVXOWWKDWKROGVIRUDW\UHDURWDU\YLVFRXVGDPSHU ZLWKDSRVLWLYHSKDVHDQJOH
E
RI VKRZV D QHJDWLYH HQHUJ\ IORZ ZKLFK PHDQV WKDW HQHUJ\ LV EHLQJ GLVVLSDWHG ,IZHZRXOGFRQVLGHUDILQLWHFDVWHUOHQJWKHPRUHVHOIH[FLWDWLRQZLOODULVH GXHWRWKHFRQWULEXWLRQRIWKHVLGHIRUFHWKDWDOVRUHVSRQGVWRWKH\DZDQJOHZLWK SKDVHODJ$WWKHVDPHWLPHKRZHYHUDSDUWRI)\ZLOOQRZUHVSRQGWRWKHVOLS DQJOHDQGFRQVHTXHQWO\GDPSVWKHRVFLOODWLRQ7KHVOLSDQJOHDULVHVDVDUHVXOW RIWKHODWHUDOVOLSVSHHGZKLFKLVHTXDOWRWKH\DZUDWHWLPHVWKHFDVWHUOHQJWK )LQDOO\DWH
DWKHPRPHQWDQGWKHVLGHIRUFHDUHH[DFWO\LQSKDVHZLWKWKH
\DZ DQJOH FI VLWXDWLRQ GHSLFWHG LQ )LJ DQG WKH VWDELOLW\ ERXQGDU\ LV UHDFKHG $W H
W WKH PRPHQW DERXW WKH VWHHU D[LV YDQLVKHV LI WKH VLPSOLILHG
VWUDLJKWWDQJHQWRUVLQJOHFRQWDFWSRLQWPRGHOLVXVHG7KLVDOVRLPSOLHVWKDWWKH ERXQGDU\RIRVFLOODWRU\LQVWDELOLW\LVDWWDLQHG :H ZLOO QRZ WXUQ WR WKH PXFK PRUH FRPSOLFDWHG V\VWHP ZLWK ODWHUDO FRPSOLDQFH RI WKH VXVSHQVLRQ LQFOXGHG 7KH WZR FRPSRQHQWV RI WKH SHULRGLF RVFLOODWRU\ PRWLRQ RI WKH ZKHHO WKDW RFFXUV RQ WKH ERXQGDU\ RI RVFLOODWRU\ LQVWDELOLW\PD\EHGHVFULEHGDVIROORZV
2
' '
\ W W
D
VLQ
P
D
P
7KHTXDQWLW\
%
VLQ
UHSUHVHQWVWKHDPSOLWXGHUDWLR LQGLFDWHVWKHSKDVHOHDGRIWKH
ODWHUDOPRWLRQZLWKUHVSHFWWRWKH\DZPRWLRQDQG HTXDOV
WRU
V
V7KHHQHUJ\
:WKDWIORZVIURPWKHURDGWRWKHZKHHOGXULQJRQHF\FOHLVHTXDOWRWKHZRUN H[HFXWHGE\WKHVLGHIRUFHDQGWKHDOLJQLQJWRUTXHWKDWDFWRQWKHPRYLQJZKHHO SODQH+HQFHZHKDYH
:
'
)
G\ 2 \
G
%
0
]
G G
G
:KHQ:!HQHUJ\IORZVLQWRWKHV\VWHPDQGLIWKHV\VWHPLVFRQVLGHUHGWREH XQGDPSHG N
(
WKH FRQFOXVLRQ PXVW EH WKDW WKH PRWLRQ LV XQVWDEOH
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
NWR
7+(:+((/6+,00<3+(120(121
D
'
& %
P
%
D
V
FRV
&
V
% %
FRV
V
% &
D
V
VLQ
&
V
VLQ
%
:LWKWKHH[SUHVVLRQV IRU WKHIRUFHDQGWKHPRPHQWWKHLQWHJUDO EHFRPHV
'
D
P
:
&
)
%
& % %
V
% & &
D W
W
FRV
D
&
& % D
W
VLQ
V
V
V
2Q WKH ERXQGDU\ RI VWDELOLW\ : ZDYHOHQJWK
DQG IRU D JLYHQ SDWK IUHTXHQF\
DUHODWLRQVKLSEHWZHHQ V
V
RU
DQG UHVXOWVIURP 7R
HVWDEOLVK WKH IXQFWLRQDO UHODWLRQVKLS LW LV KHOSIXO WR VZLWFK WR &DUWHVLDQ FRRUGLQDWHV [
'
S
FRV
\
S
'
VLQ
7KHH[SUHVVLRQIRUWKHHQHUJ\QRZUHDGV D
'&
P
:
&
)
%
V
[
% % % & %
S
\
D
S
W
[
& % D
S
W
V
\
& %
D W
S
V
V
$SSDUHQWO\IRUDJLYHQYDOXHRI:WKLVUHSUHVHQWVWKHGHVFULSWLRQRIDFLUFOH)RU :
DQGDIWHUWKHLQWURGXFWLRQRIWKHZDYHOHQJWK DVSDUDPHWHUWKHIXQFWLRQ
WDNHVWKHIRUP
[
% % % & %
S
\
D
S
W
[
S
& % D
W
\
& %
D W
S
'
:KHQWKHPRGHVKDSHRIWKHPRWLRQRIWKHZKHHOSODQHGHILQHGE\
DQG RUE\
[SDQG\SLVVXFKWKDW LVVDWLVILHGWKHPRWLRQILQGVLWVHOIRQWKHERXQGDU\ RIVWDELOLW\7KHFHQWUHRI%HVVHOLQN¶VHQHUJ\FLUFOHLVORFDWHGDW
[SF
\
SF
'&
% & D
W
& % D
W
)URP LWFDQEHDVFHUWDLQHGWKDWLQGHSHQGHQWRI WKHFLUFOHVZLOODOZD\V SDVVWKURXJKWKHWZRSRLQWVRQWKH[SD[LV [S
D DQG[S
W:HKDYHQRZ
VXIILFLHQWLQIRUPDWLRQWRFRQVWUXFWWKHFLUFOHVZLWK DVSDUDPHWHU,Q)LJ WKHVHFLUFOHVKDYHEHHQGHSLFWHGWRJHWKHUZLWKFLUFOHVWKDWDULVHZKHQGLIIHUHQW W\UHPRGHOV DUHXVHG$VH[SHFWHGWKHFLUFOHVIRUWKHPRUHH[DFWW\UHPRGHOV GHYLDWHPRUHIURPWKRVHUHVXOWLQJIURPWKHVWUDLJKWWDQJHQWW\UHPRGHOZKHQWKH ZDYHOHQJWKEHFRPHVVPDOOHU7KDWFLUFOHVDULVHDOVRIRURWKHUPDVVOHVVPRGHOV
7+(:+((/6+,00<3+(120(121
90 120
exact
150
30
5D 10D
v.Schl. ex. 180 -20D
60
v.Schlippe
-10D
5D
-(σ +D)
20D
: >0 : >0
W 330
60D
phasor
0 20D
10D
40D
210
straight tangent
λ = 80D
240
ξ
η
300
270 )LJ 0RGHVKDSHSORW DPSOLWXGHUDWLR SKDVH ZLWK%HVVHOLQN¶V]HURHQHUJ\FLUFOHVIRU GLIIHUHQWZDYHOHQJWKVDQGW\UHPRGHODSSUR[LPDWLRQVLQGLFDWLQJPRGHVKDSHVIRU ZKLFKVKLPP\EHJLQVWRVKRZXS LQVWDELOLW\RFFXUVLIDWDSDUWLFXODUYDOXHRIWKH PRWLRQZDYHOHQJWK WKHSKDVRUHQGSRLQWJHWVLQVLGHWKHFRUUHVSRQGLQJFLUFOH $W VPDOOHUZDYHOHQJWKVWKHFLUFOHEHORQJLQJWRWKHVWUDLJKWWDQJHQWDSSUR[LPDWLRQEHJLQV WRDSSUHFLDEO\GHYLDWHIURPWKHH[DFWRU9RQ6FKOLSSHUHSUHVHQWDWLRQ
90
90
ξ
ξ
9 mode 1
W
- (σ +D)
9
mode 1
mode 2
mode 2
0
η
λ
W
-(σ +D)
0
η
λ 270
A
270
B
)LJ =HURHQHUJ\FLUFOHVIRUWKHFDVHV$DQG%RI)LJLQGLFDWLQJWKHPRGHVKDSHRI PRGHDWWKHERXQGDU\RIVWDELOLW\LQFDVH$DQGWKHVDPHIRUPRGHLQFDVH%
7+(:+((/6+,00<3+(120(121
RI WKH W\UH FDQ EH HDVLO\ SURYHQ ZKHQ FRQVLGHULQJ WKH IDFW WKDW DW D JLYHQ ZDYHOHQJWKZHKDYHDFHUWDLQDPSOLWXGHDQGSKDVHUHODWLRQVKLSRIWKHIRUFH)\ DQGWKHPRPHQW0¶ZLWKUHVSHFWWRWKHWZRLQSXWPRWLRQYDULDEOHV\ DQG ] WKHFLUFOHVSDVVWKURXJKWKHSRLQW
7KDW
D IROORZVIURPWKHHDUOLHUILQGLQJWKDW
WKH UHVSRQVH RI DOO µWKLQ¶ W\UH PRGHOV ZKHQ WKH ZKHHO LV VZLYHOOHG DURXQG D YHUWLFDOD[LVWKDWLVSRVLWLRQHGDGLVWDQWDQFH
DLQIURQWRIWKHZKHHOD[OH FI
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
ZLWK
WKH J\URVFRSLF FRXSOLQJ LQFOXGHG ,Q )LJ WKH FDVHV $ DQG % KDYH EHHQ LQGLFDWHG DW D WUDLO H
%HFDXVH RI WKH J\URVFRSLF DFWLRQ WKH FRQGLWLRQ IRU
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
ZLWK IUHTXHQF\
D WKHQFURVVHVWKHFLUFOHDW9
DQG FRUUHVSRQGLQJ
$ DQGILQDOO\HQGVXS
7+(:+((/6+,00<3+(120(121
Im
= 6
9 _
mode 1
pair
9_ = 14,
_ = -0.073+0.59L V
12, 11.2, 10, 9.05, A 8, 6,
A
14 6
0
α_ + ω_ L
_= V
\Dψ
λ_ =149
= -0.050+0.62L = -0.039+0.63L = -0.019+0.64L = 0.0 +0.65L = 0.024+0.66L = 0.079+0.66L
=121 =111 = 98 = 87 A = 76 = 57
1
Re
_W
14
zero energy circles
=6
9 _
phasor and circle pair
)LJ 6\VWHP RI )LJ ZLWK J\URFRSLF FRXSOLQJ PDNLQJ VWDELOLW\ VSHHGGHSHQGHQW &LUFOH VHJPHQWV DQG SKDVRUV RI PRGH VKDSHV RI PRGH IRU D VHULHV RI VSHHG YDOXHV FRYHULQJ WKH UDQJH LQ ZKLFK SRLQWV $ DQG % RI )LJ DUH ORFDWHG $W
9
LQFUHDVLQJVSHHGWKHSKDVRUFURVVHVWKHFLUFOHDW
LQWRWKHVWDEOHGRPDLQ
Im
mode 2 -(σ _ +D _)
_W
-6
λ_ = 12.6, = 16.5, = 18.4, = 20.0, B = 21.9, = 23.1, = 25.8,
V _=
-2
α_ + ω_
Re 2
0
L
L, 9 _ =6 = -0.036+3.05L, 8 = -0.024+3.09L, 9.05 = -0.014+3.14L, 10 = 0.0 +3.21L, 11.2 = 0.008+3.27L, 12 = 0.026+3.41L, 14
V _ = -0.051+2.99
9 _ = 14 12 11.2 B 10 9.05 8
B
6
\Dψ
)LJ &LUFOHVDQGSKDVRUVRIPRGHVKDSHVRIPRGHIRUVDPHVHULHVRIVSHHGYDOXHVDV
9
LQ)LJ$WLQFUHDVLQJVSHHGWKHSKDVRUFURVVHVWKHFLUFOHDW LWVLQVLGHZKLFKPHDQV HQWHULQJWKHXQVWDEOHGRPDLQ
WRZDUGV
7+(:+((/6+,00<3+(120(121
DWDVSHHG9
RXWVLGHWKHFLUFOHPHDQLQJWKDWWKLVPRGHLVQRZVWDEOH$WWKH
VDPHWLPHIROORZLQJPRGHLQ)LJZHREVHUYHWKDWILUVWWKHPRGHSRLQW OLHV RXWVLGH WKH FLUFOH DW 9
QRZ IRU WKH GLIIHUHQW IUHTXHQF\
ZDYHOHQJWK D FURVVHVWKHFLUFOHDW9
DQG
% DQGPRYHVWRWKHLQVLGH
RI WKH FLUFOH LQWR WKH XQVWDEOH GRPDLQ )URP WKH HLJHQYDOXHV V WKH FKDQJH LQ GDPSLQJDQGIUHTXHQF\FDQEHREVHUYHG7KHPRGHVKDSHVDUHTXLWHGLIIHUHQWIRU WKH WZR PRGHV 0RGH VKRZV D UHODWLYHO\ VPDOO ODWHUDO GLVSODFHPHQW \ ZLWK UHVSHFWWRWKH\DZDQJOHPXOWLSOLHG ZLWKKDOIWKHFRQWDFWOHQJWKD VOLJKWO\ODJVEHKLQG
$WILUVW\
ZKLFKWXUQVLQWRDOHDGDWKLJKHUVSHHGV0RGHVKRZV
DKLJKHUIUHTXHQF\DQGVPDOOHUZDYHOHQJWKZKLOH\ LVUHODWLYHO\ODUJHDQGODJV EHKLQG
([HUFLVH=HURHQHUJ\FLUFOHDSSOLHGWRWKHVLPSOHWUDLOLQJZKHHOV\VWHP &RQVLGHUWKHWKLUGRUGHUV\VWHPRI)LJGHVFULEHGE\WKH(TV EXWZLWKRXW
N
GDPSLQJ
(
:
(VWDEOLVKWKHUHGXFHGH[SUHVVLRQIRUWKHZRUN
IRUPXOD 6KRZWKDWWKHVWDELOLW\FRQGLWLRQ
+
IURPWKHJHQHUDO
! FRUUHVSRQGVWR
:
ZKLFK
PHDQVWKDWWKHPRGHVKDSHSRLQWOLHVRXWVLGHWKH]HURHQHUJ\FLUFOH
7UDQVIRUPDWLRQRI)RUZDUG0RWLRQ(QHUJ\LQWR6KLPP\(QHUJ\
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
2
SODQHPRYHVVLGHZD\VRYHUDGLVWDQFHG\DQGURWDWHVDERXWWKHYHUWLFDOD[LVRYHU WKHLQFUHPHQWRIWKH\DZDQJOHG
$WWKHVDPHWLPHWKHW\UHODWHUDOGHIOHFWLRQ
LVFKDQJHGDQGZKHQVOLGLQJLQWKHFRQWDFWSDWFKLVFRQVLGHUHGVRPHHOHPHQWV ZLOO VOLGH RYHU WKH JURXQG &RQVHTXHQWO\ ZH H[SHFW WKDW WKH VXSSOLHG GULYLQJ HQHUJ\G(LVHTXDOWRWKHVXPRIWKHFKDQJHVRI VHOI H[FLWDWLRQHQHUJ\G:W\UH SRWHQWLDOHQHUJ\G8DQGW\UHGLVVLSDWLRQHQHUJ\G' ,Q)LJWKHW\UHLVVKRZQLQDGHIOHFWHGVLWXDWLRQ:HKDYH WKHSXOOLQJ
2
IRUFH )G DFWLQJ LQ IRUZDUG [ GLUHFWLRQ IURP YHKLFOH WR WKH ZKHHO D[OH DQG
7+(:+((/6+,00<3+(120(121
[
[
_
2
_
V
ψ
_
\
)H
0H
&
9J
φJ sliding zone
_
\
)G α
pulling force
9
α
[
0]
)J
adhesion zone
\
)\
)LJ2QWKHEDODQFHRIGULYLQJVHOIH[FLWDWLRQDQGGLVVLSDWLRQHQHUJ\
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
FF
)RU WKH VWUDLJKW WDQJHQW VWULQJ GHIRUPDWLRQ ZLWK
k
NWKHVORSHGLIIHUHQFHLVHTXDOWR
WUDQVLHQWVOLSDQJOHRUGHIRUPDWLRQJUDGLHQW D
NV7KHIRUFHEHFRPHV )J
FF
D
J
N7KHDYHUDJHVOLGLQJVSHHG
9JRYHUWKHYDQLVKLQJVOLGLQJUDQJHLVHTXDOWRKDOIWKHGLIIHUHQFHLQVORSHRIWKH VWULQJ LQ IURQW DQG MXVW SDVVHG WKH NLQN WLPHV WKH IRUZDUG VSHHG 9 7KLV PXOWLSOLHGZLWKWKHWLPHQHHGHGWRWUDYHOWKHGLVWDQFHGVJLYHVWKHVOLGLQJGLVWDQFH VJ:HILQGIRUWKHDVVRFLDWHGGLVVLSDWLRQHQHUJ\
G'
'
) V
J J
'
)
J J GV
'
F
F
%
D
%
D
GV
'
&
)
GV
'
)
GV
\
7KH VDPH UHVXOW DSSHDUV WR KROG IRU WKH EUXVK PRGHO :KHQ FRQVLGHULQJ WKH WKHRU\RI&KDSWHUZHILQGIRUVPDOOVOLSDQJOH DWWKHWUDLOLQJHGJH D
]RQH ZLWK UHVSHFW WR WKH ZKHHO SODQH EHFRPHV 9J
9
J
NDOHQJWKRIWKHVOLGLQJ]RQH
NDQGDVORSHRIWKHFRQWDFWOLQHLQWKLVVKRUWVOLGLQJ J
7KH DYHUDJH VOLGLQJ VSHHG
N ZKLFKPDNHVWKHVOLGLQJGLVWDQFHHTXDOWRVJ
7KHDYHUDJHVLGHIRUFHDFWLQJLQWKHVOLGLQJUDQJHDSSHDUVWREH )J 7KLVUHVXOWVLQWKHGLVVLSDWHGHQHUJ\ FRQVLGHULQJWKDW
J
&)
N GV
N
LVRIILQLWHPDJQLWXGH J
7+(:+((/6+,00<3+(120(121
DQGWKXVPXFKODUJHUWKDQ G'
'
)
J VJ
'
&
)
N
J GV
'
&
)
'
GV
)
\ &) GV
'
)
\
GV
ZKLFKREYLRXVO\LVWKHVDPHH[SUHVVLRQDVIRXQGIRUWKHVWULQJPRGHO(T $WVWHDG\VWDWHVLGHVOLSSLQJFRQGLWLRQVZLWKFRQVWDQW\DZDQJOH ZHREYLRXVO\KDYHDGUDJIRUFH)G WUDYHOOHGVHTXDOV'
)\ &) V
)\
N
)\
7KHGLVVLSDWLRQHQHUJ\DIWHUDGLVWDQFH
N V
)\
V
)G VZKLFKLVDUHVXOWWKDWZDVWR
EHH[SHFWHG 7KH LQFUHPHQW RI WKH ZRUN : GRQH E\ WKH VLGH IRUFH DQG DOLJQLQJ WRUTXH DFWLQJ RQ WKH VLGHZD\V PRYLQJ DQG \DZLQJ ZKHHO SODQH FRQVLGHUHG LQ WKH SUHYLRXVVXEVHFWLRQ LV
G:
'
H G\2
)
%
'
HG
0
G\2
)
\
GV
%
0
]
G
GV
GV
7KHFKDQJHLQW\UHGHIOHFWLRQ QRWHWKDWWKHW\UHDOPRVWFRPSOHWHO\DGKHUHVWR WKHJURXQG DULVHVWKURXJKVXFFHVVLYHO\PRYLQJWKHZKHHOSODQHVLGHZD\VRYHU
2
WKHGLVWDQFHVG\DQG
GVDQGURWDWLQJRYHUWKH\DZDQJOHG
DQGILQDOO\UROOLQJ
IRUZDUGV LQ WKH GLUHFWLRQ RI WKH ZKHHO SODQH RYHU WKH GLVWDQFH GV 7KHVH FRQWULEXWLRQVFRUUHVSRQGZLWKWKHVXFFHVVLYHIRXUWHUPVLQWKHH[SUHVVLRQIRUWKH FKDQJHLQSRWHQWLDOHQHUJ\
'
G8
)
& G2 % \
\
GV
GV
&
0
]
G GV
GV
&
)
\
GV
7KHLQFUHDVHLQGULYLQJHQHUJ\LV
G(
'
)
\
GV
,QWRWDOZHPXVWKDYHWKHEDODQFHRIHQHUJLHV
G(
'
G:
%G %G 8
'
ZKLFKDIWHULQVSHFWLRQLQGHHGDSSHDUVWREHVDWLVILHGFRQVLGHULQJWKHH[SUHVVLRQV RIWKHWKUHHHQHUJ\FRPSRQHQWV ,QWHJUDWLRQRYHURQHF\FOHRIWKHSHULRGLFRVFLOODWLRQZLOOVKRZWKDW8
VR
WKDWIRUWKHHQHUJ\FRQVXPSWLRQRYHURQHF\FOHUHPDLQV (
'
:
%
'
$WVWHDG\VWDWHVLGHVOLSZLWKVOLSDQJOH
WKHH[FLWDWLRQHQHUJ\:
DQGWKH
GULYLQJHQHUJ\EHFRPHV (
'
'
'
)
\ &)
V
'
)
\
V
'
)
G
V
7KHUHVXOWSUHVHQWHGWKURXJK(T GHPRQVWUDWHVWKDWWKHSURSXOVLRQHQHUJ\ LVSDUWO\XVHGWRFRPSHQVDWHWKHHQHUJ\ORVWE\GLVVLSDWLRQLQWKHFRQWDFWSDWFK DQGSDUWO\WRSURYLGHWKHHQHUJ\WRVXVWDLQWKHXQVWDEOHVKLPP\RVFLOODWLRQV
7+(:+((/6+,00<3+(120(121
1RQ/LQHDU6KLPP\2VFLOODWLRQV
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
D[LV I
N
)\H0]PD\EHUHSODFHGE\WKHHTXLYDOHQWWRWDODOLJQLQJVWLIIQHVV
PXOWLSOLHGZLWKWKHWUDQVLHQWVOLSDQJOH &
VWHHULQJV\VWHPWKHIULFWLRQDOFRXSOHP HTXLYDOHQWYLVFRXVGDPSLQJFRXSOHN DUH IXQFWLRQV RI WKH DPSOLWXGHV RI
N,IGU\IULFWLRQLVFRQVLGHUHGLQWKH
0
.VJQ
0 PD\EHUHSODFHGE\WKH
0 7KHHTXLYDOHQWFRHIILFLHQWV& N DQG 0 RU RI
DQG N
UHVSHFWLYHO\ 7KH
IXQFWLRQVDUHGHWHUPLQHGE\FRQVLGHULQJDKDUPRQLFYDULDWLRQRI
NDQG
DQG
WKHQWDNLQJWKHILUVWKDUPRQLFRIWKH)RXULHUVHULHVRIWKHSHULRGLFUHVSRQVHRIWKH FRUUHVSRQGLQJRULJLQDOQRQOLQHDUWHUPV
N
:HKDYHZLWK
&
('
Rm
I
N VLQ R
( VLQ
DQG
R
VLQ
G
DQG
N
('
R
m
P
( 0 FRV
G
'
.
R
$VH[SHFWHGWKHIXQFWLRQRIWKHHTXLYDOHQWWRWDODOLJQLQJVWLIIQHVV&
N VWDUWV R
IRUYDQLVKLQJDPSOLWXGHDWWKHYDOXH&) H&0 DWDVORSHHTXDOWR]HURDQGWKHQ
7+(:+((/6+,00<3+(120(121
JUDGXDOO\GHFD\VWR]HURZLWKWKHDPSOLWXGH
GDPSLQJFRHIILFLHQWN
R
N WHQGLQJWRLQILQLW\7KHHTXLYDOHQW R
DSSHDUVWRYDU\LQYHUVHO\SURSRUWLRQDOO\ZLWK
WKH VWHHU DQJOH DPSOLWXGH ZKLFK PHDQV WKDW LW EHJLQV DW YHU\ ODUJH GDPSLQJ OHYHOV DQG GHFUHDVHV VKDUSO\ ZLWK LQFUHDVLQJ VZLYHO DPSOLWXGH DFFRUGLQJ WR D K\SHUEROD ,I FOHDUDQFH LQ WKH ZKHHO EHDULQJ DERXW WKH VWHHU D[LV ZRXOG EH FRQVLGHUHG LQVHULHVZLWKWKHVWHHUGDPSHU WKHPDWWHUEHFRPHVPRUHFRPSOH[
(
DQGDQHTXLYDOHQWVWHHUVWLIIQHVVF PXVWEHLQWURGXFHG,IWKHWRWDODQJOHRISOD\
(
WKHHTXLYDOHQWFRHIILFLHQWVN DQGF DUHIRXQGWREHH[SUHVVHG
LVGHQRWHGE\
E\WKHIXQFWLRQV
N
.
('
R
&
R
R!
R!
DQG
( ' .
F
R
,I
R
&
R
WKHRVFLOODWLRQWDNHVSODFHLQVLGHWKHIUHHVSDFHRIWKHFOHDUDQFHDQGWKH
R
HTXLYDOHQWGDPSLQJDQGVWLIIQHVVEHFRPHHTXDOWR]HUR2EYLRXVO\WKHFOHDUDQFH DOOHYLDWHVWKHLQLWLDOVWURQJGDPSLQJHIIHFWRIWKHGU\IULFWLRQ
,IQRWDOOQRQOLQHDULWLHVDUHFRQVLGHUHGZHPD\HJUHSODFHN E\NDQGF
(
E\]HURRUF )RUWKHWKLUGRUGHUV\VWHPRI6HFWLRQWKHQRQOLQHDUYHUVLRQRI(TV LVUHSODFHGE\HTXLYDOHQWOLQHDUGLIIHUHQWLDOHTXDWLRQVFRQWDLQLQJWKHFRHIILFLHQWV
(
& F DQGN 7KHGDPSLQJGXHWRWUHDGZLGWKLVNHSWOLQHDU7KHFKDUDFWHULVWLF HTXDWLRQZKLFKLVVLPLODUWR EHFRPHV 9
S
%^ %^
% ( %
9 N
9
( 9 % ( ` S
N
(%
%
( ( ( ( F % & H & D ` S % & % F '
(
ZLWKQRQGLPHQVLRQDOTXDQWLWLHVDFFRUGLQJWR & F DQGN EHLQJWUHDWHG OLNH &0 F DQG N UHVSHFWLYHO\ :KHQ WKH VKLPP\ PRWLRQ KDV EHHQ IXOO\ GHYHORSHG DQG WKH OLPLWF\FOH LV DWWDLQHG WKH VROXWLRQ RI WKH HTXLYDOHQW OLQHDU V\VWHPUHSUHVHQWVDKDUPRQLFRVFLOODWLRQZLWKSDWKIUHTXHQF\
N DQG R
V
DQGDPSOLWXGHV
7KH DPSOLWXGHV DUH REWDLQHG E\ XVLQJ WKH FRQGLWLRQ WKDW DW WKLV R
VXVWDLQHGRVFLOODWLRQWKH+XUZLW]GHWHUPLQDQW+Q RXU WKLUGRUGHU V\VWHP DW WKH OLPLWF\FOH +
&RQVHTXHQWO\ZHKDYHIRU
+HQFH WKH HTXDWLRQ WKDW LV
HVVHQWLDOIRUILQGLQJWKHPDJQLWXGHRIWKHOLPLWF\FOHUHDGV
^9
%
( 9 % ( ` ^ N( 9 % ( %
N
( % &( H & D ` '
F
'
9
( % F(
&
7KHIUHTXHQF\RIWKHSHULRGLFRVFLOODWLRQWKDWRFFXUVZKHQWKHUHODWLRQ LV
7+(:+((/6+,00<3+(120(121
VDWLVILHGFDQHDVLO\EHIRXQGZKHQLQ SLVUHSODFHGE\L
V
( ( & % F
' 9
:HREWDLQ
( 9 % (
%
V
N
0RUHRYHU ZH QHHG WKH UDWLR RI WKH DPSOLWXGHV
N DQG R
R
)URP WKH VHFRQG
HTXDWLRQRI ZKLFKLVQRWFKDQJHGLQWKHOLQHDULVDWLRQSURFHVVZHJHWZLWK
"N
Y
R
R
'
% & D
%
H
V
V
(TXDWLRQV SURYLGHVXIILFLHQWLQIRUPDWLRQWRFRPSXWHWKHDPSOLWXGHDQG WKHIUHTXHQF\RIWKHOLPLWF\FOH 7KH VWDELOLW\ RI WKH OLPLWF\FOH RI WKH ZHDNO\ QRQOLQHDU V\VWHP PD\ EH DVVHVVHGE\WDNLQJWKHGHULYDWLYHRIWKH+XUZLW]GHWHUPLQDQWZLWKUHVSHFWWRWKH DPSOLWXGH,IG+G
!WKHOLPLWF\FOHLVVWDEOHDQGDWWUDFWVWKHWUDMHFWRULHVLI
R
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
7+(:+((/6+,00<3+(120(121
)H 0 \ ]
tyre characteristic
α ψ*
steer damping / play
N ψ
ψ
ψo
limit amplitudes
.
N
ψ
ψo
.
δ ψ
.
ψ* ψ δ
ψo
ψo S
.
ψ limit cycles
S
U
N
N
.
.
S U
ψ
stability
Unstable or Stable
U - S or S
S - U - S or S
U-S or U-S-U-S or U-S
)LJ /LQHDUDQGQRQOLQHDUV\VWHPVFRQVLGHUHGDQGWKHLUDXWRPRPRXVPRWLRQSURSHUWLHV LQWHUPVRIVWDELOLW\DQGSRVVLEOHOLPLWF\FOHV
WKHUHVWSRVLWLRQRUWKHODUJHVWDEOHOLPLWF\FOHLVDSSURDFKHG)LJXUHVKRZV IRU WKLV FDVH WKH VROXWLRQV LQ WKH WKUHHGLPHQVLRQDO VWDWH VSDFH 7KH XQVWDEOH OLPLWF\FOHDSSHDUVWROLHRQDWXEHVKDSHGVXUIDFHZKLFKLVKHUHWKHVHSDUDWUL[ LQWKHVROXWLRQVSDFH:HVHHWKDWZKHQWKHLQLWLDOFRQGLWLRQVDUHWDNHQRXWVLGH WKHWXEHWKHVWDEOHOLPLWF\FOHLVUHDFKHG:KHQWKHVWDUWLQJFRQGLWLRQLVLQVLGH WKHWXEHRQHRIWKHLQGLFDWHGSRVVLEOHUHVWSRVLWLRQVZLOOEHXOWLPDWHO\DWWDLQHG
viscous damping
ψo 0.15
dry friction
ψo
dry friction and play
ψo
0.15
1
2
3
0.15
δB = 0.1
0.1
0.015
0.1
0.01 0.05
0.05
0.0027
0.05
0 0
0
0.1
0.2
0.3
NB
0
0
0.005
0.01
0
.B
0
.B
0.01
)LJ /LPLW DPSOLWXGHV DV D IXQFWLRQ RI GDPSLQJ IRU LQFUHDVLQJ QXPEHU RIQRQOLQHDU
9
HOHPHQWV QRQOLQHDUW\UHGU\IULFWLRQDQGSOD\
H
7+(:+((/6+,00<3+(120(121
.
ψ stable rest positions unstable limit-cycle stable limit-cycle
ψ 0.1
0.2
0.2
initial conditions
0.4
separatrix ’tube’
trajectory
α
)LJ 6ROXWLRQVLQWKHWKUHHGLPHQVLRQDOVWDWHVSDFH6\VWHPZLWKQRQOLQHDUW\UHDQGGU\
9
IULFWLRQ
H
N
.
/LQHSLHFHIRUSRVVLEOHUHVW
SRVLWLRQVXQVWDEOHDQGVWDEOHOLPLWF\FOH7XEHVKDSHGVHSDUDWUL[VHSDUDWLQJVSDFH RIWUDMHFWRULHVOHDGLQJWRDUHVWSRVLWLRQIURPWKHVSDFHRIVROXWLRQFXUYHVOHDGLQJWR WKHVWDEOHOLPLWF\FOH
.
ψ
ψ
2δ unstable centre position stable unstable limit-cycle stable )LJ /LPLWF\FOHVDQGWUDMHFWRULHVIRUWKHV\VWHPZLWKQRQOLQHDUW\UHGU\IULFWLRQDQGSOD\ LQWKHZKHHOEHDULQJV &DVHRI)LJ
7+(:+((/6+,00<3+(120(121
7KH LQWURGXFWLRQ RI SOD\ DERXW WKH VWHHULQJ D[LV ZLWK
EHLQJ KDOI WKH
FOHDUDQFHVSDFHDSSHDUVLQGHHGWREHDEOHWRUHOD[WKHDFWLRQRIWKHGU\IULFWLRQ )RUVPDOOSOD\DQGHQRXJKGU\IULFWLRQ FDVHLQ)LJ WKHVPDOOVWDEOHOLPLW F\FOHLVUHDFKHGDXWRPDWLFDOO\$QDGGLWLRQDOGLVWXUEDQFHPD\FDXVHWKHPRWLRQ WRJHWRYHUWKHµQRVH¶DQGUHDFKWKHODUJHVWDEOHOLPLWF\FOH7KHSORWRI)LJ VKRZVIRUWKLVFDVHWKHWKUHHOLPLWF\FOHVDQGWUDMHFWRULHVDVSURMHFWHGRQWKH
0
SODQH,IWKHSOD\LVODUJHURUWKHGDPSLQJOHVVWKHODUJHOLPLWF\FOHLVUHDFKHG ZLWKRXWDQH[WHUQDOGLVWXUEDQFH ,W LV RI LQWHUHVW WR VHH KRZ WKH OLPLWDPSOLWXGHV WUDYHO
9
R YDU\ ZLWK WKH VSHHG RI
)LJXUH GHSLFWVWZRFDVHV YLVFRXVGDPSLQJDQGGU\IULFWLRQ7KH
DUHDLQVLGHZKLFKWKHDPSOLWXGHLQFUHDVHVGXHWRLQVWDELOLW\PD\EHGHVLJQDWHG DV WKH DUHD RI VHOIH[FLWDWLRQ 7KH GLDJUDPV VKRZ WKH FRXUVHV RI WKH OLPLW DPSOLWXGH IRU WKH WZR FRQILJXUDWLRQV ZLWK OLQHDU GDPSLQJ
.
ZLWK
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
H_ = 0.5 N_ = 0.25
ψo
N_ = 0 κ_ *= 1
._ = 0
κ_ *= 1
._ = 0.002
area area of of selfselfexcitation excitation
0
H_ = 0
._ = 0
0.1
0.05
9_
0
10
0
9_
0
10
)LJ %RXQGDU\RIDUHDRIVHOIH[FLWDWLRQUHSUHVHQWLQJWKHFRXUVHRIOLPLWDPSOLWXGHVRI VWHHU DQJOH
DV IXQFWLRQ RI VSHHG IRU V\VWHP ZLWK QRQOLQHDU W\UH DQG YLVFRXV
GDPSLQJ OHIWDQGULJKW DQGGU\IULFWLRQ ULJKW
FI)LJ
7+(:+((/6+,00<3+(120(121
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
ψo 6
3
o
combined oscillations
synchronous oscillations (frequency of unbalance) combined oscillations (natural + unbalance freq.)
area of selfexcitation
o
periodic oscillations
0 0
25
50
75
100
9
[km/h]
)LJ 7KHUHVSRQVHRIWKHWHQWKRUGHUV\VWHPUHSUHVHQWLQJDOLJKWWUXFNZLWKQRQOLQHDU W\UHDQGGU\IULFWLRQWRZKHHOXQEDODQFHPRPHQW7KHDUHDRIVHOIH[FLWDWLRQRIWKH DXWRQRPRXVV\VWHPKDVEHHQLQGLFDWHG$WQHDUO\NPKYLROHQWVKLPP\GHYHORSV 7KHQ D UDQJH RI VSHHG ZLWK VRFDOOHG V\QFKURQRXV RVFLOODWLRQV RFFXU :KHQ WKH V\VWHPLVGHWXQHGWRRPXFKFRPELQHGRVFLOODWLRQV EHDWV VKRZXS
7+(:+((/6+,00<3+(120(121
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
Q1 : wheel revolutions Q2 : steering oscillations
o
6
ψo
ψo Q
o
3
8Hz
4Hz
Q2 Q1
0 0
25
50
75
9 [km/h]
)LJ )URQW ZKHHO VWHHU RVFLOODWLRQ DPSOLWXGH DV D UHVSRQVH WR ZKHHO XQEDODQFHDV PHDVXUHGRQDOLJKWWUXFN7KHVSHHGLVJUDGXDOO\UHGXFHGDIWHUYLROHQWVKLPP\ZDV GHYHORSHG DW FD NPK $IWHU D VKRUW UDQJH RI VSHHG H[KLELWLQJ V\QFKURQRXV RVFLOODWLRQV VLQJOH IUHTXHQF\ FRPELQHG YLEUDWLRQV RFFXU ZLWK WZR GLVWLQFW IUHTXHQFLHV
7+(:+((/6+,00<3+(120(121
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
&KDSWHU 6,1*/(&217$&732,1775$16,(17 7<5(02'(/6
,QWURGXFWLRQ
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
6,1*/(&217$&732,1775$16,(1702'(/
0RGHO'HYHORSPHQW
7KHPRGHOFRQVLVWVRIDFRQWDFWSDWFK SRLQW WKDWLVVXVSHQGHGZLWKUHVSHFWWR WKH ZKHHO ULP E\ D ORQJLWXGLQDO FLUFXPIHUHQWLDO DQG D ODWHUDO VSULQJ 7KHVH VSULQJV UHSUHVHQW WKH FRPSOLDQFH RI WKH W\UH FDUFDVV 7KH FRQWDFW SRLQW PD\ PRYH VOLS ZLWK UHVSHFW WR WKH JURXQG LQ ODWHUDO DQG ORQJLWXGLQDO GLUHFWLRQV 7KURXJK WKLV UHODWLYH PRWLRQ WKH VLGH IRUFH ORQJLWXGLQDO IRUFH DQG DOLJQLQJ WRUTXH DUH JHQHUDWHG 7R GHWHUPLQH WKHVH IRUFHV DQG PRPHQWV FRQWDFW SDWFK ODWHUDO VOLS SRVVLEO\ LQFOXGLQJ SO\VWHHU DQG ORQJLWXGLQDO VOLS DUH GHILQHG &DPEHU FRQWDFW OLQH FXUYDWXUH SRVVLEO\ LQFOXGLQJ FRQLFLW\ DQG WXUQ VOLS LV WKRXJKWWRH[LVWZKLFKLVUHSUHVHQWHGE\DQHIIHFWLYHWRWDOFDPEHUDQJOH7KHVH FRQWDFWSDWFKVOLSTXDQWLWLHVPD\WKHQEHXVHGDVLQSXWLQWKHVWHDG\VWDWHW\UH VOLSPRGHOHJWKH0DJLF)RUPXODWRFDOFXODWHWKHWUDQVLHQWIRUFHDQGPRPHQW YDULDWLRQWKDWDFWXSRQWKHFRQWDFWSDWFK
/LQHDU0RGHO
)LJXUHGHSLFWVWKHPRGHOLQWRSYLHZ$WWKHLQVWDQWFRQVLGHUHGWKHZKHHO VOLSSRLQW6 DWWDFKHGWRWKHZKHHOULPDWDOHYHOQHDUWKHURDGVXUIDFH DQGWKH FRQWDFWSRLQW6¶DUHGHILQHGWREHORFDWHGLQWKHSODQHWKURXJKWKHZKHHOD[LVDQG QRUPDOWRWKHURDG7KHVHSRLQWV ZKLFKPD\EHWKRXJKWWROLHRQWZRSDUDOOHO VOLSFLUFOHV PRYHRYHUWKHURDGVXUIDFHZLWKWKHZKHHODQGFRQWDFWSDWFKVOLS YHORFLWLHVUHVSHFWLYHO\,QWKHILJXUHWKH[DQG\FRPSRQHQWVRIWKHVOLSYHORFLWLHV
wheel spin axis
wheel slip speed 9 (of point 6attached to rim) V
9
U
α
6
F
Y
speed of rolling
[\
speed of contact centre
&
9
U
9
V
6
speed of 6
(attached to contact patch) tyre single contact point
9
X
0 )
]
\
α
9
[
wheel plane
9
)
[
)
)LJ 6LQJOHFRQWDFWSRLQWW\UHPRGHOVKRZLQJODWHUDODQGORQJLWXGLQDOFDUFDVV
X
Y
GHIOHFWLRQV DQG 7RSYLHZ
&
6,1*/(&217$&732,1775$16,(1702'(/
KDYHEHHQVKRZQDVQHJDWLYHTXDQWLWLHV7KHGLIIHUHQFHRIWKHYHORFLWLHVFDXVHWKH FDUFDVV VSULQJV WR GHIOHFW &RQVHTXHQWO\ WKH WLPH UDWHV RI FKDQJH RI WKH ORQJLWXGLQDODQGODWHUDOGHIOHFWLRQVXDQGYUHDG
GX GW
'&
9
V[
&
9
9
V[
DQG
GY GW
'&
9
V\
&
V\
,IZHDVVXPHVPDOOYDOXHVRIVOLSZHPD\ZULWHIRUWKHVLGHIRUFHDFWLQJIURP URDGWRFRQWDFWSDWFKZLWK&) GHQRWLQJWKHFRUQHULQJRUVLGHVOLSVWLIIQHVV
)
'
\
&
)
'&
9
&
V\
)
_9 [ _
,W LV DVVXPHG KHUH WKDW WKH GLIIHUHQFH EHWZHHQ WKH ZKHHO FHQWUH ORQJLWXGLQDO YHORFLW\9[DQGWKHORQJLWXGLQDOYHORFLW\9F[RIWKHFRQWDFWFHQWUHLVQHJOLJLEOH 9
F[
.
9
[
&RQVHTXHQWO\ZHPD\HPSOR\9[LQWKHSUHVHQWFKDSWHU:LWKWKHODWHUDOW\UH VWLIIQHVVDWURDGOHYHO&)\ZHKDYHIRUWKHHODVWLFLQWHUQDOIRUFHWKDWEDODQFHVWKH VLGHVOLSIRUFH )
'
\
&
)\
Y
DQGZHFDQZULWHIRU(T ZLWK DIWHUKDYLQJLQWURGXFHGWKHUHOD[DWLRQ OHQJWKIRUVLGHVOLS
'
&
)
&
)\
WKHGLIIHUHQWLDOHTXDWLRQIRUWKHODWHUDOGHIOHFWLRQGXHWRVLGHVOLSY ODWHURQZH ZLOODOVRKDYHDODWHUDOGHIOHFWLRQGXHWRFDPEHU
GY GW :KHUH
%
_9[_ Y
'
_9 [_
'&
LV WKH ZKHHO VOLS DQJOH
9
V\
.9
V\
_9[_ 7KH VLGH IRUFH LV REWDLQHG E\
PXOWLSO\LQJY ZLWK&)\ ,QDVLPLODUZD\ZHFDQGHDOZLWKWKHORQJLWXGLQDOIRUFHUHVSRQVH:LWKWKH ORQJLWXGLQDOW\UHVWLIIQHVV&)[DWURDGOHYHODQGWKHORQJLWXGLQDOVOLSVWLIIQHVV&) ZHREWDLQIRUWKHUHOD[DWLRQOHQJWK
'
&
)
&
)[
6,1*/(&217$&732,1775$16,(1702'(/
DQG ZH FDQ GHULYH ZLWK (T WKH GLIIHUHQWLDO HTXDWLRQ IRU WKH IRUH DQG DIW GHIOHFWLRQX
GX GW ZLWK
%
'
_9[_ X
'&
_9 [_
9
V[
.9
WKHORQJLWXGLQDOZKHHOVOLSUDWLR
V[
_9[_7KHORQJLWXGLQDOIRUFHPD\
EHREWDLQHGE\PXOWLSO\LQJXZLWKWKHVWLIIQHVV&)[ 1H[WZHFRQVLGHUZKHHOFDPEHUDVLQSXW)RUDVXGGHQO\DSSOLHGFDPEHU DQJOH DERXWWKHOLQHRILQWHUVHFWLRQDWJURXQGOHYHO ZHDVVXPHWKDWDFRQWDFW LVLPPHGLDWHO\IHOWDWWKHFRQWDFW
OLQHFXUYDWXUHDQGWKXVWKHFDPEHUWKUXVW&)
SDWFK$VDUHDFWLRQDFRQWDFWSDWFKVLGHVOLSDQJOH XSWKHODWHUDOFDUFDVVGHIOHFWLRQY
NLVGHYHORSHGWKDWEXLOWV
Y $JDLQ(T DSSOLHV7KHVLGHIRUFHWKDW
DFWVRQWKHZKHHOQRZEHFRPHV )
\
'
&
)\
'
Y
%
&
)
&
)
:LWKZKHHOVLGHVOLSNHSWHTXDOWR]HUR9V\
N
DQG9 V \
_9[_
N (T FDQEH
ZULWWHQLQWKHIRUP
GY
%
GW
_9[_ Y
'
&
)
&
_9 [_
)
7KLV HTXDWLRQ VKRZV WKDW DFFRUGLQJ WR WKLV VLPSOH PRGHO WKH FDPEHU IRUFH UHOD[DWLRQ OHQJWK
LV HTXDO WR WKH UHOD[DWLRQ OHQJWK IRU VLGH VOLS
7KLV
WKHRUHWLFDO UHVXOW LV VXEVWDQWLDWHG E\ FDUHIXO VWHS UHVSRQVH H[SHULPHQWV SHUIRUPHGE\+LJXFKL RQDIODWSODQNWHVWULJ FI6HF
nLQFOXGLQJWXUQVOLSDQGFDPEHU
$VLPLODUHTXDWLRQUHVXOWVIRUWKHWRWDOVSLQ
G Yn GW
%
_9[_ Yn
'
n
&
)
&
_9 [_ n
VLQ `
)
ZLWKDFFRUGLQJWR(T
n'
&
9
^0
& & g
[
WKDW VKRZV WKDW WKH WXUQ VOLS YHORFLW\
0
FDQ EH FRQYHUWHG LQWR DQ HTXLYDOHQW
FDPEHUDQJOH 7KH IRUFHV DQG PRPHQW DUH REWDLQHG IURP WKH GHIOHFWLRQV X DQG Y E\ ILUVW DVVHVVLQJWKHWUDQVLHQWVOLSTXDQWLWLHV
N NDQG N DQGIURPWKHVHZLWKWKHVOLS
VWLIIQHVVHVWKHIRUFHVDQGPRPHQW $FFRUGLQJ WR WKH DGRSWHG VWHDG\VWDWH PRGHO (T ( WKH PRPHQW UHVSRQVH WR FDPEHU DQG WXUQ VOLS LV WKH VXP RI WKH UHVLGXDO WRUTXH 0]U DVVXPHGO\ PDLQO\ GXH WR ILQLWH WUHDG ZLGWK DQG W )\ DVVXPHGO\ FDXVHG E\ FDPEHULQGXFHGVLGHVOLSFIGLVFXVVLRQODWHURQ EHORZ(T $ILUVWRUGHU
6,1*/(&217$&732,1775$16,(1702'(/
DSSUR[LPDWLRQIRUWKHUHVSRQVHRI0]UZLWKWKHVKRUWUHOD[DWLRQOHQJWKHTXDOWR KDOI WKH FRQWDFW OHQJWK D PD\ EH HPSOR\HG 7KLV KRZHYHU ZLOO EH VDYHG IRU &KDSWHU ZKHUHVKRUWZDYHOHQJWKVUHVSRQVHVDUHFRQVLGHUHG+HUHZHVXIILFH ZLWKWKHDVVXPSWLRQWKDWWKHPRPHQWGXHWRWUHDGZLGWKUHVSRQGVLQVWDQWDQHRXVO\ WRFDPEHUDQGWXUQVOLS )RUWKHOLQHDUVPDOOVOLSFRQGLWLRQZHILQG
Y
. WDQ '
'
X
'
&
'
)
\
)
[
)
\
&
)
DQGVLPLODUIRU
)
'
Y
)
&
'
&
)
&
)
0
]
'&
&
0
'&
W )
\
0
]U
'
&
0
%
W
&
)
n7KHSQHXPDWLFWUDLOGXHWRVLGHVOLSLVGHQRWHGKHUHZLWKW
7KHWRWDODOLJQLQJWRUTXHEHFRPHV FI(T ( DQG)LJ 0
]
'&
W
)\
%
)
\
%
0
]U
,QDQDOWHUQDWLYHPRGHOXVHGIRUPRWRUF\FOHG\QDPLFVVWXGLHVWKHWHUPVZLWK W &) RUW ) LQ DUHRPLWWHGFIGLVFXVVLRQEHORZ(T 7KH HTXDWLRQV PD\ EH ZULWWHQ LQ WHUPV RI WKH WUDQVLHQW VOLS TXDQWLWLHV )RU H[DPSOH ZH PD\ H[SUHVV Y LQ WHUPV RI
N E\ XVLQJ WKH ILUVW
HTXDWLRQRI ,QVHUWLRQLQ JLYHV
G
%_
9
GW
_
[
'
_9 _ [
'&
9
V\
,IZHUHFRJQLVHWKHIDFWWKDWWKHUHOD[DWLRQOHQJWKLVDIXQFWLRQRIWKHYHUWLFDOORDG DQGLIWKHDYHUDJHVOLSDQJOHLVXQHTXDOWR]HURDQDGGLWLRQDOWHUPVKRZVXSLQ WKHOLQHDULVHGHTXDWLRQ YDULDWLRQRI
_9[_
%
GLIIHUHQWLDWLRQRIY
G
GW
%
G
NDQGRI) W VPDOO ZKLFKUHVXOWVIURPWKH ]
NZLWKUHVSHFWWRWLPH7KHQ(T EHFRPHV
G) ]
G)] GW
'
_9[_
'&
9
V\
2EYLRXVO\ZKHQXVLQJ(T DUHVSRQVHWRDYDULDWLRQRIWKHYHUWLFDOORDG FDQQRWEHH[SHFWHG,IWKHORDGYDULHV(T LVLQDGHTXDWHDQGWKHRULJLQDO HTXDWLRQ VKRXOGEHXVHGRUWKHFRUUHVSRQGLQJHTXDWLRQ :LWK HTXDWLRQ PD\EHZULWWHQGLUHFWO\LQWHUPVRI)\,IZHPD\ FRQVLGHUWKHFDUFDVVODWHUDOVWLIIQHVV&)\YLUWXDOO\LQGHSHQGHQWRIWKHZKHHOORDG )]ZHREWDLQE\XVLQJ(T
G )\ GW
%_
9
_ )\
[
'
_9[_ )\VV
6LQFHZHKDYHWKHVDPHUHOD[DWLRQOHQJWKIRUERWKWKHUHVSRQVHVWRVLGHVOLSDQG
6,1*/(&217$&732,1775$16,(1702'(/
U
Ω
UH
9
[
&
)[
6
HΩ
9U 9V[ [
&
)κ
9
[
X
)LJ0HFKDQLFDOPRGHORIWUDQVLHQWWDQJHQWLDOW\UHEHKDYLRXU
FDPEHUWKLVHTXDWLRQDSSHDUVWRKROGIRUWKHFRPELQHGOLQHDUUHVSRQVHWRWKH LQSXWV
DQG RU
n ,Q WKH ULJKWKDQG PHPEHU )
\VV
GHQRWHV WKH VWHDG\VWDWH
LQVWDQWDQHRXV UHVSRQVHWRWKHVHLQSXWVDQGSRVVLEO\DFKDQJLQJYHUWLFDOORDGDW D JLYHQ VOLS FRQGLWLRQ 0XOWLSOLFDWLRQ ZLWK WKH SQHXPDWLF WUDLO SURGXFHV WKH
n
PRPHQW0¶ LI ]
$VLPLODUGLIIHUHQWLDOHTXDWLRQPD\EHZULWWHQIRUWKH
)[UHVSRQVHWR 7KHHTXDWLRQV KDYHEHHQZULWWHQLQWKHIRUP 9[QRWLQGHQRPLQDWRU DQG9V[\XVHGDVULJKWKDQGPHPEHU WKDWPDNHVWKHPDSSOLFDEOHIRUVLPXODWLRQV RIVWRSSLQJDQGVWDUWLQJIURP]HURVSHHGRFFXUUHQFHV$WVSHHG9[ WXUQV LQWR DQ LQWHJUDWRU X EHFRPHV )[
&)
N
&) X
I
(T
9V[GW :LWK (T WKH ORQJLWXGLQDO IRUFH ZKLFKZLWK LVHTXDOWR & )[ X7KLVLVWKH
FRUUHFW H[SUHVVLRQ IRU WKH W\UH WKDW DW VWDQGVWLOO DFWV OLNH D ORQJLWXGLQDO RU WDQJHQWLDOVSULQJ:KHQWKHZKHHOVWDUWVUROOLQJWKHW\UHJUDGXDOO\FKDQJHVLQWR D GDPSHU ZLWK UDWH &)
_9[_ )LJXUH GHSLFWV D FRUUHVSRQGLQJ PHFKDQLFDO
PRGHOZLWKVSULQJDQGGDPSHULQVHULHV,WVKRZVWKDWDWORZVSHHGWKHGDPSHU EHFRPHVYHU\VWLIIDQGWKHVSULQJGRPLQDWHV$WKLJKHUIRUZDUGYHORFLWLHVWKH VSULQJEHFRPHVUHODWLYHO\VWLIIDQGWKHGDPSHUSDUWGRPLQDWHVWKHEHKDYLRXURI WKHW\UH$VLPLODUPRGHOPD\EHGUDZQIRUWKHWUDQVLHQWODWHUDOEHKDYLRXU ,W PD\EHQRWHGWKDWWKHHTXDWLRQ LVQRWVXLWHGIRUPRYLQJQHDURUDW9[
,Q 6HFWLRQ WKH XVH RI WKH WUDQVLHQW PRGHOV IRU WKH UHVSRQVH WR ODWHUDO DQG ORQJLWXGLQDOZKHHOVOLSVSHHGDWDQGQHDU]HURVSHHGZLOOEHGHPRQVWUDWHG $SSDUHQWO\(T ZLWK IDLOVWRGHVFULEHWKHUHVSRQVHWRYDULDWLRQV LQZKHHO\DZDQJOH
I
&)n &) G
DWYDQLVKLQJVSHHG7KHQWKHODWHUDOGHIOHFWLRQEHFRPHVYn
:LWK
nN
&) &)n Yn
ZHKDYH
nN
LQGLFDWLQJDQ
LQVWDQWDQHRXVUHVSRQVHRI)\ZKLFKKRZHYHUVKRXOGUHPDLQ]HUR:HUHIHUWR &KDSWHU 6HFWLRQ (T HWF WKDWVXJJHVWVDIXUWKHUGHYHORSHGPRGHO WKDWFDQKDQGOHWKLVVLWXDWLRQFRUUHFWO\&RQVHTXHQWO\LWPXVWEHFRQFOXGHGWKDW WKHSUHVHQWWUDQVLHQWPRGHOFDQQRWEHHPSOR\HGWRVLPXODWHSDUNLQJPDQRHXYUHV XQOHVV)\LVVXSSUHVVHGDUWLILFLDOO\LQWKHORZHUVSHHGUDQJH
6,1*/(&217$&732,1775$16,(1702'(/
6HPL1RQ/LQHDU0RGHO
)RUWKHH[WHQVLRQRIWKHOLQHDUWKHRU\WRFRYHUWKHQRQOLQHDUUDQJHRIWKHVOLS FKDUDFWHULVWLFVLWPD\EHWHPSWLQJWRHPSOR\(T DQGXVHWKHLQVWDQWDQHRXV QRQOLQHDUIRUFHUHVSRQVHDVLQSXWLQWKHGLIIHUHQWLDOHTXDWLRQ7KHLQSXWVWHDG\ VWDWH VLGH IRUFH LV FDOFXODWHG HJ ZLWK WKH 0DJLF )RUPXOD XVLQJ WKH FXUUHQW ZKHHOVOLSDQJOH
7KLVPHWKRGKRZHYHUPD\OHDGWRLQFRUUHFWUHVXOWVDVGXH
WRWKHSKDVHODJLQVLGHIRUFHUHVSRQVHWKHFXUUHQW YDU\LQJ ZKHHOORDGPD\QRW FRUUHVSRQG WR WKH FDOFXODWHG PDJQLWXGH DWWDLQHG E\ WKH VLGH IRUFH ,Q OLPLW FRQGLWLRQVWKHW\UHPD\WKHQEHSUHGLFWHGWREHVWLOOLQDGKHVLRQZKLOHLQUHDOLW\ IXOOVOLGLQJRFFXUV$EHWWHUDSSURDFKLVWKHXWLOLVDWLRQRIWKHRULJLQDOHTXDWLRQ DQGFDOFXODWHWKHVLGHIRUFHDIWHUZDUGVE\XVLQJWKHUHVXOWLQJWUDQVLHQWVOLS
NDVLQSXWLQWKH0DJLF)RUPXOD
DQJOH
,QJHQHUDOZHKDYHWKHWKUHHHTXDWLRQV DQGSRVVLEO\ DQGWKH ILUVWHTXDWLRQVRI SURGXFLQJ
N NDQG NRUnNZKLFKDUHXVHGDVLQSXW nGLUHFWO\LQWKHH[SUHVVLRQV
LQWKHQRQOLQHDUIRUFHDQGPRPHQWIXQFWLRQV RU
IRU0U HJWKHHTXDWLRQVRIWKH0DJLF)RUPXODW\UHPRGHO &KDSWHU ) )
[
\
0
' ' '& ' ' N% % @
]
0
]U
0
]
) )
[
\
W
0
0
)
)
)
]
\
]U
]
]
0
]U
V
)
]
)
(
[
ZKHUHLIUHTXLUHG PD\EHUHSODFHGE\
nDVWKHVSLQDUJXPHQW
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
VO
ZKHUH WKH SHDNKRUL]RQWDOIRUFHRFFXUV
$SSUR[LPDWHO\ ZH PD\ DGRSW WKH IROORZLQJ OLPLWLQJ DOJRULWKP ZLWK WKH HTXLYDOHQWVOLSDQJOHDFFRUGLQJWR(T (
6,1*/(&217$&732,1775$16,(1702'(/
LI
_
WKHQ
LI 9V[
UHT
_!
LI 9V\
HOVH
VO
DQG _9
%_ %_
9 9
[
_
9
ORZ
' 0'
_X
X X0
_Y
Y
[
[
Y
HOVH (T DSSOLHV HOVH (T DSSOLHV
(TV DSSO\
ZLWKURXJKO\ VO
'
' \ &)
([SHULHQFHLQDSSO\LQJWKHPRGHOKDVLQGLFDWHGWKDWVWDUWLQJIURPVWDQGVWLOOJLYHV ULVHWRRVFLOODWLRQVZKLFKDUHSUDFWLFDOO\XQGDPSHG'DPSLQJLQFUHDVHVZKHQ VSHHG LV EXLOW XS 7R DUWLILFLDOO\ LQWURGXFH VRPH GDPSLQJ DW YHU\ ORZ VSHHG ZKLFKZLWKWKHDFWXDOW\UHLVHVWDEOLVKHGWKURXJKPDWHULDOGDPSLQJRQHPLJKW HPSOR\ WKH IROORZLQJ H[SUHVVLRQ IRU WKH WUDQVLHQW VOLS
N DV VXJJHVWHG E\
%HVVHOLQNLQVWHDGRI(T
'
&
X
N
9ORZ
&
9
V[
)
7KHGDPSLQJFRHIILFLHQWN9ORZVKRXOGEHJUDGXDOO\VXSSUHVVHGWR]HURZKHQWKH VSHHGRIWUDYHO9[DSSURDFKHVDVHOHFWHGORZYDOXH9ORZ%H\RQGWKDWYDOXHWKH PRGHO VKRXOG RSHUDWH DV XVXDO ,Q &KDSWHU 6HF DQ DSSOLFDWLRQ ZLOO EH JLYHQ$VLPLODUHTXDWLRQPD\EHHPSOR\HGIRUWKHODWHUDOWUDQVLHQWVOLS $QRWKHU H[WUHPH VLWXDWLRQ LV WKH FRQGLWLRQ DW ZKHHO ORFN $W VWHDG\ VWDWH (T UHGXFHVWR
_9 _ X [
'&
9
[
ZKLFKLQGLFDWHVWKDWWKHGHIOHFWLRQXDFFRUGLQJWRWKHVHPLOLQHDUWKHRU\EHFRPHV DVODUJHDVWKHUHOD[DWLRQOHQJWK
$YRLGLQJWKHGHIOHFWLRQVIURPEHFRPLQJWRR
ODUJHZKLFKLVRILPSRUWDQFHDWHJUHSHWLWLYHEUDNLQJFDOOVIRUDQHQKDQFHG QRQOLQHDU PRGHO $QRWKHU VKRUWFRPLQJ RI WKH PRGHO WKDW LV WR EH WDFNOHG FRQFHUQV WKH H[SHULPHQWDOO\ REVHUYHG SURSHUW\ RI WKH W\UH WKDW LWV UHOD[DWLRQ OHQJWKGHSHQGVRQWKHOHYHORIVOLS$WKLJKHUOHYHOVRIVLGHVOLSWKHW\UHVKRZV D TXLFNHU UHVSRQVH WR DGGLWLRQDO FKDQJHV LQ VLGH VOLS 7KLV LQGLFDWHV WKDW WKH UHOD[DWLRQOHQJWKVGHFUHDVHZLWKLQFUHDVLQJVOLS
)XOO\1RQ/LQHDU0RGHO
)LJXUH UHSHDWHGKHUHDV)LJVKRZVWKHGHIOHFWHGVWULQJPRGHOSURYLGHG ZLWKWUHDGHOHPHQWVDWYDULRXVOHYHOVRIVWHDG\VWDWHVLGHVOLS&OHDUO\WKLVPRGHO SUHGLFWVWKDWWKHµLQWHUVHFWLRQ¶OHQJWK
(
GHFUHDVHVZLWKLQFUHDVLQJ WKDWLV ZKHQ
6,1*/(&217$&732,1775$16,(1702'(/
9 α
[
α stretched string
σ*
sliding range
D
tread elements
D
σ
\ )LJ
7KHVWULQJW\UHPRGHOZLWKWUHDGHOHPHQWVDWLQFUHDVLQJVOLSDQJOHVVKRZLQJJURZLQJ VOLGLQJUDQJHDQGGHFUHDVLQJµLQWHUVHFWLRQ¶OHQJWK
WKHVOLGLQJUDQJHJURZV,QWKHVLQJOHSRLQWFRQWDFWPRGHOZHPD\LQWURGXFHD VLPLODU UHGXFWLRQ RI WKH FRUUHVSRQGLQJ OHQJWK 7KH LQWHUVHFWLRQ OHQJWK
(
LV
GHILQHGKHUHDVWKHUDWLRRIWKHODWHUDOGHIOHFWLRQY DQGWKHWUDQVLHQWODWHUDOVOLS WDQ
Y
('
WDQ
,Q(T WKHUHOD[DWLRQOHQJWK
GY GW
%
(
'
_9 _ Y
[
LVUHSODFHGE\
'&
_9 _ WDQ
[
9
(
DQGZHJHW
V\
ZLWKDSSDUHQWO\
('
&
)\
)
\
WDQ
'
R &
)
)
\
WDQ
ZLWKWKHLQLWLDOUHOD[DWLRQOHQJWK DW
R' WRZKLFK
. N
R &
)
\ % &) )
_) _
_WDQ
I %
_
)
&
)
&
(
)\
DSSURDFKHVZKHQ
N67RDYRLGVLQJXODULW\RQHPD\XVHWKHODVW DQGDGGWR NWKHVKLIW WRDUULYHDW NDV
H[SUHVVLRQRI ZLWKVPDOO
)
I
LQGLFDWHGLQ)LJ)LJXUHSUHVHQWVWKHFKDUDFWHULVWLFRIWKHLQWHUVHFWLRQ OHQJWKWRJHWKHUZLWKWKHVLGHIRUFHFKDUDFWHULVWLFIURPZKLFKLWLVGHULYHG$OVR LQWKHHTXDWLRQIRUWKHFDPEHUGHIOHFWLRQUHVSRQVH WKHUHOD[DWLRQOHQJWK
6,1*/(&217$&732,1775$16,(1702'(/
~σα
σα R
)\
σα* ~σα*
σα
α
0 )LJ&KDUDFWHULVWLFVRIW\UHVLGHIRUFHµLQWHUVHFWLRQOHQJWK¶
(
LVUHSODFHGE\
(
DQGUHOD[DWLRQOHQJWK
(
$PRUHGLUHFWZD\WRZULWH(T LV\LHOGHGE\HOLPLQDWLQJ
ZLWKWKHXVH
RI(T
GY
%_
9 _ WDQ
GW
[
'
7KHWUDQVLHQWVOLSDQJOH
_9 _ WDQ
[
'&
9
V\
NLVREWDLQHGIURPWKHGHIOHFWLRQY E\XVLQJWKHLQYHUVH
N N FKDUDFWHULVWLFFI+LJXFKL
SRVVLEO\DGDSWHG)\
Y
'
)
\
&
)\
7RDYRLGGRXEOHYDOXHGVROXWLRQVDIRUWKLVSXUSRVHDGDSWHGFKDUDFWHULVWLFIRU
N N LQ RU PD\EHUHTXLUHGVKRZLQJDSRVLWLYHVORSHLQWKHVOLS NRUIRU (ZKHQ(T LVXVHGLVREWDLQHG
)\
UDQJHRILQWHUHVW7KHYDOXHIRU
WKURXJK LWHUDWLRQV RU E\ XVLQJ LQIRUPDWLRQ IURP WKH SUHYLRXV WLPH VWHS FI +LJXFKL 3DFHMND 7DNDKDVKL 7KHXOWLPDWHYDOXHRIWKHIRUFH )\LVILQDOO\REWDLQHGIURP(T E\XVLQJWKHFRPSXWHGWDQ
N
Y
(
:ULWLQJWKHHTXDWLRQ HQWLUHO\LQWHUPVRIWKHWUDQVLHQWVOLSDQJOHE\ XVLQJ ZLWKY
&
)\
M
)
\
M WDQ
Y DQGUHPHPEHULQJWKDW)\
%_
G WDQ
GW
9 _ WDQ
[
'&
9
)\
V\ &
N) GLUHFWO\\LHOGV
&
)\
M M
]
) )
\
G)
]
GW
M
]
M
7KHDGGLWLRQDOLQSXWG)] G WUHTXLUHVLQIRUPDWLRQRIWKHVORSH )\ )]DWJLYHQ
YDOXHV RI WKH VOLS DQJOH ,I WKH YHUWLFDO ORDG UHPDLQV FRQVWDQW WKH ODVW WHUP YDQLVKHVDQGZHKDYHWKHRIWHQXVHGHTXDWLRQRIWKHUHVWULFWHGIXOO\QRQOLQHDU PRGHO
6,1*/(&217$&732,1775$16,(1702'(/
G WDQ
%_
9 _WDQ
GW ZLWK
'
M
&
[
)
9
V\
\
M WDQ
)\
'&
,IZHFRQVLGHUDQDYHUDJHVOLSDQJOH
R
DQGDVPDOOYDULDWLRQ
a
RIWDQ
DQGWKH
FRUUHVSRQGLQJODWHUDOVOLSYHORFLWLHV(T EHFRPHVDIWHUKDYLQJVXEWUDFWHG WKHDYHUDJHSDUW
G
a
%_
a
9 _
GW
[
'&
a
9
V\
ZKLFK LQGLFDWHV WKDW WKH VWUXFWXUH RI LV UHWDLQHG DQG WKDW
UHSUHVHQWVWKHDFWXDOUHOD[DWLRQOHQJWKRIWKHOLQHDULVHGV\VWHPDWDJLYHQORDG DQGVOLSDQJOH,WVFKDUDFWHULVWLFKDVEHHQGHSLFWHGLQ)LJDVZHOO2EYLRXVO\ WKHUHOD[DWLRQOHQJWKLVDVVRFLDWHGZLWKWKHVORSHRIWKHVLGHIRUFHFKDUDFWHULVWLF ,WDOVRVKRZVWKDWWKHUHOD[DWLRQOHQJWKEHFRPHVQHJDWLYHEH\RQGWKHSHDNRIWKH VLGH IRUFH FKDUDFWHULVWLF ZKLFK PDNHV WKH VROXWLRQ RI EXW DOVR RI WKH RULJLQDOHTXDWLRQ RU XQVWDEOHLIWKHSRLQWRIRSHUDWLRQOLHVLQWKDW UDQJH RI VLGH VOLS :H PD\ KRZHYHU OLPLW LQVWDELOLW\ DQG H[FHVVLYH FRPSXWDWLRQ WLPH
GRZQZDUGV WR DYRLG ERWK PD[
PLQ
7KH WUDQVLHQW
UHVSRQVHRIWKHYDULDWLRQRIWKHIRUFHSURFHHGVLQSURSRUWLRQZLWKWKHYDULDWLRQ RI
a
a
'M
M
a \ )\ WDQ
DV )
:KHQWKH(T LVXVHGDQDOJHEUDLFORRSGRHVQRWRFFXUDVLVWKHFDVH ZKHQ(T LVHPSOR\HG7KHUHOD[DWLRQOHQJWK IURPWKHDOUHDG\DYDLODEOHWDQ
FDQEHGLUHFWO\GHWHUPLQHG
N+RZHYHUVLQFHWKHODVWWHUPRI KDVEHHQ
RPLWWHGHTXDWLRQ KDVEHFRPHLQVHQVLWLYHWR)]YDULDWLRQV 6LPLODUIXQFWLRQVDQGGLIIHUHQWLDOHTXDWLRQVFDQEHGHULYHGIRUWKHWUDQVLHQW UHVSRQVHWRORQJLWXGLQDOVOLS:HREWDLQIRUWKHGLVWDQFHIDFWRUV
('
)
&
[
'
)[
R &
)
)
[
.
[ % &) )
_) _
R
_ _
&
)
%
)
DQG
'
&
M M
)
)[
[
,WLVRILQWHUHVWWRQRWHWKDWWKHH[WUHPHFDVHRIZKHHOORFNFDQQRZEHKDQGOHG FRUUHFWO\$VZHKDYHVHHQEHORZ(T WKHGHIOHFWLRQWKHQEHFRPHVHTXDO WR PLQXV WKH UHOD[DWLRQ OHQJWK ZKLFK DFFRUGLQJ WR WKH IXOO\ QRQOLQHDU PRGHO EHFRPHV X
(
:LWK DQG
N
WKHGHIOHFWLRQWDNHVWKH YDOXH WKDW
6,1*/(&217$&732,1775$16,(1702'(/
ZRXOGDFWXDOO\RFFXUDWZKHHOORFN X
)[&)[
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
E
7KHIXOO\QRQOLQHDUPRGHODFFRUGLQJWR(TV RU RUDVLQ
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
DVSUHVHQWHGLQ)LJ7KHUHVXOWLQJYDULDWLRQRI
ZLWKVOLSDQJOH
OHYHOLVFRPSDUHGZLWKWKHVORSHRIWKHVWHDG\VWDWHVLGHIRUFHFKDUDFWHULVWLFERWK QRUPDOLVHGPDNLQJWKHLUYDOXHVHTXDOWRXQLW\DWYDQLVKLQJVOLSDQJOH
6,1*/(&217$&732,1775$16,(1702'(/
)\
test model )]= 4000N
o
10
2000
[N] 2
1500
o
α = 1o
1000
500
0
0
0.5
1
V
1.5
2
[m]
2.5
)LJ 6LGHIRUFHUHVSRQVHWRVPDOODQGODUJHVWHSFKDQJHLQVOLSDQJOHDVDVVHVVHGE\IODW SODQNH[SHULPHQWVDQGFRPSXWHGZLWKWKHPRGHOGHILQHGE\(TV RU
∆)\
)]= 4000N
600
final α=
∆α= 0.5o
o
0.5
o
[N]
1
o
400
1.5
o
2
200 o
3o 4o 5
0 0
0.5
1
V
1.5
[m]
)LJ 6LGH IRUFH UHVSRQVH WR VPDOO LQFUHPHQW RI VOLS DQJOH
2
2.5 DW GLIIHUHQW OHYHOVRIVOLS
DQJOHLQGLFDWLQJWKHGHFUHDVHRIUHOD[DWLRQOHQJWKDWLQFUHDVLQJVOLGHVOLSOHYHOZKLFK LVLQDJUHHPHQWZLWKH[SUHVVLRQ
6,1*/(&217$&732,1775$16,(1702'(/
σα
M) \ Mα
1.0
σα
+
+
0.6
[m] 0.4
+ +
0.2
0
6000N 4000N 2000N
+
0.8
)]= 4000N
normalised value
)] +
+
+
4
5
0
0
1
2
3
final slip angle
4
α
0
5
[o ]
)LJ &RUUHVSRQGHQFH EHWZHHQ
)\
[o]
)LJ 5HOD[DWLRQOHQJWKPHDVXUHGDW
DV
YDULRXVZKHHOORDGVGHFD\LQJZLWK
DVVHVVHGE\H[SHULPHQWV
LQFUHDVLQJVOLSDQJOH
)]= 4000N
M)\
2 3 final slip angle α
YV
VORSH DQG UHOD[DWLRQ OHQJWK
0.5
1
∆α=0.5o ∆)] =400N
σα
M)]
(initial) )]=4000N
0.6
[m]
0.3
0.4 0.2
0.2
0.1 0
0
)LJ
1
α
2
3
)
[o]
4
5
)
7KHVORSHRI \YV ]DVDIXQFWLRQ RI
VOLS
DQJOH
DSSHDULQJ
LQ
(T DVDVVHVVHGE\WHVWV
0 0
1
2 3 (final) slip angle α
4
5
[o ]
)LJ 5HOD[DWLRQOHQJWKVUHVXOWLQJ IURPVPDOOVWHSFKDQJHVLQVOLS DQJOHRUZKHHOORDGDVIXQFWLRQ RI VOLS DQJOH OHYHO +LJXFKL
7KH H[FHOOHQW DJUHHPHQW VXSSRUWV WKH WKHRUHWLFDO ILQGLQJV $GGLWLRQDO H[SHULPHQWVKDYHEHHQFRQGXFWHGE\+LJXFKLLQYHVWLJDWLQJWKHUHVSRQVHWRVPDOO LQFUHPHQWV LQ YHUWLFDO ORDG DW FRQVWDQW VOLS DQJOH $V VKRZQ E\ (T WKH
M) M) SOD\VWKHUROHDVLQSXWSDUDPHWHU)LJXUH VKRZVLWV
SDUWLDOGHULYDWLYH
\
]
YDULDWLRQ ZLWK VOLS DQJOH OHYHO )LJXUH FRPSDUHV WKH UHVXOWLQJ UHOD[DWLRQ OHQJWKYDOXHVZLWKWKRVHDVVRFLDWHGZLWKFKDQJHVLQVOLSDQJOHDWFRQVWDQWORDG $SSDUHQWO\ WKH UHOD[DWLRQ OHQJWK IRU VOLS DQJOH FKDQJH LV ODUJHU WKDQ WKH RQH EHORQJLQJWRORDGFKDQJH+RZHYHUGXHWRWKHILQLWHPDJQLWXGHRIWKHLQFUHPHQWV
E
DQG1UHVSHFWLYHO\ WKHDFWXDOGLIIHUHQFHVDUHH[SHFWHGWREHVPDOOHU ,QWKHGLDJUDPWKHFXUYHIRU WKHFXUYHIRU
)
E
PD\EHEHWWHUVKLIWHGWRWKHOHIWRYHU ZKLOH
PD\EHUHGXFHGDOLWWOHLQKHLJKWXVLQJLQIRUPDWLRQIURP)LJ ]
DFFRXQWLQJIRUDGHFUHDVHRIWKHDYHUDJHORDGOHYHORI1
6,1*/(&217$&732,1775$16,(1702'(/
+LJXFKLDOVRLQYHVWLJDWHGWKHUHVSRQVHVWRFKDQJHVLQFDPEHUDQJOHDQGWR DOLPLWHGH[WHQWDOVRWRWKHUHODWHGWXUQVOLS7KHFKDQJHLQFDPEHUZDVFRUUHFWO\ HVWDEOLVKHGE\URWDWLQJWKHURDGVXUIDFH WKHSODQN DERXWWKHOLQHRILQWHUVHFWLRQ RIZKHHOFHQWUHSODQHDQGSODQNVXUIDFH VWHHUDQJOHLVNHSWHTXDOWR]HUR 7KH VWHSUHVSRQVHWRWXUQVOLS
n
W
5
LVREWDLQHGE\LQWHJUDWLQJWKHUHVSRQVHWRD
SXOVH FKDQJHLQWXUQVOLSWKDWLV ORDGW\UHWZLVW
DQGWKHQUROOLQWKHQHZ
ZKHHO SODQH GLUHFWLRQ DQG PXOWLSO\ WKH UHVXOW ZLWK
5
7KH UHVSRQVHV WR
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
&
FDPEHUVWLIIQHVVZLOOEH
)
g & nU )
H
:LWKWKHW\UHHIIHFWLYHUROOLQJUDGLXV
HTXDO WR DSSUR[LPDWHO\ P DQG WKH HVWLPDWHG VWHDG\VWDWH OHYHOV UHDFKHG LQ )LJV DQG D IRU
5
P DQG
UHGXFWLRQIDFWRUDSSUR[LPDWHO\
g
E
UHVSHFWLYHO\ ZH ILQG IRU WKH
7KHUHVSRQVHRIWKHDOLJQLQJWRUTXHWRFDPEHULVVLPLODUWRWKHUHVSRQVHWR WXUQVOLS FI)LJ 7KHPRPHQWTXLFNO\UHDFKHVDPD[LPXPDIWHUZKLFKD VORZHUGHFD\WRWKHVWHDG\VWDWHOHYHORFFXUV7KHVWULQJPRGHOZLWKWUHDGZLGWK HIIHFW SUHGLFWV WKH VDPH IRU WKH UHVSRQVH WR WXUQ VOLS DV FDQ EH FRQFOXGHG E\ DGGLQJ WKH FXUYHV IRU
0¶ ]
DQG
0
]
UHVSRQVH WR
n )LJ LQ DQ DSSURSULDWH
SURSRUWLRQ 7KH PRGHO GHYHORSHG LQ WKH SUHVHQW FKDSWHU JHQHUDWHV D VLPLODU UHVSRQVHEXWWKURXJKDGLIIHUHQWPHFKDQLVP(TXDWLRQ ( WKDWIRU WDNHVWKHIRUP
)
[
0 ] ' 0 ]N % 0]U ' &W )\ % 0]U LVXVHGDIWHUKDYLQJFRPSXWHGWKHWUDQVLHQWVOLSDQGFDPEHUDQJOHV
NDQG N
6,1*/(&217$&732,1775$16,(1702'(/
1: ) ] 2: γ
-200
)]= 4000N
0
-2
-200
)\
o
o
-5
-400
-10
[N] -600
-800
o
γ = -15 o 0
0.5
1
V
1.5
[m]
2
2.5
)LJD6LGHIRUFHUHVSRQVHWRVWHSFKDQJHLQFDPEHUDQJOH
1: )]2: γ )]= 4000N
0
-10
0
-2
o
-20
-5
] -30
o
-10
o
[Nm]
γ = -15 o
-40
-50
0
0.5
1
V
1.5
[m]
2
2.5
)LJE0RPHQWUHVSRQVHWRVWHSFKDQJHLQFDPEHUDQJOH
1.0
)]
+
6000N 4000N 2000N
σγ
0.8
+
+
+
+
0.6
[m] 0.4 0.2 0 -15 )LJ
-10
γ
-5
[o]
0
5HOD[DWLRQOHQJWKIRUVWHSUHVSRQVHVWRGLIIHUHQWFDPEHUDQJOHV DEVFLVVD DWYDULRXV ZKHHOORDGV FDPEHUDIWHUORDGLQJ
6,1*/(&217$&732,1775$16,(1702'(/
$VFDQEHVHHQLQ)LJRUWKHDOLJQLQJWRUTXHFDXVHGE\FDPEHUDW ]HURVOLSDQJOHLVDWWULEXWHGWRWKHUHVLGXDOWRUTXH 0]U DQGWRWKHFRXQWHUDFWLQJ PRPHQWHTXDOWRWKHDOLJQLQJVWLIIQHVV&0 WLPHV DW
ZKLFKLVWKHVDPHDVW
)\
7KHUHVLGXDOWRUTXHUHVSRQGVTXLFNO\ZLWKDUHODWLYHO\VKRUWUHOD[DWLRQ
OHQJWK DERXWHTXDOWRKDOIWKHFRQWDFWOHQJWK RUDVKDVEHHQVXJJHVWHGDERYHIRU WKH SUHVHQW PRGHO LQVWDQWDQHRXVO\ $V WKH VLGH IRUFH UHVSRQGV VORZO\ ZLWK UHOD[DWLRQ OHQJWK
DQG W
)\ LV RSSRVLWH LQ VLJQ ZLWK UHVSHFW WR WKH UHVLGXDO
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
W
)\LVHTXDOWRWKHPRPHQWGXHWRWKH\DZ
WRUVLRQLQGXFHGVLGHVOLS+RZHYHULQUHDOLW\RQO\DSDUWRIWKHFDPEHUIRUFH)\ UHVXOWVIURPFDPEHUVSLQLQGXFHGVLGHVOLS,WLVH[SHFWHGWKDWWKHDERYHDQDO\VLV LVSDUWO\WUXH3HUKDSVHYHQIRUDQDSSUHFLDEOHSDUW7KHUHPDLQLQJSDUWWKDWLV UHVSRQVLEOH IRU WKH µKXPS¶ LQ WKH PRPHQW UHVSRQVH PXVW WKHQ EH GXH WR WKH WUDQVLHQWDV\PPHWULFODWHUDOW\UHGHIRUPDWLRQWKDWYDQLVKHVZKHQWKHVWHDG\VWDWH FRQGLWLRQLVUHDFKHG)LJXUH GHSLFWVWKHQDWXUHRIWKLVWUDQVLHQWGHIOHFWLRQ ,QDQDOWHUQDWLYHPRGHOXVHGIRUPRWRUF\FOHG\QDPLFVVWXGLHVGHDOLQJZLWK SRVVLEO\ ODUJH FDPEHU DQJOHV WKH RWKHU H[WUHPH LV XVHG DQG WKH WHUP W UHSODFHGE\W
)\
)\ LV
WKDWLV ZLWKWKHVLGHIRUFHDWWULEXWHGWRWKH WUDQVLHQW VLGHVOLS
DQJOHDORQH FI&KDSWHU
1RQ/DJJLQJ3DUW
7KHIRUFHUHVSRQVHWRFDPEHUFKDQJHH[KLELWVDSHFXOLDUIHDWXUH,WWXUQVRXWWKDW GLUHFWO\ DIWHU WKDW WKH ZKHHO LV FDPEHUHG D VLGH IRUFH LV GHYHORSHG LQVWDQWDQHRXVO\7KLVREYLRXVO\LVFDXVHGE\WKHQRQV\PPHWULFGLVWRUWLRQRI WKHFURVVVHFWLRQRIWKHORZHUSDUWRIWKHW\UH7KLVLQLWLDOµQRQODJJLQJ¶VLGHIRUFH VKRZQLQ)LJDDWDGLVWDQFHUROOHGV
DSSHDUVLQWKHFDVHFRQVLGHUHGWR
DFWLQDGLUHFWLRQRSSRVLWHWRWKHVWHDG\VWDWHVLGHIRUFH$OVRHTXDOGLUHFWLRQV WXUQRXWWRRFFXUDVKDVEHHQUHSRUWHGLQWKHOLWHUDWXUHIRUDPRWRUF\FOHW\UHFI 6HJHODQG:LOVRQ ,QWKHVHH[SHULPHQWVWKHFDPEHUDQJOHLVDSSOLHGDIWHU WKDWWKHW\UHLVORDGHG:HPD\FRQGXFWDQH[SHULPHQWZKHUHWKHIUHHZKHHOLV
6,1*/(&217$&732,1775$16,(1702'(/
1: ) ] 2:γ
100%
)]
1: γ 2: ) ]
+
6000N 4000N 2000N
50%
)\,non-lag + )\,ss 0 +
+
+
+
+
+
+
-50%
-100% -15
γ
-10
-5
[o ]
0
)LJ 1RQODJJLQJ SDUW RI VLGH IRUFH UHVSRQVH IRU WKH FDVHV DSSO\LQJ FDPEHUDIWHU KDYLQJ ORDGHG WKH W\UH DQG ORDGLQJ WKH W\UH QRUPDO WR WKH URDG DIWHU KDYLQJ FDPEHUHGWKHZKHHOµLQWKHDLU¶
ILUVWFDPEHUHGDQGWKHQPRYHGWRWKHURDGLQQRUPDOGLUHFWLRQ7KLVDSSHDUVWR UHVXOWLQDVRPHZKDWODUJHUQRQODJJLQJSDUW)LJXUHJLYHVWKHSHUFHQWDJH RIWKHQRQODJJLQJSDUWZLWKUHVSHFWWRWKHVWHDG\VWDWHVLGHIRUFHIRUGLIIHUHQW ZKHHO ORDGV DQG IRU WKH WZR FRQVLGHUHG ZD\V RI UHDFKLQJ WKH ORDGHG DQG FDPEHUHGFRQGLWLRQEHIRUHUROOLQJKDVVWDUWHG7KHKLJKHUORDGHGW\UHDSSHDUVWR JHQHUDWHHTXDOO\GLUHFWHGLQLWLDODQGVWHDG\VWDWHVLGHIRUFHUHVSRQVHVWRDFKDQJH LQFDPEHUDQJOH 7RLQWURGXFHWKLVW\UHSURSHUW\LQWRWKHPRGHOZHPD\GHILQHDQRQODJJLQJ
g
IUDFWLRQ
1/
7KHSRUWLRQRIWKHLQSXWFDPEHUDQJOH
ODJJLQJ UHVSRQVH LV
g
1/
7KH UHPDLQLQJ SDUW
/
1/
WKDWJLYHVULVHWRWKHQRQ
g
1/
\LHOGV WKH ODJJHG
UHVSRQVH,QWKHULJKWKDQGPHPEHURI(T PXVWQRZEHUHSODFHGE\ DQGWKHVXP
N
1/
/
LVWREHXVHGDVLQSXWLQWR(T ZKLOHWKHFDPEHUDQJOH
LWVHOI DFWV DV WKH GLUHFW LQSXW LQWR (T IRU WKH UHVLGXDO WRUTXH LI ZH
QHJOHFWWKHDVVRFLDWHGVKRUWUHOD[DWLRQOHQJWK 7KHIROORZLQJHTXDWLRQVDSSO\LQVWHDGRI(T
GY GW
%
_9 _ Y
[
' & g1/
&
)
&
)
_9 _
[
IURPZKLFKWKHWUDQVLHQWFDPEHUDQJOHLVREWDLQHG
/'
&
)
Y
&
)
7KHQRQODJJLQJSDUWLV
1/ ' g1/
VRWKDWWKHXOWLPDWHYDOXHIRUWKHLQSXWLQ(T EHFRPHV
6,1*/(&217$&732,1775$16,(1702'(/
'
/ % 1/
)RUWKHOLQHDUFDVHZHZRXOGREWDLQIRUWKHFDPEHUIRUFHDQGPRPHQWUHVSRQVHV )
' ) ] ' & &
\
0
W
&
)
g
)RUWKHIUDFWLRQ
1/
%
&
%
0
W &
)
ZHPD\WDNHWKHUDWLRRIWKHRXWSXWVLGHIRUFHDVGHSLFWHGLQ
)LJ DWVPDOOFDPEHU 7KHQRQODJJLQJIUDFWLRQLVORDGGHSHQGHQW7KLVDOVR KROGVIRUWKHQRQODJJLQJIUDFWLRQWKDWLVIRXQGWRRFFXUZKHQORDGLQJRIWKHW\UH SHUIRUPHGE\PRYLQJWKHZKHHOLQDGLUHFWLRQSHUSHQGLFXODUWRWKHURDGVXUIDFH IROORZVDIWHUWKDWWKHIUHHXQORDGHGZKHHOKDVEHHQWLOWHG:KHQDYHKLFOHPRYHV RYHUDQXQGXODWHGURDGVXUIDFHWKHREVHUYHGW\UHSURSHUWLHVUHJDUGLQJFDPEHU DQGVLGHVOLSIRUFHUHVSRQVHWRZKHHOORDGYDULDWLRQVZLOOEHRILPSRUWDQFH,QWKH IROORZLQJH[HUFLVHDQGLQ6HFWKLVSUREOHPLVDGGUHVVHG
([HUFLVH&DPEHUHGZKHHOPRYLQJXSDQGGRZQ &RQVLGHUDZKHHOWKDWUXQVDWDFRQVWDQWFDPEHUDQJOHDW]HURVOLSDQJOHRQDVPRRWK URDGVXUIDFH7KHZKHHOD[OHLVPRYHGYHUWLFDOO\DVGHSLFWHGLQ)LJ7KHIRUFHV DUHPHDVXUHGLQWKHZKHHOKXE
8VH(TV DQGGHYHORSDQH[SUHVVLRQIRUWKHVLGHIRUFHUHVSRQVHWRDUDPS
FKDQJHLQYHUWLFDOORDGGHILQHGDV V
V
)]
)]R
V VU
)]
)]R
)]VVU
V!VU
)]
)]R
)]
9W
7KHFDPEHUDQJOHDQGORDGYDULDWLRQDUHDVVXPHGWR
γ
EH UHODWLYHO\ VPDOO DQG WKH HTXDWLRQV FDQ EH NHSW OLQHDU ZLWK UHOD[DWLRQ OHQJWK
WUHDWHG DV D
FRQVWDQW:HPD\VLPSOLI\WKHHTXDWLRQVWR a a
GY
a
% ' & g1/ a
Y
GV
a
/ 'E
&)
R
1/ ' g1/
'
F )
Y
ZKHUHF
)
/%
&)\DQGE
&)
]
]
]R
R
&)
R
:LWK
1/
ZHKDYHIRUWKHWRWDOVLGHIRUFH
)
\'
&
)
\'
) R / % &)
a
ZLWK
a
&
)] )LJ&DPEHUHGZKHHO
7KHFKDQJHLQVLGHIRUFHEHFRPHVOLQHDUL]HG
)
)\
a
1/
PRYHGXSDQGGRZQ ([HUFLVH
6,1*/(&217$&732,1775$16,(1702'(/
a a
)
&
DQG
)
'
]'
)
a
)
)
]
&
)R
]R
] & ) ]R
7DNHIRUWKHSDUDPHWHUVWKHYDOXHV
PE
UDG)]R
F
P&)
1
)]
R
1UDG
1VU
P
V
'UDZDJUDSKRIWKHFKDQJHLQVLGHIRUFHZLWKWUDYHOOHGGLVWDQFH IRUWKHWKUHHFDVHV
g
1/
7U\WRILQGDW\SLFDOIHDWXUHRIWKHUHVXOWLQJYDULDWLRQRIWKHFRXUVHRIWKHVLGHIRUFH FKDQJHWKDWPD\HQDEOH\RXWRDVVHVVWKHSHUFHQWDJHQRQODJJLQJSDUW
g1/IURPDWHVW
UHVXOW*LYHWKHUHOHYDQWIRUPXOD 8VHWKHVDPHVHWRIHTXDWLRQVDQGDVVXPHWKHLQFUHPHQWLQYHUWLFDOORDGWRFKDQJH VLQXVRLGDOO\
]'
a
)
)
] VLQ V V
'HULYHIRUDJLYHQFRQVWDQWFDPEHUDQJOHWKHIUHTXHQF\UHVSRQVHIXQFWLRQ a
L
+
)\)]
V
'
)
\ L V
a
)
] L V
3ORWWKHDPSOLWXGHUDWLRDQGWKHSKDVHODJYHUVXVSDWKIUHTXHQF\ YDOXHVRI
g
V
IRUWKHVDPHWKUHH
1RWHW\SLFDOIHDWXUHRISKDVHODJGHSHQGLQJRQVLJQRI 1/
g
1/
IURPZKLFK
WKHQRQODJJLQJSHUFHQWDJHPD\EHDVVHVVHG'HULYHWKHIRUPXODIRUWKHSDWKIUHTXHQF\ ZKHUHSKDVHODJHLWKHUUHDFKHVDPD[LPXPRUEHFRPHV GHJUHHV
7KH*\URVFRSLF&RXSOH
,Q &KDSWHU WKH J\URVFRSLF FRXSOH WKDW DULVHV DV D UHVXOW RI WKH WLPH UDWH RI FKDQJHRIWKHDYHUDJHW\UHWLOWGHIOHFWLRQDQJOHKDVEHHQLQWURGXFHG7KLVDQJOH LVFRQVLGHUHGWREHSURSRUWLRQDOZLWKWKHODWHUDOW\UHGHIOHFWLRQYRUWKHVLGHIRUFH )\(TXDWLRQ PD\EHZULWWHQLQWHUPVRIWKHGHIOHFWLRQZLWKWKHXVHRI
]J\U '
0
F
J\U P W UH
GY
GW
ZKHUHDWIUHHUROOLQJWKHZKHHOVSHHGRIUHYROXWLRQHTXDOVWKHIRUZDUGYHORFLW\ GLYLGHGE\WKHHIIHFWLYHUROOLQJUDGLXVRIWKHW\UH,QJHQHUDOZHKDYH
'
9
[ & 9 V[ U
H
6,1*/(&217$&732,1775$16,(1702'(/
)RUDUDGLDOSO\VWHHOEHOWHGFDUW\UHWKHQRQGLPHQVLRQDOFRHIILFLHQWFJ\UKDVEHHQ HVWLPDWHGWRWDNHWKHYDOXH7KHTXDQWLW\PWUHSUHVHQWVWKHPDVVRIWKHW\UH ([WHQGLQJWKHH[SUHVVLRQ \LHOGVIRUWKHWRWDODOLJQLQJPRPHQW 0
]'
0N
]
%
0
]U % 0]J\U
(QKDQFHG1RQ/LQHDU7UDQVLHQW7\UH0RGHO
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
9U
9
V
Ω
9
α
X
9
rim mass P
F
[
Y
*
9V
)
[
*
6
0]
)\
)
9*
UH
9
rim
PF
*
V[
9
V[
6
)[
)LJ (QKDQFHGWUDQVLHQWW\UHPRGHOLQWRSDQGVLGHYLHZVKRZLQJFDUFDVVFRPSOLDQFHV DQGFRQWDFWSDWFKPDVV
6,1*/(&217$&732,1775$16,(1702'(/
HQDEOHVWUDLJKWIRUZDUGFRPSXWDWLRQVDPDVVSRLQWLVWKRXJKWWREHDWWDFKHGWR
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
ODWHUDOVSHHGFRPSRQHQWV9 V[\UHDG
0 (
)
0 (
)
F V[ % N F[ X0 % FF[ X '
P 9
F V\ % N F\ Y0 % FF\ Y '
P 9
[ )]
\ )] & )\1/
7KHIRUFHVDFWLQJIURPJURXQGWRFRQWDFWSDWFKVKRZQDWWKHULJKWKDQGVLGHVDUH FRPSXWHGIURPWKHVWHDG\VWDWHIRUPXODH7KHQRQODJJLQJFDPEHUIRUFHSDUW )\1/LVDVVXPHGWRDFWGLUHFWO\RQWKHZKHHOULP:HPD\DSSUR[LPDWHWKHQRQ ODJJLQJIRUFHSDUWE\DOLQHDUUHODWLRQZLWK XVLQJWKHFDPEHUWKUXVWVWLIIQHVV &) WKHQRQODJJLQJIUDFWLRQ
g
1/
FI6HF DQGWKHZHLJKWLQJIXQFWLRQ*\
( WRWDNHFDUHRIWKHSUHVHQFHRIDIRUHDQGDIWIRUFH)[ )
\1/ '
\ g1/ &)
*
,WVKRXOGEHQRWHGWKDW
g
1/
FKDQJHVZLWKZKHHOORDGFI)LJ
7RHQDEOHFDOFXODWLRQVQHDURUDWVWDQGVWLOOZHPD\DGGDGGLWLRQDOILUVWRUGHU GLIIHUHQWLDO HTXDWLRQV ZLWK UHOD[DWLRQ OHQJWK
F
)URP WKHVH HTXDWLRQV WKH
WUDQVLHQWVOLSTXDQWLWLHVUHVXOWZKLFKDFWDVLQSXWLQWKHVWHDG\VWDWHVOLSIRUFH IRUPXODH:HKDYH
F F
G
%_
[_
%_
[_
GW G GW
9
9
'& '&
9
9
(
V\
(
V[
,IQHHGHGZHPD\LQWKHULJKWKDQGPHPEHURI UHSODFHDUJXPHQW WKHWUDQVLHQWVSLQVOLS
F
G n
GW
%_
9
ZLWK
nNWKDWUHVXOWVIURPDQHTXDWLRQVLPLODUWR
[_ n ' _9[_ n
ZKHUHLQWKHULJKWKDQGPHPEHU
nLVWREHUHSODFHGE\WKHH[SUHVVLRQ
6,1*/(&217$&732,1775$16,(1702'(/
7KHFRQWDFWUHOD[DWLRQOHQJWK
F
PD\EHJLYHQDVPDOOEXWILQLWHYDOXHIRU
LQVWDQFHHTXDOWRKDOIWKHFRQWDFWOHQJWKDRDWQRPLQDOORDGWKDWFRUUHVSRQGVWR RXU ILQGLQJV LQ &KDSWHU 7KH HTXDWLRQV GR QRW UHVSRQG WR ORDG FKDQJHV)RUWKLVZHUHO\RQWKHHIIHFWRIWKHFDUFDVVFRPSOLDQFHZKLFKJLYHV DGHTXDWHUHVXOWV(IIHFWLYHO\WKHUHVXOWLQJODWHUDOFRPSOLDQFHRIWKHVWDQGLQJW\UH LV
&
'
)\
F
F\
%
F
&
)
DQGWKHHIIHFWLYHUHVXOWLQJW\UHUHOD[DWLRQOHQJWK IRUVLGHVOLSUHVSRQVH &
)
'
&
&
)
'
)\
F
F\
%
F
7KLVHTXDWLRQPD\LQIDFWEHHPSOR\HGWRDVVHVVWKHODWHUDOFDUFDVVVWLIIQHVVDW JURXQG OHYHO FF\ )URP WKH PHDVXUHG W\UH UHOD[DWLRQ OHQJWK
DQG FRUQHULQJ
VWLIIQHVV&) WKHODWHUDOVWLIIQHVVRIWKHVWDQGLQJW\UH&)\IROORZV%\WDNLQJ
F
HTXDO WR KDOI WKH FRQWDFW OHQJWK D WKH FDUFDVV ODWHUDO VWLIIQHVV FF\ FDQ EH GHWHUPLQHG7KHUHOD[DWLRQOHQJWKIRUORDGYDULDWLRQVUHPDLQVHTXDOWR &
)
)] '
F
F\
ZKLFKLVDOLWWOHVPDOOHUWKDQ
$OWKRXJKWKLVUHVXOWVIURPSUDFWLFDOPRGHOOLQJ
FRQVLGHUDWLRQV ZH PD\ LQ IDFW FRPH FORVH WR WKH PHDVXUHPHQW UHVXOWV RI )LJ ,I RQH LV QRW LQWHUHVWHG WR LQFOXGH WKH DELOLW\ WR VLPXODWH QHDU RU DW ]HUR IRUZDUGVSHHGWKHFRQWDFWUHOD[DWLRQOHQJWK
F
PD\EHGLVUHJDUGHGDQGWDNHQ
HTXDOWR]HUR,Q&KDSWHU WKHFRQWDFWUHOD[DWLRQOHQJWK
F
IRUPVDQHVVHQWLDO
HOHPHQWLQWKHPRGHOGHYHORSHGIRUVKRUWZDYHOHQJWKEHKDYLRXU 7KHGHIOHFWLRQUDWHVQHHGHGLQ(TV DUHHTXDOWRWKHGLIIHUHQFHLQVOLS YHORFLWLHV X 0
'
Y 0
'
9
(
V[ & 9V[
V\ & 9V\
9
(
$VKDVEHHQPHQWLRQHGEHIRUHWKHZKHHOVOLSYHORFLW\9VZLWKFRPSRQHQWV9V[\ LV GHILQHG DV WKH KRUL]RQWDO YHORFLW\ RI WKH VOLS SRLQW 6 WKDW LV WKRXJKW WR EH DWWDFKHGWRWKHZKHHOULPDGLVWDQFHUHWKHHIIHFWLYHUROOLQJ UDGLXVEHORZWKH ZKHHOFHQWUHLQWKHZKHHOFHQWUHSODQH 9
V[ '
9
V\ '
9
[ & UH
9
\& U
0
6,1*/(&217$&732,1775$16,(1702'(/
ZKHUH9[\GHQRWHWKHKRUL]RQWDO SDUDOOHOWRURDGSODQH FRPSRQHQWVRIWKHZKHHO FHQWUHYHORFLW\,Q(T WKHFDPEHUDQJOHLVDVVXPHGWREHVPDOODQGUHLV UHSODFHGE\WKHORDGHGUDGLXVU 7KHIRUFHVDFWLQJRQWKHZKHHOULPILQDOO\EHFRPH ) )
[D ' \D '
N N
F[ X0 % FF[ X
F\ Y0 % FF\ Y % )\1/
7KHDOLJQLQJWRUTXHEHFRPHV
]'
0
]U '
0 0
]'
&
W
0
)
]U
0N
]
\
%
0
)
]
]U % V @ )[ % 0]J\U
)[
)\
γ = 10o 0
100%
α= 6
κ
o
10%
travelled distance
8m
)LJ 6LGHIRUFHDQGEUDNHIRUFHUHVSRQVHWRVWHSFKDQJHVLQFRPELQHGVOLSFRPSXWHGZLWK WKHHQKDQFHGWUDQVLHQWW\UHPRGHORIWKHSUHVHQWVHFWLRQ
)\
0]
0
α
o
8
0
0
travelled distance
4m
)LJ 6LGHIRUFHDQGDOLJQLQJWRUTXHUHVSRQVHWRVXFFHVVLYHVWHSVLQVLGHVOLSDVFRPSXWHG ZLWKWKHWUDQVLHQWIXOO\QRQOLQHDUPRGHORI6HF(T
6,1*/(&217$&732,1775$16,(1702'(/
7KHODVWHTXDWLRQIRUPVDQH[WHQVLRQRI(T ( WKURXJKWKHLQWURGXFWLRQRI WKHJ\URVFRSLFFRXSOHWKDWIROORZVIURP(T 7RLOOXVWUDWHWKHSHUIRUPDQFHRIWKHPRGHOWKHFDOFXODWHGVLGHIRUFHUHVSRQVH WRVXFFHVVLYHVWHSFKDQJHVLQFDPEHUDQJOHVOLSDQJOHDQGEUDNHVOLSKDVEHHQ SUHVHQWHGLQ)LJ7KHFRQWDFWSDWFKUHOD[DWLRQOHQJWK
F
KDVQRWEHHQXVHG
LQWKLVH[DPSOH,Q&KDSWHUWKHPRGHOLVDSSOLHGLQWKHSUREOHPVRIFRQWUROOHG EUDNLQJRQDQXQHYHQURDGDQGRQVWDUWLQJIURPVWDQGVWLOOZKHUHLQWKHODWWHUFDVH F
LV HPSOR\HG 7KH OHVV GHPDQGLQJ FDVH RI WKH UHVSRQVH RI VLGH IRUFH DQG
DOLJQLQJ WRUTXH WR VXFFHVVLYH VWHSV LQ SXUH VLGH VOLS VKRZQ LQ )LJ ZDV FDOFXODWHG ZLWK WKH WUDQVLHQW PRGHO RI 6HFWLRQ (T 9HU\ VLPLODU UHVXOWVDUHIRXQGZKHQWKHHQKDQFHGPRGHOLVXVHGWKDWLV ZLWKWKHLQFOXVLRQRI WKHFRQWDFWPDVVPF
&KDSWHU $33/,&$7,2162)75$16,(17 7<5(02'(/6
7KLVFKDSWHULVGHYRWHGWRWKHDSSOLFDWLRQRIWKHVLQJOHFRQWDFWSRLQWWUDQVLHQW W\UHPRGHOVDVGHYHORSHGLQWKHSUHFHGLQJFKDSWHU7KHDSSOLFDWLRQVGHPRQVWUDWH WKHHIIHFWRIWKHW\UHUHOD[DWLRQOHQJWKRQYHKLFOHG\QDPLFEHKDYLRXU
9HKLFOH5HVSRQVHWR6WHHU$QJOH9DULDWLRQV
,Q&KDSWHU6HFWLRQWKHG\QDPLFUHVSRQVHRIWKHWZRGHJUHHRIIUHHGRP YHKLFOHPRGHOGHSLFWHGLQ)LJV WRVWHHUDQJOHLQSXWKDVEHHQDQDO\VHG$V DQ H[WHQVLRQ WR WKLV PRGHO ZH ZLOO LQWURGXFH W\UHV ZLWK ODJJHG VLGH IRUFH UHVSRQVH7KHV\VWHPUHPDLQVOLQHDUDQGZHPD\XVH(T 7KHUHOD[DWLRQ OHQJWKZLOOEHGHQRWHGE\ :HKDYHWKHQHZVHWRIHTXDWLRQV P Y 0 , U 0 9
9
'
%
D&
)
0
%
0
%
'
9U
&
)
&
'
'
' &
' & &
Y
9 Y
%
E&
)
%
&
)
DU
EU
9
)LJXUHSUHVHQWVWKHFRPSXWHGIUHTXHQF\UHVSRQVHIXQFWLRQVZKLFKKROGIRU WKHH[WHQGHGV\VWHPIRUGLIIHUHQWYDOXHVRIWKHUHOD[DWLRQOHQJWK,Q)LJWKH FRUUHVSRQGLQJVWHSUHVSRQVHIXQFWLRQVKDYHEHHQGHSLFWHG7KHGLDJUDPVVKRZ WKDW WKH LQIOXHQFH RI WKH W\UH ODJ RQ WKH YHKLFOH PRWLRQ LV UHODWLYHO\ VPDOO $W KLJKHU YDOXHV RI VSHHG WKH HIIHFW GLPLQLVKHV DQG PD\ EHFRPH QHJOLJLEOH +RZHYHUIRUFORVHGORRSYHKLFOHFRQWUROV\VWHPVWKHDGGLWLRQDOSKDVHODJFDXVHG E\WKHUHOD[DWLRQOHQJWKVPD\VLJQLILFDQWO\DIIHFWWKHSHUIRUPDQFH
$33/,&$7,2162)75$16,(177<5(02'(/
10
9 = 20m/s
σ=
U
δ
1m 0 0 0
-90
0
o
1m
0.5
o
-180
0
Q [Hz]
1
15
9 = 20m/s
σ=
D\ J
2
1m
δ
0 0 0
-90 -180
0 o
1m
0.5
o
0
1
1.5
Q
9 = 20m/s
σ=
β δ
2
[Hz]
1m 0.5 0 0 0
1m
o
-90 -180
0
o
0
1
Q
[Hz]
2
)LJ )UHTXHQF\UHVSRQVHRI\DZUDWHODWHUDODFFHOHUDWLRQDQGVLGHVOLSDQJOHWRVWHHUDQJOH RIYHKLFOHPRGHOIHDWXULQJW\UHVZLWKODJJLQJVLGHIRUFHUHVSRQVHFI)LJ
$33/,&$7,2162)75$16,(177<5(02'(/
U
δ
9 = 20m/s
4 2 0
0 -1
σ= 0.5 1m
0
D\ J
1
W
9 = 20m/s
σ=
δ
2
[s]
1m
10
0
0.5
5
0 -2.5
0
1
W
9 = 20m/s
σ=
1
2
[s]
1m
β δ
0
0.5
0 -0.2
0
1
W
[s]
2
)LJ 6WHS UHVSRQVH RI \DZ UDWH ODWHUDO DFFHOHUDWLRQ DQG YHKLFOH VLGH VOLS DQJOH WRVWHHU DQJOHRIYHKLFOHPRGHOIHDWXULQJW\UHVZLWKODJJLQJVLGHIRUFHUHVSRQVH
$33/,&$7,2162)75$16,(177<5(02'(/
$QLQWHUHVWLQJHIIHFWFDQEHVHHQWRRFFXUQHDUWKHVWDUWRIWKHVWHSUHVSRQVH IXQFWLRQV:LWKWKHUHOD[DWLRQOHQJWKLQWURGXFHGWKHUHVSRQVHVDSSHDUWRH[KLELW DPRUHUHOD[HGQDWXUH7KH\DZUDWHUHVSRQVHVKRZVVRPHLQLWLDOODJ
&RUQHULQJRQ8QGXODWHG5RDGV
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
)\ % )\ ' )\VV ' &) GV
G
V
ZKHUH GHQRWHVWKHWUDYHOOHGGLVWDQFHDQGG
V 9 W
[
G 7KHVOLSDQJOHLVDVVXPHG
WREHFRQVWDQWZKLOHWKHYHUWLFDOORDGYDULHVSHULRGLFDOO\
) ] ' )]R % ) ] ' )]R % )Ö ] VLQ V
(VVHQWLDOLVWKDWERWKWKHFRUQHULQJVWLIIQHVVDQGWKHUHOD[DWLRQOHQJWKYDU\ZLWK WKHYHUWLFDOORDG
&) ' &) ) ]
DQG
'
)
]
)LJXUHLOOXVWUDWHVWKHVLWXDWLRQ 7KH ORVV LQ FRUQHULQJ SRZHU FDQ EH GLYLGHG LQWR WKH µVWDWLF¶ ORVV DQG WKH µG\QDPLF¶ ORVV 7KH VWDWLF ORVV DULVHV GXH WR WKH FXUYDWXUH RI WKH FRUQHULQJ VWLIIQHVVYHUVXVZKHHOORDGFKDUDFWHULVWLF$VLPLODUORVVKDVEHHQH[SHULHQFHG WR RFFXU DV D UHVXOW RI ODWHUDO ORDG WUDQVIHU LQ WKH DQDO\VLV RI VWHDG\VWDWH FRUQHULQJ RI DQ DXWRPRELOH FI )LJ )LJXUH H[SODLQV WKH VLWXDWLRQ DW VLQXVRLGDOO\FKDQJLQJZKHHOORDG
$33/,&$7,2162)75$16,(177<5(02'(/
)]
&)α
)]
σ
&)α , σ
V
)LJ&RUQHULQJVWLIIQHVVDQGUHOD[DWLRQOHQJWKYDU\LQJZLWKZKHHOORDG
$V ZLOO EH VKRZQ LQ WKH DQDO\VLV EHORZ WKH G\QDPLF ORVV DSSHDUV WR EH DWWULEXWHGWRWKHUDWHRIFKDQJHRIWKHUHOD[DWLRQOHQJWKZLWKZKHHOORDG7KLVRI FRXUVHFDQRQO\RFFXUEHFDXVHRIWKHIDFWWKDWWKHFRUQHULQJVWLIIQHVVYDULHVZLWK ZKHHOORDGVLQFHWKDWUDWHRIFKDQJHIRUPVWKHLQSXWLQ(T $SSDUHQWO\ZLWK
)
ERWK
)
\
\
DQG
)
EHLQJGHSHQGHQWRQ
]
WKHSKHQRPHQRQLVQRQOLQHDULQ
ZKLOH LQWKHSUHVHQWDQDO\VLVRQO\DIIHFWV
)
]
DQGOLQHDULQ
,I ZH DVVXPH D TXDGUDWLF IXQFWLRQ IRU WKH FXUYDWXUH
E&
)
&
)
YV
)
]
UHODWLRQVKLS ZLWK WKH
R
&)D ' &)DR % D ) ] & E )]
WKHVWDWLFORVVFDQEHVLPSO\IRXQGWREHFRPH
&)
VWDWORVV '
E)Ö ] &) R
)\
)
)
\VWDW
\
α= 1o
VWDWLF ORVV
)\
DYHUDJH
0
V
)
0
V
]
)
]VWDW
)LJ7KHVWDWLFORVVLQDYHUDJHVLGHIRUFHGXHWRWKHFXUYHGIRUFHYVORDGUHODWLRQVKLS
$33/,&$7,2162)75$16,(177<5(02'(/
7KH RFFXUUHQFH RI WKH G\QDPLF ORVV PD\ EH H[SODLQHG E\ DVVXPLQJ D OLQHDU YDULDWLRQRIERWKWKHFRUQHULQJVWLIIQHVVDQGWKHUHOD[DWLRQOHQJWKZLWKORDG
' F) ] &) ' H) ]
(TXDWLRQ EHFRPHVKHUHZLWK
)\ % )\ ' H)] GV
G
F)]
RUVLPSOLILHG
] G\ % \ ' ] GV ] F) \ )
ZLWK
]
\
H
F&RQVLGHUDWUXQFDWHG)RXULHUVHULHVDSSUR[LPDWLRQ
RIWKHSHULRGLFVROXWLRQ
'
\
\
R % \V VLQ
VV % \F FRV
VV % \V VLQ
VV %
DQGLWVGHULYDWLYH
G\ GV
'
VV & \F V VLQ
V V FRV
\
VV % \V V FRV
VV &
ZKLOHWKHLQSXW
]
'
]
R % Ö] VLQ
V'
VV ZLWK
6XEVWLWXWLRQRI LQ DQGVXEVHTXHQWO\PDNLQJWKHFRHIILFLHQWV RIFRUUHVSRQGLQJWHUPVLQWKHOHIWDQGULJKWPHPEHUVHTXDOWRHDFKRWKHU\LHOGV IRUWKHDYHUDJHRXWSXW
\
R'
]
R
&
V Ö]
%
V ]R
7KHDYHUDJHVLGHIRUFH )\DYHH[SUHVVHGLQWHUPVRIWKHRULJLQDOTXDQWLWLHVQRZ UHDGV
\DYH '
)
&
) R
&
VF
%
Ö )
VF
]
)
ZLWK F
'
]R
)RUODUJH)]DPSOLWXGHVDQGVKRUWZDYHOHQJWKV
V
G G)
]
WKHIRUPXODVKRZVWKDW
WKHG\QDPLFORVVDSSURDFKHVRIWKHVLGHIRUFHJHQHUDWHGRQVPRRWKURDGV $VZHZLOOVHHWKLVILQGLQJDJUHHVYHU\ZHOOZLWKVLPXODWLRQDQGWHVWUHVXOWV 7KH DOLJQLQJ WRUTXH YDULDWLRQ PD\ EH FRPSXWHG DV ZHOO ,Q KLV VWXG\ 7DNDKDVKL HPSOR\HGWKHHTXDWLRQEDVHGRQWKHVWULQJFRQFHSWDQGDGGHG WKH J\URVFRSLF FRXSOH :H REWDLQ DFFRUGLQJ WR (TV DQG WKH ODWWHUH[SUHVVLRQFRUUHVSRQGLQJZLWK
$33/,&$7,2162)75$16,(177<5(02'(/
0.8
σ
60
1000
40
[N/o]
[Nm/o]
&)α &0α
σ
[m]
1500
&)α
0.4
500
0
0
)]
4000
0
[N] 8000
&0α
20
0
0
4000
)]
[N] 8000
)LJ /RDGGHSHQGHQF\RIW\UHSDUDPHWHUVDFFRUGLQJWRPHDVXUHPHQWVDQGWKLUGGHJUHH SRO\QRPLDOILWV
G 0
]
G V
%
]'
0
&
&
0
DQG
] J\U '
0
&
J\U
9
G)
\
GV
7KHWRWDOPRPHQWUHVSRQGLQJWRWKHVOLSDQJOHEHFRPHV 0
]'
] % 0]J\U
0
([SHULPHQWV KDYH EHHQ FRQGXFWHG WR DVVHVV WKH UHOHYDQW SDUDPHWHUV RI D 5+ W\UH DW YDULRXV YDOXHV RI WKH YHUWLFDO ORDG 7KH GLDJUDPV RI )LJ SUHVHQW WKH FKDUDFWHULVWLFV IRU WKH UHOD[DWLRQ OHQJWK WKH FRUQHULQJ VWLIIQHVV DQG WKH DOLJQLQJ VWLIIQHVV 7KH UHOD[DWLRQ OHQJWK ZDV GHWHUPLQHG E\ FXUYHILWWLQJRIWKHGDWDREWDLQHGIURPDVZHSWVLQHVWHHULQJRVFLOODWLRQWHVWDWD VSHHGRINPKLQWKHIUHTXHQF\UDQJHRIWR+]7KHJ\URVFRSLFFRHIILFLHQW
&J\UZDVDVVHVVHGLQDVLPLODUWHVW EXW FRQGXFWHGDWDVSHHGRINPK7KH UHVXOWLQJYDOXHWXUQVRXWWREH &J\U GLPHQVLRQDOFRHIILFLHQWFJ\U
î V WKDWFRUUHVSRQGVWRWKHQRQ
. FI(T
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
$33/,&$7,2162)75$16,(177<5(02'(/
experiments
linear model
1500
)]R= 3000N )$] = 0.9)]R 9= 30km/h
)\
[N]
α= 1
o
0
0.5 Hz 2.0 Hz 4.0 Hz 6.0 Hz 8.0 Hz
60 0]
[Nm] 0 0
0.5
V/λ
1
0
0.5
V/λ
1
)LJ6LGHIRUFHDQGDOLJQLQJWRUTXHUHVSRQVHWRZKHHOORDGYDULDWLRQDWDVPDOOVOLSDQJOH
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
$33/,&$7,2162)75$16,(177<5(02'(/
)\ )
α = 1o
)]R =3000N 9= 30km/h
0.6
G\QORVV
)$] )]R
\VWDW
0.90 0.73 0.36
0.95
0.4
G\QDPLF ORVV
0.90
linear model
0.2
VWDWLF ORVV 0.36 0.73 0.90
0.73
0.36
0 0
10 2.0
1.0
λ
Q [m]
[Hz]
20
0.5
)LJ'\QDPLFVLGHIRUFHORVVDWDVPDOOVOLSDQJOHIRUGLIIHUHQWORDGDPSOLWXGHUDWLRV
ZKHUHWKHW\UHDOPRVWORRVHVFRQWDFWDSSHDUVWRDSSURDFKWKHYDOXHRIFD ZKHQWKHIUHTXHQF\LQFUHDVHVWRODUJHYDOXHV7KLVLQGHHGFRUUHVSRQGVWRWKH DQDO\WLFDOUHVXOWHVWDEOLVKHGE\(T 2QWKHRWKHUKDQGWKHPRUHDGYDQFHG PRGHORI6HFWLRQSUHGLFWVDSRVVLEO\HYHQODUJHUUHGXFWLRQDWWKHYHUJHRI SHULRGLFOLIWRIIFI)LJZKHUHWKHZDYHOHQJWKFRXOGEHGLPLQLVKHGWROHYHOV DVVPDOODVWKHVWDWLFFRQWDFWOHQJWKDR ,WPD\EHREYLRXVWKDWDQDQDORJRXVPRGHO(T ZLWK
N
X
FDQEH
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
( GY GV
(
% ' Y
ZLWK
('
R
&
)
)
\
$33/,&$7,2162)75$16,(177<5(02'(/
)]
ZKLFKTXDQWLW\LVDIXQFWLRQRIERWKWKHZKHHOORDG RUGHIRUPDWLRQJUDGLHQW
N WHQGV WR ]HUR DQG
ZKHQ
DQGWKHWUDQVLHQWVOLSDQJOH
N,WUHGXFHVWRWKHUHOD[DWLRQOHQJWK
)\
N WR
&)
R
DW
N
2EYLRXVO\ RIIVHWV RI WKH VLGH IRUFH
YHUVXVVOLSDQJOHFKDUDFWHULVWLFQHDUWKHRULJLQKDYHEHHQGLVUHJDUGHGKHUH7KH WUDQVLHQWVOLSDQJOHIROORZVIURP
'
Y
(
7KHVLGHIRUFHDQGPRPHQWDUHREWDLQHGE\XVLQJWKHVWHDG\VWDWHFKDUDFWHULVWLFV )
\
0
' '
)
]
)
\VV
0
]VV
]
)
]
:LWKWKHJ\URVFRSLFFRXSOHFI(TV DQG
0
]J\U
'
F
J\U
P 9
W
GY
GV
WKHWRWDOPRPHQWEHFRPHV 0
]
'
0
]
%
0
]J\U
7KHVWHDG\VWDWHFKDUDFWHULVWLFVDVPHDVXUHGRQWKHURDGDWDQXPEHURIZKHHO ORDGVKDYHEHHQSUHVHQWHGLQ)LJ $VKDVEHHQQRWHGLQ6HFWKHDVVHVVPHQWRI
(
UHTXLUHVVRPHDWWHQWLRQ
,QWKHLQYHVWLJDWLRQGRQHE\7DNDKDVKL DQG+LJXFKL WKHVROXWLRQ ZDVREWDLQHGWKURXJKLWHUDWLRQV$OWHUQDWLYHO\ZHPD\XVHLQIRUPDWLRQIURPWKH SUHYLRXV WLPH VWHS WR DVVHVV
(
:KHQ WKH ORDG LV UDSLGO\ ORZHUHG YLROHQW
YLEUDWLRQVPD\RFFDVLRQDOO\VKRZXSLQWKHVROXWLRQ :LWKWKLVQRQOLQHDUPRGHOVLPXODWLRQVKDYHEHHQGRQHDQGH[SHULPHQWVKDYH EHHQ FDUULHG RXW IRU D VOLS DQJOH HTXDO WR VL[ GHJUHHV 3ORWV RI WKH UHVXOWLQJ
6000 )\
)]
120
)]
0]
2000
)]
[Nm]
[N]
2000 N 3000 N 4000 N 5000 N 6000 N
0 1000
0
0
5
α 10
[ o ] 15
-80
0
5
α 10
[ o ] 15
)LJ6WHDG\VWDWHVLGHIRUFHDQGDOLJQLQJWRUTXHFKDUDFWHULVWLFVDVPHDVXUHGRQWKHURDG
$33/,&$7,2162)75$16,(177<5(02'(/
experiments 6000
non-linear model
µ)]
)\
)]R = 3000N )$] = 0.83)]R 9= 30km/h
µ)]
[N]
α= 6o
0 -2000
0.5 Hz 2.0 Hz 4.0 Hz 6.0 Hz 8.0 Hz
150 0]
[Nm] 0
-100
0
V/λ
0.5
1
0
V/λ
0.5
1
)LJ 6LGHIRUFHDQGDOLJQLQJWRUTXHUHVSRQVHWRZKHHOORDGYDULDWLRQDWDODUJHVOLS DQJOH
UHVSRQVHVDUHVKRZQLQ)LJ 7KHPRGHODSSHDUVWREHVXFFHVVIXOLQSUHGLFWLQJ WKH VLGH IRUFH YDULDWLRQ FDXVHG E\ WKH ZKHHO ORDG H[FLWDWLRQ DW GLIIHUHQW IUHTXHQFLHV7KHORDGDPSOLWXGHUDWLRRIJDYHULVHWRIXOOVOLGLQJLQFHUWDLQ LQWHUYDOVRIWKHSHULRGLFPRWLRQ7KLVFDQEHVHHQWRRFFXUZKHQWKHIRUFHVLJQDO WRXFKHVWKHFXUYHUHSUHVHQWLQJWKHPD[LPDOO\DFKLHYDEOHVLGHIRUFH)],QWKLV UHVSHFWWKHPRGHOFHUWDLQO\SHUIRUPVZHOOZKLFKFDQQRWEHVDLGRIWKHDOWHUQDWLYH
)\ )
α = 6o
)]R = 3000N )$] = 0.83)]R 9= 30km/h
0.6 G\QORVV \VWDW
experiments
0.4
non-linear model
0.2
0 0
10 2.0
1.0
λ
Q [m]
[Hz]
20
0.5
)LJ7KHG\QDPLFORVVDWDODUJHFRQVWDQWVOLSDQJOHDQGDODUJHORDGDPSOLWXGHUDWLR
$33/,&$7,2162)75$16,(177<5(02'(/
PRGHO JRYHUQHG E\ (T ZLWK )\VV FDOFXODWHG ZLWK (T ZKHUH
N LV
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
/RQJLWXGLQDO)RUFH5HVSRQVHWR7\UH1RQ 8QLIRUPLW\$[OH0RWLRQVDQG5RDG8QHYHQQHVV
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
)] ' &)]
,Q&KDSWHU W\UHLQHUWLDDQGRWKHULQIOXHQFHVZLOOEHLQWURGXFHGWRLPSURYHWKH UHODWLRQVKLSDWKLJKHUIUHTXHQFLHVDQGVKRUWHUZDYHOHQJWKVRIURDGXQHYHQQHVVHV $ FRXSOLQJ EHWZHHQ YHUWLFDO GHIOHFWLRQ YDULDWLRQ DQG ORQJLWXGLQDO IRUFH DSSHDUV WR H[LVW WKDW LV GXH WR WKH UHVXOWLQJ YDULDWLRQ RI WKH HIIHFWLYH UROOLQJ
$33/,&$7,2162)75$16,(177<5(02'(/
Ω
UI UH U ρ
9 effective rolling circle
slip point
6 ) ]
9V[
)[
0\
9
)LJ7KHUROOLQJZKHHOVXEMHFWHGWRORQJLWXGLQDOVOLS VOLSVSHHG
V[
UDGLXVRIWKHW\UH )RUWKHQRQVWHDG\VWDWHDQDO\VLVZHZLOOHPSOR\WKHOLQHDUHTXDWLRQ WKDW PD\WDNHWKHIRUPZLWK9
X% GW
G
9[ DVVXPHGSRVLWLYH
9X ' 9 ' & 9V[
7KHLQSXWLVWKHORQJLWXGLQDOVOLSYHORFLW\WKDWLVGHILQHGDV
9V[ ' 9 & UH
2EYLRXVO\WKHHIIHFWLYHUROOLQJUDGLXVUHSOD\VKHUHDFUXFLDOUROH
(IIHFWLYH5ROOLQJ5DGLXV9DULDWLRQVDW)UHH5ROOLQJ
)RUDZKHHOZLWKW\UHWKDWLVXQLIRUPDQGUROOVIUHHO\DWFRQVWDQWVSHHGRYHUDQ HYHQ KRUL]RQWDO URDG VXUIDFH WKH WUDFWLYH IRUFH UHTXLUHG LV GXH WR UROOLQJ UHVLVWDQFHDORQH8QGHUWKHVHFRQGLWLRQVWKHHIIHFWLYHUROOLQJUDGLXVUHLVGHILQHG WRUHODWHVSHHGRIUROOLQJ
9 ' UH
ZLWKIRUZDUGVSHHG9
7KHFHQWUHRIURWDWLRQRIWKHZKHHOERG\6OLHVDGLVWDQFHUHEHORZWKHZKHHOVSLQ D[LV$VKDVEHHQGLVFXVVHGLQ&KDSWHUE\GHILQLWLRQWKLVSRLQWZKLFKFDQEH LPDJLQHGWREHDWWDFKHGWRWKHZKHHOULPLVVWDWLRQDU\DWWKHLQVWDQWFRQVLGHUHG LI WKH ZKHHOUROOVIUHHO\ FI)LJZLWKVOLSVSHHG9V[
,QJHQHUDO WKH
HIIHFWLYHUROOLQJUDGLXVFKDQJHVZLWKW\UHGHIOHFWLRQ :HPD\ZULWH
UH ' UI & I
$33/,&$7,2162)75$16,(177<5(02'(/
ZLWK UI GHQRWLQJ WKH IUHH XQGHIRUPHG UDGLXV WKDW PD\ YDU\ DORQJ WKH W\UH FLUFXPIHUHQFHGXHWRW\UHQRQXQLIRUPLW\:HKDYHIRUWKHORDGHGUDGLXV
U ' UI &
)LJXUHVKRZVYDULDWLRQVRIWKHHIIHFWLYHUROOLQJUDGLXVUHDQGWKHORDGHG UDGLXVUDVIXQFWLRQVRIZKHHOORDG)]IRUWZRGLIIHUHQWW\UHVDWGLIIHUHQWVWDJHV RIZHDUDQGDWGLIIHUHQWVSHHGVDVPHDVXUHGRQDGUXPVXUIDFHRIPGLDPHWHU 7KHIURPWKLVGLDJUDPGHGXFHGUHGXFWLRQRIWKHHIIHFWLYHUROOLQJUDGLXVI ZLWK UHVSHFW WR LWV WKHRUHWLFDO RULJLQDO YDOXH UI DW ]HUR ZKHHO ORDG
KDV EHHQ
SUHVHQWHGLQ)LJ7KHLQIOXHQFHRIDFKDQJHLQWUHDGGHSWKKDVEHHQVKRZQ
g
7KHVORSH DQGWKHFRHIILFLHQW LQGLFDWHWKHLQIOXHQFHRIFKDQJHVLQGHIOHFWLRQ
315
radial steel belted 310
tread depth
[mm]
UH
305
9 [km/h]
7 mm
155 20
3 mm
U
1 mm
S = 1.8 bar L
320
UH
[mm]
126 68
310
U
bias ply tread depth: 7 mm S = 1.5 bar
150
L
320 [mm]
UH
310
U
20
bias ply tread depth: 0 mm S = 1.5 bar
163 300
L
20 290 0
2000
4000
[N]
)]
)LJ 7\UHHIIHFWLYHUROOLQJUDGLXVDQGORDGHGUDGLXVDVDIXQFWLRQRIZKHHOORDGPHDVXUHG RQPGUXPDWGLIIHUHQWVSHHGVDQGIRUGLIIHUHQWWUHDGGHSWKVDQGW\UHGHVLJQ1RWH WKHGLIIHUHQWVFDOHVIRUWKHUDGLDODQGELDVSO\W\UHUDGLL
$33/,&$7,2162)75$16,(177<5(02'(/
UI UH
ρ
15
radial-ply GW=7mm
6
below road level
studs
[mm] 10
η
1
5
εG G=7mm G= 0 W
G=1mm
W
W
ribs bias-ply
W
0 0
)LJ
5
10
15
20
U ZLWKUHVSHFWWRW\UHIUHHUDGLXVU GXHWRW\UH G 6LWXDWLRQ DW QRPLQDO ORDG )
5HGXFWLRQRIHIIHFWLYHUROOLQJUDGLXV UDGLDO GHIOHFWLRQ
ρ
25 [mm] 30
H
I
DQG WUHDG GHSWK FKDQJH
W
]R
1KDVEHHQLQGLFDWHGE\FLUFOHV
DQGWUHDGGHSWKQHDUWKHQRPLQDOORDG )LJXUHWULHVWRH[SODLQWKHREVHUYDWLRQUHJDUGLQJWKHORFDWLRQRIWKHVOLS
6
6
SRLQW 3RVVLEOHORFDWLRQVKDYHEHHQLQGLFDWHG$W]HURZKHHOORDG OLHVDWURDG OHYHORUDOLWWOHORZHU GXHWRDLUDQGEHDULQJUHVLVWDQFHZKLFKVORZVWKHZKHHO GRZQDWWKHH[SHQFHRIVRPHVOLS $VPDOOZKHHOORDGFDXVHVDQXPEHURIWKH RULJLQDOO\UDGLDOO\GLUHFWHGWUHDGHOHPHQWVWRDVVXPHDYHUWLFDORULHQWDWLRQ7KH URWDWLRQ RI WKHVH WUHDG HOHPHQWV LV ORVW ZKLFK FDXVHV WKH HIIHFWLYH UDGLXV WR EHFRPH VPDOOHU 3RLQW
6
PD\ WKHQ WXUQ RXW WR OLH DERYH URDG OHYHO $W KLJKHU
6
ORDGVZHJHWWKHPRUHFRPPRQO\NQRZQVLWXDWLRQRI ORFDWHGEHORZURDGOHYHO
U
,WKDVEHHQDVVXPHGKHUHWKDWWKHEHOWZLWKUDGLXV
LVLQH[WHQVLEOH7KHQZKHQ F
WKHORZHUWUHDGHOHPHQWVKDYHDYHUWLFDORULHQWDWLRQDQGWUDQVODWHEDFNZDUGVZLWK UHVSHFW WR WKH ZKHHO FHQWUH WKH ZKHHO ZLOO LQ RQH UHYROXWLRQ FRYHU D GLVWDQFH HTXDOWRWKHFLUFXPIHUHQFHRIWKHEHOW7KLVPHDQVWKDWWKHHIIHFWLYHUROOLQJUDGLXV HTXDOVWKHUDGLXVRIWKHEHOW:KHQWKHW\UHSRVVHVVHVDULEEHGWUHDGSDWWHUQLW
GW UI
UF
U
H
U
6 at road level
U
I
H
U
F
6 above road level 6
6 below
)LJ/RFDWLRQRIWKHFHQWUHRIURWDWLRQWKHVOLSSRLQW DWWKUHHGLIIHUHQWUDGLDOGHIOHFWLRQV
$33/,&$7,2162)75$16,(177<5(02'(/
LVH[SHFWHGWKDWGXHWRWKHJUHDWHUFRKHUHQFHEHWZHHQULEHOHPHQWVLQFRPSDULVRQ WRWKHLQGHSHQGHQWVWXGVRIWKHUDGLDOSO\W\UHVRPHURWDWLRQVWLOORFFXUVLQWKH
6
ULEVLQWKHFRQWDFW]RQHZKLFKLQFUHDVHVWKHHIIHFWLYHUDGLXVDQGSXVKHVSRLQW
HDUOLHUEHORZWKHURDGVXUIDFH7KLVWKHRU\VHHPVWREHVXSSRUWHGE\WKHFXUYHV
g
RIWKHQHZELDVSO\W\UH7KHGHJUHHRIHIIHFWLYHLQFRKHUHQFH LVHVWLPDWHGIRU WKHUDGLDOSO\W\UHWREHHTXDOWRFD DQGIRUWKHELDVSO\W\UHHTXDOWRDW WKHUDWHGORDG%HFDXVHZHKDYHSORWWHGLQ)LJWKHUHGXFWLRQRIWKHHIIHFWLYH
U U
U
UDGLXVZLWKUHVSHFWWRWKHIUHHXQORDGHGUDGLXV I HWKHGHFUHDVHRI HGXHWKH ZHDULQJRIIRIWKHWUHDGUXEEHULVQRWVKRZQGLUHFWO\2QO\WKHLQGLUHFWHIIHFWRI
g
WUHDGZHDULVVKRZQWKURXJKWKHWUHDGLQFRKHUHQFHSDUDPHWHU ,I GXHWRWKH YHUWLFDO W\UH GHIOHFWLRQ WKH WUHDG EDQG LV FRPSUHVVHG DQG KDV GHFUHDVHG LQ FLUFXPIHUHQFH WKH HIIHFWLYH UDGLXV GLPLQLVKHV 7KLV SURSHUW\ ZKLFK LV PRUH DSSDUHQWWRKDSSHQZLWKWKHELDVSO\W\UHLVSDUWO\UHVSRQVLEOHIRUWKHVORSHRI WKHFXUYHV VKRZQDWODUJHUYHUWLFDOGHIOHFWLRQV7KLVVORSHGHQRWHGZLWK
LV
HVWLPDWHGWREHHTXDOWRDERXW IRU WKH QHZUDGLDOSO\W\UHZLWKDQDOPRVW LQH[WHQVLEOHEHOWDQGIRUWKHQHZELDVSO\W\UHDWWKHLUQRPLQDOORDG 7KH DQDO\VLV ZLOO EH NHSW OLQHDU DQG ZH DVVXPH VPDOO GHYLDWLRQV IURP WKH XQGLVWXUEHGFRQGLWLRQ,QWKHQHLJKERXUKRRGRIWKLVDYHUDJHVWDWHZHZULWHIRU WKHHIIHFWLYHUDGLXVWKHORDGHGUDGLXVDQGWKHUDGLDOGHIOHFWLRQ U
H'
U
HR % U H a
U
'
U
R% U a
'
R%
a
6PDOO GHYLDWLRQV IURP WKH XQGLVWXUEHG FRQGLWLRQ DUH LQGLFDWHG E\ D WLOGH 9DULDWLRQRIWKHIUHHUDGLXVPD\RFFXUDORQJWKHFLUFXPIHUHQFHRIWKHW\UH7KLV NLQGRIQRQXQLIRUPLW\ RXWRIURXQGQHVV LVFRPSRVHGRIWZRFRQWULEXWLRQV2QH
U
LVGXHWRYDULDWLRQVRIWKHFDUFDVVUDGLXV WUHDGWKLFNQHVV
I ' UF
a
U
a
%
G
F
DQGWKHRWKHUGXHWRYDULDWLRQVRIWKH
:HKDYH W
a
G
W
7KHYDOXHRIRXWRIURXQGQHVVLVFRQVLGHUHGWREHSUHVHQWDWWKHPRPHQWFRQ VLGHUHG DORQJ WKH UDGLXV SRLQWLQJ WR WKH FRQWDFW FHQWUH 7KH IROORZLQJ OLQHDU UHODWLRQ EHWZHHQ WKH HIIHFWLYH UDGLXV YDULDWLRQ DQG WKH LPSRVHG GLVWXUEDQFH TXDQWLWLHVLVIRXQGWRH[LVW
H ' UI & g G W &
a
U
a
a
a
' F % & g W & a
U
a
G
a
7KH REVHUYDWLRQ WKDW D UHGXFWLRQ LQ WKLFNQHVV RI WKH WUHDG UXEEHU RI D
)
UDGLDOSO\W\UHSURGXFHVDGHFUHDVHLQHIIHFWLYHUROOLQJUDGLXV DW
]R
DPRXQWLQJ
WRRQO\DVPDOOIUDFWLRQRIWKHUHGXFWLRQLQWUHDGGHSWKLVUHIOHFWHGE\WKHIDFWRU
g
LQ WKH ULJKWKDQG H[SUHVVLRQ RI 7KH FRXSOLQJ FRHIILFLHQW
LV
UHODWLYHO\VPDOOIRUEHOWHGW\UHV,QFRQWUDVWWRELDVSO\W\UHVWKHHIIHFWLYHUROOLQJ UDGLXVRIDEHOWHGW\UHGRHVQRWFKDQJHYHU\PXFKZLWKGHIOHFWLRQRQFHWKHLQLWLDO WUHDGWKLFNQHVVHIIHFW DWVPDOOFRQWDFWOHQJWKVZLWKWUHDGHOHPHQWVVWLOORULHQWHG
$33/,&$7,2162)75$16,(177<5(02'(/
DOPRVWUDGLDOO\ KDVEHHQRYHUFRPH
&RPSXWDWLRQRIWKH+RUL]RQWDO/RQJLWXGLQDO)RUFH5HVSRQVH
7KHKRUL]RQWDOIRUFHUHVSRQVHRQXQGXODWHGURDGVLVFRPSRVHGRIWKHIROORZLQJ WKUHHFRQWULEXWLRQV WKHKRUL]RQWDOFRPSRQHQWRIWKHQRUPDOORDGWKHYDULDWLRQ RIWKHUROOLQJUHVLVWDQFHDQGWKHORQJLWXGLQDOIRUFHUHVSRQVHWRWKHYDULDWLRQRIWKH
;
ZKHHOORQJLWXGLQDOVOLS7KHKRUL]RQWDOLQSODQHIRUFHGHVLJQDWHGZLWK
EHFRPHV
ZLWKOLQHDULVHGYDULDWLRQVFI)LJ
;
'
'& '& ZLWK
R% ;
a
;
)
UR % )[ % )]R
)
UR % )U % )
a
a
a
GZ
%
)
GV )
]R
GZ GV
GHQRWLQJ WKH DYHUDJH UROOLQJ UHVLVWDQFH IRUFH )RU WKLV RFFDVLRQ LW LV UR
DVVXPHGWKDWWKHYDULDWLRQLQUROOLQJUHVLVWDQFHIRUFHLVGLUHFWO\WUDQVPLWWHGWRWKH WUHDGWKURXJKFKDQJHVLQYHUWLFDOORDG:HZULWH
U'
a
)
a
U ]
$ )
ZKHUHWKHYDULDWLRQRIWKHYHUWLFDOORDGUHVXOWVIURPYDULDWLRQVLQGHIOHFWLRQDQG SRVVLEO\IURPFKDQJHVLQUDGLDOVWLIIQHVVDORQJWKHW\UHFLUFXPIHUHQFH
]'
a
)
&
)]R
a
%
a
R &)]
RUH[SUHVVHGLQWHUPVRIWKHYDULDWLRQLQUDGLDOVWDWLFGHIOHFWLRQ
V [
9
)
]
Ω
]
)
[
& ;
Z
dZ dV
0 \
)LJ,QSODQHH[FLWDWLRQE\URDGXQHYHQQHVVDQGE\YHUWLFDODQGIRUHDQGDIWD[OHPRWLRQV
$33/,&$7,2162)75$16,(177<5(02'(/
a
V'& R
a
& &
)]
)]R
ZHPD\ZULWH
]'
a
)
&
)]R
a
a V
&
7KHYDULDWLRQRIWKHWDQJHQWLDOVOLSIRUFHEHFRPHVZLWKWKHWUDQVLHQWORQJLWXGLQDO
¶
VOLS
'
a
)
&
'
)
X
&
)
X
ZKHUHWKHW\UHGHIOHFWLRQ IROORZVIURP(T 7KHORQJLWXGLQDOVOLSYHORFLW\ DSSHDULQJ DV LQSXW LQ LV JRYHUQHG E\ (T :LWK WKH YDULDWLRQV LQ IRUZDUG YHORFLW\ VSHHG RI UHYROXWLRQ DQG HIIHFWLYH UROOLQJ UDGLXV ZH ILQG LQ OLQHDULVHGIRUP
9
V[ '
&
a
9
R UH & UHR a
a
7KH YDULDWLRQ RI WKH HIIHFWLYH UROOLQJ UDGLXV IROORZV IURP ZKHUH WKH GHIOHFWLRQLVDUHVXOWRIWKHFRPELQDWLRQRIZKHHOYHUWLFDOGLVSODFHPHQWW\UHRXW RIURXQGQHVVDQGURDGKHLJKWFKDQJHV a
' % F% W& a
a
]
U
G
Z
7KHZKHHODQJXODUYHORFLW\UHVXOWVIURPWKHG\QDPLFHTXDWLRQ ,
Z
J
'& ,
LQZKLFK
Z
U)
[ & 0\
GHQRWHVWKHZKHHOSRODUPRPHQWRILQHUWLD LQFOXGLQJDODUJHSDUWRI
WKHW\UHWKDWYLEUDWHVDORQJZLWKWKHZKHHOULPLIWKHIUHTXHQF\OLHVZHOOEHORZ WKHLQSODQHILUVWQDWXUDOIUHTXHQF\FI&KDS
0
7KHPRPHQW
\
&
DFWVDERXWWKHWUDQVYHUVHD[LVWKURXJKWKHFRQWDFWFHQWUH
)LJDQGLVFRPSRVHGRIWKHSDUWGXHWRUROOLQJUHVLVWDQFHDQGDSDUWGXHWR H[FHQWULFLW\RIWKHWUHDGEDQGZKLFKFRQVWLWXWHVWKHILUVWKDUPRQLFRIWKHW\UHRXW RIURXQGQHVV7KLVODWWHUSDUWUXQV GHJUHHVDKHDGLQSKDVHZLWKUHVSHFWWRWKH ORDGHG UDGLXV YDULDWLRQ WKDW LV VHQVHG DW WKH FRQWDFW FHQWUH :LWK VXEVFULSW UHIHUULQJWRWKHILUVWKDUPRQLFZHKDYH 0
\'
0
U % )] U W% a
0
7KHUROOLQJUHVLVWDQFHPRPHQW WKH UROOLQJ UHVLVWDQFH IRUFH
)
U
LVKHUHVXSSRVHGWREHGLUHFWO\FRQQHFWHGZLWK
ZKLFK LV D VLPSOLI\LQJ DVVXPSWLRQ WKDW KDV D U
QHJOLJLEOH HIIHFW RQ WKH UHVXOWLQJ UHVSRQVHV $V D UHVXOW WKH UROOLQJ UHVLVWDQFH FRPSRQHQWVFDQFHORXWLQWKHULJKWKDQGPHPEHURI(T 7KLVHTXDWLRQQRZ UHGXFHVWR
$33/,&$7,2162)75$16,(177<5(02'(/
,
J
Z
'&
U
&
R)
)
a
]R U W%
&ROOHFWLRQRIWKHUHOHYDQWHTXDWLRQV DQGHOLPLQDWLRQ RIDOOYDULDEOHVH[FHSWWKHLQSXWTXDQWLWLHV
a
9 ' 0[a ] Z UaF GaW
a
a
V
Ua '
ILUVW KDUPRQLF RI
DQG WKH RXWSXW ; DQG IRU QRZ DOVR
UH a
DQG
a
UaI ' UaF % GaW
\LHOGV WKH IROORZLQJ /DSODFH
V
WUDQVIRUPHGUHSUHVHQWDWLRQRIWKHKRUL]RQWDOIRUFHYDULDWLRQZLWK GHVLJQDWLQJ WKH/DSODFHYDULDEOHRU
a
;
'
, &
)
L
R V @ UH & ,Z V a
Z
,
&
U )] @
$ &
a
&
a
ZV
a
%
@[
& ,
Z
)
]R UR H[S ò
9 V
R
%
&
V % ) ]R V @ Z 9R
)
U
V
a
R @ U
%
R
ZKHUHZLWK DQG ZHKDYHWKHYDULDWLRQVRIHIIHFWLYHUDGLXVDQGW\UH GHIOHFWLRQ
H'
a
Z
U
& % & & g W % & a
]
G
a
UF
DQG a
' % F% W& ]
a
U
a
G
Z
)URPWKLVH[SUHVVLRQ RIWKHFRPELQHG UHVSRQVHWKHLQGLYLGXDOWUDQVIHU IXQFWLRQVRUWKHIUHTXHQF\UHVSRQVHIXQFWLRQVWRWKHYDULRXVLQSXWTXDQWLWLHVPD\ EHZULWWHQH[SOLFLWO\)RUWKHODWWHUIXQFWLRQVWRREWDLQZHVLPSO\KDYHWRWDNH WKHFRHIILFLHQWVRIRQHRIWKHLQSXWYDULDEOHV LQ DIWHU DQG
V
L
KDYHEHHQLQVHUWHGLQ DQGUHSODFH E\
L
RU
R
GHSHQGLQJRQWKH
W\SHRILQSXWYDULDEOH )LJXUHSUHVHQWVERWKWKHRUHWLFDODQGH[SHULPHQWDOGDWDWKHODWWHUEHLQJ REWDLQHG IURP WHVWV SHUIRUPHG RQ D P GUXP WHVW VWDQG SURYLGHG ZLWK D VSHFLDOO\ GHVLJQHG ULJ 3DFHMND 9DQ GHU %HUJ DQG -LOOHVPD 7KH SDUDPHWHUVRIWKHW\UHVWHVWHGKDYHEHHQOLVWHGLQ7DEOH7KHUHVSRQVHWR
]
KDV EHHQ IRXQG E\ PRYLQJ WKH D[OH XS DQG GRZQ E\ PHDQV RI D K\GUDXOLF DFWXDWRU7KHYDULDWLRQLQFDUFDVVUDGLXV
U
F
KDVEHHQDFKLHYHGE\PRXQWLQJ D
UHDVRQDEO\XQLIRUP ZKHHODQGW\UHZLWKPPHFFHQWULFLW\RQWKHVKDIWDQG VXEVHTXHQWO\EDODQFLQJWKHURWDWLQJV\VWHP,QWKLVWHVWWKHD[OHZDVKHOGIL[HG 7KH VSHFLDO DUUDQJHPHQW UHVXOWHG LQ D ODUJHU SRODU PRPHQW RI LQHUWLD
G
NJP 7KHUHVSRQVHWR
,
Z
YDULDWLRQVFRXOGEHREWDLQHGE\SDUWO\EXIILQJRIIWKH W
W\UHWUHDG2QRQHVLGHPPOHVVWUHDGUXEEHUUHPDLQHGZLWKUHVSHFWWRWKHRWKHU
&
VLGHRIWKHW\UH$VDVLGHHIIHFWWKHDYHUDJHVOLSVWLIIQHVV
)
LQFUHDVHG7KHQHZ
YDOXHZDVHVWLPDWHG$OOW\UHSDUDPHWHUVKDYHEHHQGHWHUPLQHGWKURXJKVSHFLDO
$33/,&$7,2162)75$16,(177<5(02'(/
H[SHULPHQW
WKHRU\ radial ply bias ply 37
)~[ ]
29
50
H[SHULPHQW
mm
27 21 12
21
o
21 27
φ o
-180
0
ω/ 2π
20
ω/ 2π
0
180
φ 0
20 Hz
U~F ~ GW
U~F
o
12 21 27 37
o
-120
0 50 0
12 21 29 37
29 37
12
o
-150
ΩRUHR
U~F ~ GW
~
12 37
0 -120
U
UI 37 N/ mm 29 21 100
21 12 27 21
N
~ ~ 8 8 2.5mm, 9 I = UF+ G W UF= GW =
)~[
9 [m/s]
WKHRU\
~
~
GW
~
GW
U~F
U~F
Ω R/ 2π
0
U~F
20
0
Ω R/ 2π
20 Hz
)LJ ([SHULPHQWDODQGWKHRUHWLFDOIUHTXHQF\UHVSRQVHSORWVRIWKHORQJLWXGLQDOIRUFH
)[
VW
WRYHUWLFDOD[OHRVFLOODWLRQV]DQGWRRXWRIURXQGQHVV KDUPRQLFRIYDULDWLRQVLQ FDUFDVVUDGLXVUFRULQWUHDGWKLFNQHVVGW $YHUDJHYHUWLFDOORDG)]R
1
WHVWV6RPHWLPHVVOLJKWFKDQJHVRIWKHVHYDOXHVFRXOGUHVXOWLQDEHWWHUPDWFKRI WKHUHVSRQVHGDWD ,Q JHQHUDO D JRRG FRUUHVSRQGHQFH ZLWK H[SHULPHQWDO UHVXOWV FRXOG EH HVWDEOLVKHG$OVRWKHSKDVHUHODWLRQVKLSZKLFKLVYHU\VHQVLWLYHWRWKHVWUXFWXUH RIWKHPRGHODQGFKDQJHVLQSDUDPHWHUVWXUQV RXWWREHKDYHVDWLVIDFWRULO\$ QHJDWLYHSKDVHDQJOH
kLQGLFDWHVWKDWWKHUHVSRQVHODJVEHKLQGWKHLQSXW
7DEOH7\UHLQSODQHSDUDPHWHUYDOXHVIRUWKHUDGLDODQGELDVSO\W\UHV
a
5DGLDOVWHHOEHOWHG SL ,Z )]R
a
)]R
] RU U NJP UIR
I
&)[&) &)[ 1)UR
%LDVSO\ SL ,Z
EDUWUHDGGHSWK DW G
PUR
1P&) 1&)]
] RU U NJP UIR
I
&)[&) &)[ 1)UR
PP
. P
g
W
PP
PUR
1P&) 1&)]
1 1DW GÖ
1P$U
EDUWUHDGGHSWK
a
W
. P
1
1P$U
PP
$33/,&$7,2162)75$16,(177<5(02'(/
7R H[SODLQ WKHVH ILQGLQJV ZKLFK WXUQ RXW WR EH TXLWH GLIIHUHQW IRU WKH WZR W\SHVRIW\UHZHVKDOOQRZDQDO\VHWKHUHVSRQVHVWRYHUWLFDOD[OHPRWLRQV]DQG
a
RXWRIURXQGQHVV U
F
a
U
F ILUVWKDUPRQLF LQJUHDWHUGHWDLO
)UHTXHQF\5HVSRQVHWR9HUWLFDO$[OH0RWLRQV
$IWHUUHSODFLQJLQ(T WKH/DSODFHYDULDEOHVE\L
ZLWK
GHQRWLQJWKH
IUHTXHQF\RIWKHH[FLWDWLRQZHILQGIRUWKHUHVSRQVHWRYHUWLFDOD[OHPRWLRQVWKH IROORZLQJIUHTXHQF\UHVSRQVHIXQFWLRQZLWK;
)[ & ' & $U&)] & ) ] UR a
)[
L & % L
ZLWKWKHQRQGLPHQVLRQDOIUHTXHQF\
'
R
DQGWKHVSHHGGHSHQGHQWGDPSLQJUDWLR
'
,Z9
R'
&) U R
9
R
ZKHUHWKHQDWXUDOIUHTXHQF\RIWKHW\UHZKHHOURWDWLRQZLWKUHVSHFWWRWKHIRRW SULQWKDVEHHQLQWURGXFHG
&)[ UR ' ,Z
R'
&) U R ,Z
$QLQWHUHVWLQJREVHUYDWLRQLVWKDWDQLQFUHDVHLQVOLSVWLIIQHVVRUUHOD[DWLRQOHQJWK GHFUHDVHV WKH QRQGLPHQVLRQDO GDPSLQJ FRHIILFLHQW
$ KLJKHU VSHHG 9
REYLRXVO\SURGXFHVPRUHGDPSLQJ )RU WKH H[WUHPH FDVH
6 ZH ILQG DV H[SHFWHG RQO\ WKH FRQWULEXWLRQ WR
UROOLQJUHVLVWDQFH
)[ ] a
6
' & $U& )]
,IZHQHJOHFWWKHVPDOOFRQWULEXWLRQIURPWKHUROOLQJUHVLVWDQFHWKHILUVWWHUPRI YDQLVKHV 7KH PDJQLWXGH RI WKH UHPDLQLQJ WHUP UHGXFHV WR D VLPSOH H[SUHVVLRQLIWKHIUHTXHQF\LVFRQVLGHUHGWREHVPDOOZLWKUHVSHFWWRWKHQDWXUDO IUHTXHQF\
$33/,&$7,2162)75$16,(177<5(02'(/
)[ ]
&) UR
a
/ 0 0
'
/ 0 0
©
R
,Z
'
UR
9 '
,Z
UR
9Q
ZLWK9WKHVSHHGRIWUDYHOLQPVDQGQWKHIUHTXHQF\RIWKHYHUWLFDOD[OHPRWLRQ LQ +] ,W LV QRWHG IURP (T WKDW WKH ULVH RI WKH UHVSRQVH LQ WKH ORZHU IUHTXHQF\UDQJHZLWKWKHIUHTXHQF\QDQGWKHVSHHG9LVLQSDUWLFXODULQIOXHQFHG E\WKHPDJQLWXGHRIWKHIDFWRU
PXOWLSOLHGZLWKWKHSRODUPRPHQWRILQHUWLD,Z
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
R
7KH
PD[LPXPUHVSRQVHDPSOLWXGHDPRXQWVWR
)[ ] a
/ 0 0
'
/ 0 0
PD[
&) % $U& )] UR
9 = 40m/s
60
)-[ ]
[N/mm]
10 57Hz
0
0
0.5
1.0
o
ω/ωΩο1.5
-120
φ
10 o
40
-180
o
-240
)LJ 7KHRUHWLFDOIUHTXHQF\UHVSRQVHIXQFWLRQRIWKHORQJLWXGLQDOIRUFH)[WRYHUWLFDOD[OH PRWLRQV]IRUWKHELDVSO\FDUW\UH
$33/,&$7,2162)75$16,(177<5(02'(/
7KLV PD[LPXP LV GHWHUPLQHG DJDLQ SULPDULO\ E\ WKH HIIHFWLYH UROOLQJ UDGLXV JUDGLHQW
EXW QRZ PXOWLSOLHG ZLWK VOLS VWLIIQHVV &) )RU WKH UDGLDO W\UH WKH
PD[LPXPUHVSRQVHEHFRPHVFD1PPDWWKHQDWXUDOIUHTXHQF\+]DQGIRU WKHELDVSO\W\UHWKLVEHFRPHVFD1PPDW+])LJXUHVKRZVERWKWKH DPSOLWXGHDQGSKDVH ODJ UHVSRQVHFDOFXODWHGIRUWKHELDVSO\W\UH)LJXUH KDVDOUHDG\LQGLFDWHGWKDWIRUWKHUDGLDOW\UHDPXFKORZHUUHVSRQVHLVH[SHFWHG WRRFFXU
)UHTXHQF\5HVSRQVHWR5DGLDO5XQ2XW )URPHTXDWLRQ ZHGHULYHIRUWKHIUHTXHQF\UHVSRQVHWRUDGLDO FDUFDVV
a
UXQRXWFRQVLGHULQJRQO\LWVILUVWKDUPRQLF U
I
ZKHHOVSHHGRIUHYROXWLRQ
)[ ' & $ U&)] % L UF a
QRQGLPHQVLRQDOIUHTXHQF\
& &)[ & )]R UR & % L UR
R
a
' Ua ' UaF ZLWKIUHTXHQF\HTXDOWRWKH
7KLVH[SUHVVLRQGLIIHUVLQWZRPDLQUHVSHFWVIURPWKHRQHIRUWKHUHVSRQVHWRD[OH YHUWLFDO PRWLRQV )LUVWO\ ZH KDYH QRZ WKH IDFWRU LQVWHDG RI
DQG
VHFRQGO\DQDGGLWLRQDOWHUPZLWK)]RVKRZVXSZKLFKLVGXHWRWKHH[FHQWULFLW\ FI(T )LJXUHSUHVHQWVWKHUHVSRQVHSORWVIRUWKHELDVSO\W\UH )RUVSHHGVYHU\ORZ
)[ UF a
a
6
6 WKHH[SUHVVLRQUHGXFHVWR
) ' & $U&)] & L ] UR
7KHDPSOLWXGHUDWLRDWWDLQVDPLQLPXPQHDU
)]R UR & &)[
'
ZKLFKPHDQVIRUWKHUDGLDOW\UH QHDUWKHIUHTXHQF\Q W\UH QHDUQ
R
R
DQGIRUWKHELDVSO\
:LWKSDUDPHWHUVFRUUHVSRQGLQJWRWKHH[SHULPHQWFRQGXFWHG
,Z
NJP UHVXOWLQJLQQDWXUDOIUHTXHQFLHVRIDQG+] ZHREWDLQ
IRUWKHVHIUHTXHQFLHV DQG+]UHVSHFWLYHO\7KHPLQLPXPYDOXHDSSHDUV WREHFRPHYHU\FORVHWR$U&)]
DQG1PUHVSHFWLYHO\$VFDQEHVHHQ
LQWKHXSSHUULJKWKDQGGLDJUDPRI)LJWKHH[SHULPHQWDOO\DVVHVVHGFXUYH DOVRIHDWXUHVVXFKDGLSLQWKHORZIUHTXHQF\UDQJHDQGWKHDFFRPSDQ\LQJVKDUS GHFUHDVHLQSKDVHOHDG$WWKHQDWXUDOIUHTXHQF\Q
R
WKHDPSOLWXGHUDWLR
FRPHVFORVHWRLWVPD[LPXPYDOXH7KLVPD[LPXPLVDSSUR[LPDWHO\HTXDOWR
)[ UF a
/ 0 0
a
. & $U & )] % &
/ 0 0
PD[
&) ) ]R & UR UR
$33/,&$7,2162)75$16,(177<5(02'(/
80
)-[ UF
[N/mm]
48.5Hz 0
0
0.5
Ω/ωΩο1.5
1.0
o
180
φ 0 )LJ 7KHRUHWLFDOIUHTXHQF\UHVSRQVHFXUYHRIORQJLWXGLQDOIRUFHWRW\UHRXWRIURXQGQHVV VW
KDUPRQLF IRUWKHELDVSO\FDUW\UH7KHVSHHGRIWUDYHO9
UH
7KHVHFRQGWHUPE\IDUGRPLQDWHVWKLVH[SUHVVLRQ7KHUDGLDOW\UHDWWDLQV 1PP DW+] DQGWKHELDVSO\W\UH1PP DW+] &RPSDUHGZLWK WKHPD[LPXPORQJLWXGLQDOIRUFHUHVSRQVHWRYHUWLFDOD[OHPRWLRQV(T ZH QRWHDPXFKVWURQJHUVHQVLWLYLW\RIWKHUDGLDOW\UHWRRXWRIURXQGQHVV FD 1PP WKDQWRYHUWLFDOD[OHPRWLRQV FD1PP 7KLVLVGXHWRWKHIDFWRU RFFXUULQJLQ ZKLFKLQ DSSHDUVDV
)RUWKHELDVSO\W\UHZLWKD
PXFKPRUHFRPSUHVVLEOHWUHDGEDQGZHKDYH1PPYHUVXV1PP
([HUFLVH5HVSRQVHWRW\UHVWLIIQHVVYDULDWLRQV 'HULYHIURP(TV WKHIUHTXHQF\UHVSRQVHIXQFWLRQRI )[WRWKH VWDWLF GHIOHFWLRQ YDULDWLRQ × V WKDW LV WR YDULDWLRQV LQ UDGLDO W\UH VWLIIQHVV IRU IL[HG D[OH
KHLJKW ]
7KHQUHOHDVHWKHYHUWLFDOD[OHPRWLRQDQGILQGWKHIUHTXHQF\UHVSRQVH
suspension spring/damper wheel/axle mass tyre radial stiffness
F
N
F
P
z
&)]
z
)]R
P
~
)]
]V
)
~
& &
N
&
)]R
)LJ :KHHOVXVSHQVLRQPDVVV\VWHPH[FLWHGE\W\UHVWLIIQHVVYDULDWLRQV ([HUFLVH
$33/,&$7,2162)75$16,(177<5(02'(/
IXQFWLRQ IRU ] WR × V &RQVLGHU )LJ WKDW UHSUHVHQWV WKH VLPSOLILHG D[OH
ZKHHOVXVSHQVLRQV\VWHP6PDOOPRWLRQVDUHFRQVLGHUHGVRWKDWRQO\OLQHDUWHUPVFDQ EHWDNHQLQWRDFFRXQW8VHWKHSDUDPHWHUYDOXHVRIWKHELDVSO\W\UHRI7DEOHDQG IXUWKHUPRUHWKHYDOXHVIRUWKHPDVVWKHVXVSHQVLRQVWLIIQHVVDQGWKHGDPSLQJ P
NJF
1PN
1VP
1RZILQGWKHWRWDOIUHTXHQF\UHVSRQVHRI)[WR × VIRUWKHUHOHDVHGD[OHPRWLRQ,WPD\ EHQRWHGWKDWWKHH[FLWDWLRQE\DYDULDWLRQRIWKHUDGLDOVWLIIQHVVLVHTXLYDOHQWWRWKH H[FLWDWLRQE\DYHUWLFDOIRUFH a
)
]V
'
a R
&
)]
' &&
a )]R
V
UHSUHVHQWLQJWKHYDULDWLRQRIWKHµVWDWLF¶W\UHORDGDWFRQVWDQWD[OHKHLJKWZKLOHWKHW\UH UROOV
([HUFLVH6HOIH[FLWHGZKHHOKRS ,QYHVWLJDWH WKH VWDELOLW\ RI WKH IRXUWKRUGHU V\VWHP RI )LJ XVLQJ WKH +XUZLW] FULWHULXP FI(T (PSOR\(TV DVVXPH0\ DQGFRQVLGHU D IODW URDG DQG D XQLIRUP W\UH $VVHVV WKH UDQJH RI WKH VORSH
RI WKH ZKHHO D[OH
JXLGDQFH VXUIDFH ZKHUH LQVWDELOLW\ ZLOO RFFXU IRU WKH XQGDPSHG ZKHHO VXVSHQVLRQ V\VWHP7KHHIIHFWLYHUROOLQJUDGLXVFRXSOLQJFRHIILFLHQW LVWKHGHFLVLYHSDUDPHWHU 3ORWWKHVWDELOLW\ERXQGDU\LQWKHWDQ YHUVXV GLDJUDP9DU\ EHWZHHQWKHYDOXHV DQG,QGLFDWHWKHDUHDVZKHUHVWDELOLW\SUHYDLOV7DNHWKHSDUDPHWHUYDOXHV
U U & F & R
)]
P
HR
]
P
)[
NJ,Z
& & )
P
)[
î 1P
PUR NJP
7KDWDPRGHUDWHRVFLOODWRU\LQVWDELOLW\PD\LQGHHGVKRZXSZLWKVXFKDV\VWHPKDV EHHQ GHPRQVWUDWHG H[SHULPHQWDOO\ LQ SURIHVVRU 6.&ODUN¶V ODERUDWRU\ DW WKH 8QLYHUVLW\RI0LFKLJDQLQ
9
F ]
Ω
P
]
,Z
&)]
U H
)
[
β
)] )LJ2QWKHSRVVLEO\XQVWDEOHZKHHOKRSPRWLRQ ([HUFLVH
$33/,&$7,2162)75$16,(177<5(02'(/
)RUFHG6WHHULQJ9LEUDWLRQV
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
DUHQRWFRXSOHGZLWKWKHYHUWLFDOD[OHPRWLRQ
ZKHQVPDOOGLVSODFHPHQWVDUHFRQVLGHUHGDQGW\UHFRQWDFWIRUFHVDUHGLVUHJDUGHG )LUVWZHZLOOH[DPLQHWKHG\QDPLFVRIWKHIUHHV\VWHPQRWWRXFKLQJWKHURDG $IWHUWKDWWKHW\UHLVORDGHGDQGW\UHWUDQVLHQWPRGHOVIRUWKHLQSODQHDQGRXW RISODQHEHKDYLRXUDUHLQWURGXFHGDQGWKHV\VWHPVUHVSRQVHWRZKHHOXQEDODQFH ZLOOEHDVVHVVHGDQGGLVFXVVHG
9o
E F
ψ
[
O
O
N
ψ ,]
P PXQ
F[
ψ
,Z\
ω02 ω22
N[
F[ ω[2= P
ω22
QDWXUDOIUHTXHQFLHV
ω12 ω]2
\L
Fψ
PE2
1
F]
]
2
3
ω [2 ωψ2
2 \2
E
,]
E =0.8 β --L]
3
]HURDQG
0
ω]2= P
2
ωψ2 ω]2 ω12
UXQ
ωψ2=
E =1.5
ω 02
0 1
\1
.
ψ
9o
FHQWUHVRIURWDWLRQ
)LJ &RQILJXUDWLRQ RI D VLPSOH VWHHULQJVXVSHQVLRQ V\VWHP DQG WKH UHVXOWLQJQDWXUDO IUHTXHQFLHVDQGYLEUDWLRQDOPRGHV
$33/,&$7,2162)75$16,(177<5(02'(/
'\QDPLFVRIWKH8QORDGHG6\VWHP([FLWHGE\:KHHO8QEDODQFH
7KH VLPSOH V\VWHP GHSLFWHG LQ )LJ SRVVHVVHV WZR KRUL]RQWDO GHJUHHV RI IUHHGRP WKH URWDWLRQ DERXW WKH YHUWLFDO VWHHULQJ D[LV
DQG WKH IRUH DQG DIW
VXVSHQVLRQ GHIOHFWLRQ [ 7KH ILJXUH SURYLGHV GHWDLOV DERXW WKH JHRPHWU\ VWLIIQHVVHVGDPSLQJDQGLQHUWLD7KHPDVV PUHSUHVHQWV WKH WRWDOPDVVRIWKH
KRUL]RQWDOO\PRYLQJSDUWV7KHOHQJWKL]GHQRWHVWKHUDGLXVRILQHUWLD L ]
,] P
LVSURYLGHGZLWKDQXQEDODQFH
7KHZKHHOULPWKDWUHYROYHVZLWKDVSHHG
PDVVPXQ LQWKHZKHHOFHQWUHSODQHDWDUDGLXVUXQ 7KHFHQWULIXJDOIRUFHKDVD FRPSRQHQWLQIRUZDUGGLUHFWLRQ
)XQ [ ' & P XQ UXQ
W
VLQ
7KHHTXDWLRQVRIPRWLRQRIWKLVIRXUWKRUGHUV\VWHPUHDG
P[@ & PE % N [ [J % F[ [ ' )XQ [ P L] % E & PE[@ % N 0 % F ' &O)XQ [
:LWKGDPSLQJGLVUHJDUGHGWKHPDJQLWXGHRIWKHIUHTXHQF\ UHVSRQVHIXQFWLRQ EHFRPHV
8
P XQUXQ
' O
&
F
&
&
LQZKLFKZHKDYHWKH]HURIUHTXHQF\
'
O
O& E
[
DQGWKHWZRQDWXUDOIUHTXHQFLHV
'
%
[%
[&
%
[
%
[&
ZKHUHZHKDYHLQWURGXFHGWKHµXQFRXSOHG¶QDWXUDOIUHTXHQFLHV
['
F[ P
DQG
DQGWKHFRXSOLQJIDFWRU
'
F
P L ] % E
$33/,&$7,2162)75$16,(177<5(02'(/
' E
L]
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
DQG
LQFUHDVH ZLWK LQFUHDVLQJ
ORQJLWXGLQDOVXVSHQVLRQVWLIIQHVV7KHORZHUQDWXUDOIUHTXHQF\ZLWKDFHQWUHRI URWDWLRQORFDWHGDWWKHLQVLGHRIWKHNLQJSLQDSSURDFKHVWKHXQFRXSOHGQDWXUDO IUHTXHQF\
)URP LWLVVHHQWKDWD]HURGRHVQRWRFFXULIWKHXQEDODQFHDUPOHQJWK
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
ZKHUH,ZGHQRWHVWKHZKHHOSRODUPRPHQWRILQHUWLDDQGUWKHZKHHOUDGLXV
'\QDPLFVRIWKH/RDGHG6\VWHPZLWK7\UH3URSHUWLHV,QFOXGHG
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
$33/,&$7,2162)75$16,(177<5(02'(/
)RU WKH H[WHQGHG V\VWHP ZLWK URDG FRQWDFW WKH IROORZLQJ FRPSOHWH VHW RI OLQHDUHTXDWLRQVDSSO\
P [@ % [ [ [J % [ [ & PE & ) [ ' )XQ [ P] ]@ % ] ] ]J % ] ] ' )XQ ] 0 % , % & PE[ @ % O)[ & 0 ] ' &O)XQ [ ,Z J % UR ) [ ' )J [ % 9R )[ ' & &) 9V[ )J \ % 9R )\ ' & &) 9V\ 9V[ ' [J & O 0 & UR & R % R ] 9V\ ' & 9R 0 ] ' & W )\ & ( 0 & &J\U9R )J \ 9R )XQ [ ' & P XQ UXQ R VLQ RW )XQ ] ' PXQ UXQ R FRV RW 9R ' UR R
7DEOH3DUDPHWHUYDOXHVRIZKHHOVXVSHQVLRQV\VWHPDQGW\UHFRQVLGHUHG
P
NJ
P]
NJ
,
NJP
,Z
NJP
PXQ
NJ
]
UDGV
P
)]R
1
UDGV
P
&)]
N1P
P
&)
N1
&)
N1UDG
[
]
W
( &J\U
1P
x
V
7KHW\UHVLGHIRUFHGLIIHUHQWLDOHTXDWLRQ KDVEHHQXVHGDQGVLPLODUIRUWKH ORQJLWXGLQDOIRUFH7KHORQJLWXGLQDOVOLSVSHHGIROORZVIURP ZLWK 1RWHWKDWPHFKDQLFDOFDVWHUKDVQRWEHHQFRQVLGHUHGVRWKDWWKHODWHUDOVOLSVSHHG LVVLPSO\H[SUHVVHGE\ 7KHDOLJQLQJWRUTXHLVEDVHGRQ(T ZLWKIRU
0]UWKHVSLQPRPHQW0 ]DFFRUGLQJWR(T DQGWKHJ\URVFRSLFFRXSOHIURP (TV RU 7KH UROOLQJ UHVLVWDQFH PRPHQW KDV EHHQ QHJOHFWHG LQ
$33/,&$7,2162)75$16,(177<5(02'(/
F
[
o
1
ψ8
5
3 x10 N/m
2.4
4 F
[
o
F[
4
1.8
0.5
6
1.2
9
0.6
0
0
5
10
Q]
15
20
Q
[Hz]
25
30
)LJ 6WHHUYLEUDWLRQDPSOLWXGHGXHWRZKHHOXQEDODQFHDVDIXQFWLRQRIZKHHOIUHTXHQF\ RIUHYROXWLRQQ
R IRUYDULRXVYDOXHVRIWKHORQJLWXGLQDOVXVSHQVLRQVWLIIQHVV
DQG WKH DYHUDJH HIIHFWLYH UROOLQJ UDGLXV KDV EHHQ WDNHQ HTXDO WR WKH DYHUDJHD[OHKHLJKWRUORDGHGUDGLXVUR LQUHDOLW\UHLVXVXDOO\VOLJKWO\ODUJHUWKDQ
UR ,Q7DEOHWKHVHWRISDUDPHWHUYDOXHVXVHGLQWKHFRPSXWDWLRQVKDYHEHHQ OLVWHG 7KH PRPHQW RI LQHUWLD DERXW WKH VWHHULQJ D[LV LV GHQRWHG ZLWK , DQG
HTXDOVP E L ] 7KHDPSOLWXGHRIWKHVWHHUDQJOHWKDWRFFXUVDVDUHVSRQVHWR D ZKHHO XQEDODQFHPDVVRINJKDVEHHQSORWWHGDVDIXQFWLRQRIWKHZKHHO VSHHG RI UHYROXWLRQ LQ )LJ 7R H[DPLQH WKH LQIOXHQFH RI WKH ORQJLWXGLQDO VXVSHQVLRQFRPSOLDQFHDVHULHVRIYDOXHVRIWKHORQJLWXGLQDOVWLIIQHVVF[KDVEHHQ
40
10 30 40
30
9=50m/s
Q
20
Q3
60
Q2
L
Q1
[Hz]
Q
10
0
9=10m/s 9
60
20
0
2
4
F[
6
[N/m]
]
8
10 x10
5
)LJ 9DULDWLRQRIWKHWKUHHQDWXUDOIUHTXHQFLHVZLWKORQJLWXGLQDOVXVSHQVLRQVWLIIQHVVDW GLIIHUHQWVSHHGVWRJHWKHUZLWKWKHFRQVWDQWYHUWLFDOQDWXUDOIUHTXHQF\7KHFLUFOHVRQ WKHKRUL]RQWDOD[LVPDUNWKHVWLIIQHVVFDVHVRI)LJ
$33/,&$7,2162)75$16,(177<5(02'(/
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¶V
RI
)LJ 7KH ]HUR IUHTXHQF\ FORVHO\ IROORZV WKH IRUPXOD RI WKH IUHH V\VWHP$WWKHORZHVWVWLIIQHVVWKH]HURIUHTXHQF\QDOPRVWFRLQFLGHVZLWKWKH VHFRQGQDWXUDOIUHTXHQF\QDQGVXSSUHVVHVWKHVHFRQGSHDN ,WPD\EHRILQWHUHVWWRILQGRXWKRZWKHVHYHUDOW\UHSDUDPHWHUVDIIHFWWKH UHVSRQVH FKDUDFWHULVWLFV RI )LJ ,Q )LJ WKH UHVXOW RI PDNLQJ WKHVH
x
SDUDPHWHUVHTXDOWR]HURKDVEHHQGHSLFWHGIRUWKHVWLIIQHVVFDVHF[ 1P 1HJOHFWLQJWKHIDFWRU FDVH PHDQLQJWKDWWKHHIIHFWLYHUROOLQJUDGLXVZRXOG QRWFKDQJHZLWKORDGDSSHDUVWRKDYHDFRQVLGHUDEOHHIIHFWLQGLFDWLQJWKDWZLWK WKH YHUWLFDO PRWLRQ GRHV DPSOLI\ WKH VWHHULQJ RVFLOODWLRQ 2PLWWLQJ WKH J\URVFRSLF W\UH PRPHQW ZKLOH UHLQVWDWLQJ
DSSHDUV DV H[SHFWHG WR
HIIHFWLYHO\ GHFUHDVH WKH VWHHU GDPSLQJ 7KLV LV VWUHQJWKHQHG E\ DGGLWLRQDOO\ GHOHWLQJWKHPRPHQWGXHWRWUHDGZLGWK 2PLWWLQJWKHUHOD[DWLRQOHQJWKV
3
F[= 3.0 x10 5 N/m
2 5 4
1o
ψ8
0: original
6
1
o
1: η =0 2: &J\U =0
6 3 5 2 4 0
0
0.5
3: 4: 5: 6:
+: κ*=0 +: σα=σκ =0 +: & &=0 )α 0α +: &)κ=0
1
0
0
5
10
Q
15
20
[Hz]
25
30
)LJ,QIOXHQFHRIYDULRXVW\UHSDUDPHWHUVRQWKHVWHHUDQJOHDPSOLWXGHUHVSRQVHFXUYH
$33/,&$7,2162)75$16,(177<5(02'(/
ORZHUVWKHSHDNVWKXVUHPRYLQJWKHQHJDWLYHGDPSLQJGXHWRW\UHFRPSOLDQFH 'HOHWLQJLQDGGLWLRQWKHVLGHIRUFHDQGWKHDOLJQLQJWRUTXH UDLVHVWKHSHDNV DJDLQLQGLFDWLQJWKDWVRPHHQHUJ\LVORVWWKURXJKWKHVLGHVOLS'LVUHJDUGLQJWKH KRUL]RQWDO W\UH IRUFHV DOWRJHWKHU EULQJV XV EDFN WR WKH KRUL]RQWDOO\ IUHH V\VWHP WUHDWHG LQ WKH SUHYLRXV VXEVHFWLRQ $V SUHGLFWHG E\ WKH DQDO\VLV WZR VKDUSUHVRQDQFHSHDNVDULVHDVZHOODVWKHGLSDWWKH]HURIUHTXHQF\
$%6%UDNLQJRQ8QGXODWHG5RDG
7KHDLPRIWKLVVHFWLRQLVWRLQYHVWLJDWHWKHLQIOXHQFHRIG\QDPLFHIIHFWVGXHWR YHUWLFDODQGORQJLWXGLQDOZKHHOYLEUDWLRQVH[FLWHGE\URDGLUUHJXODULWLHVXSRQWKH EUDNLQJSHUIRUPDQFHRIWKHW\UHDQGDQWLORFNEUDNLQJV\VWHP7KHVHGLVWXUELQJ IDFWRUVDIIHFWWKHDQJXODUYHORFLW\RIWKHZKHHODQGFRQVHTXHQWO\PD\LQWURGXFH GLVLQIRUPDWLRQ LQ WKH VLJQDOV WUDQVPLWWHG WR WKH $%6 V\VWHP XSRQ ZKLFK LWV SURSHUIXQFWLRQLQJLVEDVHG 2QH PD\ WDNH WKH VLPSOH YLHZ WKDW WKH SULPDU\ IXQFWLRQ RI WKH DQWLORFN GHYLFHLVWRFRQWUROWKHEUDNHVOLSRIWKHZKHHOWRFRQILQHWKHZKHHOVOLSZLWKLQD QDUURZUDQJHDURXQGWKHVOLSYDOXHDWZKLFKWKHORQJLWXGLQDOW\UHIRUFHSHDNV,Q WKHVDPHUDQJHLWIRUWXQDWHO\WXUQVRXWWKDWWKHODWHUDOIRUFHWKDWWKHW\UHGHYHORSV DV D UHVSRQVH WR WKH ZKHHO VOLS DQJOH LV XVXDOO\ VXIILFLHQW WR NHHS WKH YHKLFOH VWDEOHDQGVWHHUDEOH ,QRUGHUWRNHHSWKHWUHDWPHQWVLPSOHZHZLOOUHVWULFWWKHYHKLFOHPRWLRQWR VWUDLJKWOLQHEUDNLQJDQGFRQVLGHUDTXDUWHUYHKLFOHZLWKZKHHODQGD[OHWKDWLV VXVSHQGHGZLWKUHVSHFWWRWKHYHKLFOHERG\WKURXJKDYHUWLFDODQGDORQJLWXGLQDO VSULQJ DQG GDPSHU 7KH DQDO\VLV LV EDVHG RQ WKH VWXG\ RI 9DQ GHU -DJW HW DO
,Q3ODQH0RGHORI6XVSHQVLRQDQG:KHHO7\UH$VVHPEO\
7KHYHKLFOHLVDVVXPHGWRPRYHDORQJDVWUDLJKWOLQHZLWKVSHHG97KHIRUZDUG DFFHOHUDWLRQ RI WKH YHKLFOH ERG\ LV FRQVLGHUHG WR EH SURSRUWLRQDO ZLWK WKH ORQJLWXGLQDOW\UHIRUFH)[9HUWLFDODQGKRUL]RQWDOYHKLFOHERG\SDUDVLWLFPRWLRQV ZLOOEHQHJOHFWHGZLWKUHVSHFWWRWKRVHRIWKHZKHHOD[OH&RQVHTXHQWO\WKHUROH RIWKHZKHHOVXVSHQVLRQLVUHVWULFWHGWRD[OHPRWLRQDORQH7KHVHVLPSOLILFDWLRQV HQDEOHXVWRFRQFHQWUDWHRQWKHLQIOXHQFHRIWKHFRPSOH[LQWHUDFWLRQVEHWZHHQ PRWLRQVRIWKHD[OHDQGWKHW\UHXSRQWKHEUDNLQJSHUIRUPDQFHRIWKHW\UH)LJXUH GHSLFWVWKHV\VWHPWREHVWXGLHG:HKDYHD[OHGLVSODFHPHQWV[DQG]DQG DYHUWLFDOURDGSURILOHGHVFULEHGE\ZDQGLWVVORSHGZGV7RVXLWWKHOLPLWDWLRQV
$33/,&$7,2162)75$16,(177<5(02'(/
F ]
9
N ]
Ω
. 9[
N
[
%
0
FF]
F [
P F
P ,Z
* 9 V[
Z dZ dV
)
[
)
]
.
]
0 \
)LJ6\VWHPFRQILJXUDWLRQIRUWKHVWXG\RIFRQWUROOHGEUDNLQJRQXQGXODWHGURDGVXUIDFH
RIWKHW\UHPRGHOHPSOR\HGWKHZDYHOHQJWK RIWKHURDGXQGXODWLRQLVFKRVHQ UHODWLYHO\ ODUJH 7KH ZKHHO DQJXODU YHORFLW\ LV
DQG WKH EUDNLQJ WRUTXH LV
GHQRWHG ZLWK 0% 7KH XQVSUXQJ PDVV DQG WKH SRODU PRPHQW RI LQHUWLD RI WKH ZKHHODUHOXPSHGZLWKDODUJHSDUWRIWKRVHRIWKHW\UHDQGDUHGHQRWHGZLWKP DQG,Z 7KURXJK WKH W\UH UDGLDO GHIOHFWLRQ WKH QRUPDO ORDG LV JHQHUDWHG 6LQFH ZH LQWHQGWRFRQVLGHUSRVVLEO\ODUJHVOLSIRUFHVWKHDQDO\VLVKDVWRDFFRXQWIRUQRQ OLQHDUW\UHFKDUDFWHULVWLFSURSHUWLHV7RDGHTXDWHO\HVWDEOLVKWKHIRUHDQGDIWW\UH FRQWDFW IRUFH ZH ZLOO HPSOR\ WKH HQKDQFHG WUDQVLHQW W\UH PRGHO GLVFXVVHG LQ 6HFWLRQ 7KH FRQWDFW UHOD[DWLRQ OHQJWK
F
ZLOO EH GLVUHJDUGHG KHUH ,Q WKH
PRGHODFRQWDFWSDWFKPDVVPFLVLQWURGXFHGWKDWFDQPRYHZLWKUHVSHFWWRWKH ZKHHO ULP LQ WDQJHQWLDO GLUHFWLRQ WKHUHE\ SURGXFLQJ WKH ORQJLWXGLQDO FDUFDVV GHIOHFWLRQX7KHFRQWDFWSDWFKPDVVPD\GHYHORSDVOLSVSHHGZLWKUHVSHFWWRWKH
URDGGHQRWHGE\9 V[7KHWUDQVLHQWVOLSLVGHILQHGE\ 9!
'&
(
V[
9
9
7KLVORQJLWXGLQDOVOLSUDWLRLVXVHGDVLQSXWLQWKHVWHDG\VWDWHORQJLWXGLQDOW\UH FKDUDFWHULVWLF7KHLQWHUQDOW\UHIRUFHWKDWDFWVLQWKHFDUFDVVDQGRQWKHZKHHO D[OHZLOOEHGHVLJQDWHGDV)[DZKLOHWKHWDQJHQWLDOFRQWDFWIRUFHLVGHQRWHGZLWK )[$FFRUGLQJWRWKHWKHRU\WKLVVOLSIRUFHLVJRYHUQHGE\WKHVWHDG\VWDWHIRUFH YV VOLS UHODWLRQVKLS )[VV ZKLFK PD\ EH PRGHOOHG ZLWK WKH 0DJLF )RUPXOD
$33/,&$7,2162)75$16,(177<5(02'(/
4000 )
]
4000 N
)[
3000 N
2000 2000 N
[N] 1000 N
0 0
0.5
κ
[-]
1
)LJ 6WHDG\VWDWHW\UHEUDNHIRUFHFKDUDFWHULVWLFVDWGLIIHUHQWORDGVDVFRPSXWHGZLWK WKHDLGRIWKH0DJLF)RUPXODDQGWKHVLPLODULW\WHFKQLTXH
6LQFHZHKDYHWRFRQVLGHUWKHLQIOXHQFHRIDYDU\LQJQRUPDOORDGWKHUHODWLRQVKLS PXVW FRQWDLQ WKH GHSHQGHQF\ RQ )] 7R KDQGOH WKLV WKH VLPLODULW\ PHWKRG RI &KDSWHULVXVHG:HKDYHWKHIROORZLQJHTXDWLRQV
[ VV '
)
) )
]
)
]R
[R HT
ZLWKDUJXPHQW
HT '
)
]R
)
]
) )]
&
&
)R
7KHPDVWHUFKDUDFWHULVWLF)[R GHVFULEHGE\WKH0DJLF)RUPXODKROGVDWWKH UHIHUHQFHORDG)]RZKLFKLVWDNHQHTXDOWRWKHDYHUDJHORDG,WVDUJXPHQW UHSODFHGE\WKHHTXLYDOHQWVOLSYDOXH
LV
$OVRWKHORQJLWXGLQDOVOLSVWLIIQHVV&) HT
R
LVGHILQHGDWWKHUHIHUHQFHORDG,Q)LJWKHVWHDG\VWDWHIRUFHFKDUDFWHULVWLFV KDYHEHHQSUHVHQWHGIRUDVHWRIYHUWLFDOORDGV
,QSODQHEUDNLQJG\QDPLFVHTXDWLRQV
7KHV\VWHPKDVDVLQSXWWKHURDGSURILOHZDQGWKHEUDNHWRUTXH0%7KHOHYHO ZDQGWKHIRUZDUGVORSHGZGVDUHJLYHQDVVLQXVRLGDOIXQFWLRQVRIWKHWUDYHOOHG GLVWDQFHV7RSDUWO\OLQHDULVHWKHHTXDWLRQVWKHURDGIRUZDUGVORSHLVDVVXPHG WREHVPDOO7KHEUDNH WRUTXHXOWLPDWHO\UHVXOWVIURPDFRQWURODOJRULWKPEXW PD\LQWKHSUHVHQWDQDO\VLVEHFRQVLGHUHGDVDJLYHQIXQFWLRQRIWLPH 7KH IROORZLQJ HTXDWLRQV DSSO\ IRU WKH ZKHHO URWDWLRQDO G\QDPLFV WKH KRUL]RQWDO DQG YHUWLFDO D[OH PRWLRQV DQG WKH WDQJHQWLDO PRWLRQ RI WKH FRQWDFW
$33/,&$7,2162)75$16,(177<5(02'(/
SDWFKPDVV ,
J
Z
%
U)
[D % 0\ ' & 0%
@ % N [ [J % F[ [ & ) [D & )]
GZ
@ % N ] ]J % F] ] % )] & )[D
GZ
P[
P]
GV
GV
'
'
(
F V[ % )[D & ) [ '
J P 9
ZLWKDX[LOLDU\HTXDWLRQVIRUWKHW\UHORDGHGUDGLXV
' R&
U
U
WKHUDGLDOGHIOHFWLRQ
' & ]
Z
WKHHIIHFWLYHUROOLQJUDGLXV U
H ' UH
WKHORQJLWXGLQDOFDUFDVVGHIOHFWLRQUDWH
'
X 0
9
(
V[ & 9V[
WKHZKHHOVOLSYHORFLW\
V[ '
9
9
% 0& [
U
H
%
] 0
GZ GV
DQGWKHWUDQVLHQWVOLS
'&
9
(
V[
9
0RUHRYHUWKHIROORZLQJFRQVWLWXWLYHUHODWLRQVDUHWREHDFFRXQWHGIRU ) )
]'
[D '
0 )
\'
['
)
] X
)
[D X X0
\ ) ] )[
[ VV )]
0 )
7KHIXQFWLRQ LVUHSUHVHQWHGE\(TV DQGLOOXVWUDWHGLQ)LJ7KH UHPDLQLQJFRQVWLWXWLYHUHODWLRQVDUHVLPSOLILHGWROLQHDUH[SUHVVLRQV
)UHTXHQF\UHVSRQVHRIZKHHOVSHHGWRURDGXQHYHQQHVV
$V HVVHQWLDO HOHPHQW RI WKH DQDO\VLV ZH ZLOO DVVHVV WKH UHVSRQVH RI WKH ZKHHO DQJXODUYHORFLW\YDULDWLRQWRURDGXQGXODWLRQVDWDJLYHQFRQVWDQWEUDNHWRUTXH
$33/,&$7,2162)75$16,(177<5(02'(/
)RUWKLVWKHVHWRIHTXDWLRQVPD\EHOLQHDULVHGDURXQGDJLYHQSRLQWRIRSHUDWLRQ FKDUDFWHULVHG E\ WKH DYHUDJH ORDG ) ]R WKH FRQVWDQW EUDNH WRUTXH 0%R DQG WKH DYHUDJHVOLSUDWLR
]'
a
) a
)
[D '
a
0 a
)
F
F
\'
['
F]
a
%
a
F[X
R
H
%
:HILQGIRUWKHFRQVWLWXWLYHUHODWLRQV a
][ X
N
a
J
F[ X
%
a
H
[] )]
U ] % %U ) [D a
a
$ )
[] )] % &[ a
&
a
ZKHUHZLWKWKHXVHRI(TV
&
[] '
M)[ VV
M)]
'
)
[R
)
]R
%
R
G&)
) R G)]
&
&
)
]R
G)[R
G HT
DQG
[
&
'
G)[R
G HT
)RUWKHYDULDWLRQLQHIIHFWLYHUROOLQJUDGLXVZHKDYH
H'&
a
U
a
7KH OLQHDULVHG HTXDWLRQV RI PRWLRQ DUH REWDLQHG IURP WKH (TV E\ VXEWUDFWLQJWKHDYHUDJHVWHDG\VWDWHYDOXHVRIWKHYDULDEOHTXDQWLWLHVDQGQHJOHFW SURGXFWVRIVPDOOYDULDWLRQV 7KHIUHTXHQF\UHVSRQVHRI
WRZKDVEHHQFDOFXODWHGIRUGLIIHUHQWYDOXHVRI
WKHORQJLWXGLQDOVXVSHQVLRQVWLIIQHVVF[7KHDYHUDJHFRQGLWLRQLVJLYHQE\ WKH NPKDYHUWLFDOORDG)]R
YHKLFOHVSHHG9R
1DQGDEUDNHWRUTXH0%R
1P WKDW FRUUHVSRQGV WR DQ DYHUDJH VOLS UDWLR
8
XQGXODWLRQDPSOLWXGH Z
R
7KH URDG
P
6RPH RI WKH W\UH FRQVWLWXWLYH UHODWLRQV ZHUH GHWHUPLQHG GLUHFWO\ IURP PHDVXUHPHQWVZKLOHIRUVRPHRIWKHSDUDPHWHUVYDOXHVKDYHEHHQHVWLPDWHG7KH VPDOO LQWHUDFWLRQ SDUDPHWHUV
H[] DQG H ][ DQG DOVR WKH UROOLQJ UHVLVWDQFH
SDUDPHWHUV$UDQG%UKDYHEHHQGLVUHJDUGHG7DEOHOLVWVWKHSDUDPHWHUYDOXHV XVHGLQWKHFRPSXWDWLRQV 7DEOH3DUDPHWHUYDOXHVIRUZKHHOW\UHVXVSHQVLRQV\VWHP
,Z
NJP
P
PF
F[
N1P
F]
N1P
)]R
1
NJ
N[
N1VP
N]
N1VP
UR
P
NJ
FF[
N1P
FF]
N1P
NF[
N1VP
$33/,&$7,2162)75$16,(177<5(02'(/
9= 60 km/h Z8 = 1 mm
0.4
5 2.5
%= 300 Nm
0
~ Ω8 5
F[ = 1.0 x10
0.2
N/m 100
[rad/s]
0
0
5
10
Q
15
[Hz]
20
)LJ )UHTXHQF\UHVSRQVHRIDPSOLWXGHRIZKHHOVSLQIOXFWXDWLRQVWRURDGXQGXODWLRQV
9
0%
DWDFRQVWDQWVSHHG DQGEUDNHWRUTXH
F[
VXVSHQVLRQVWLIIQHVV
Z
DQGIRUGLIIHUHQWYDOXHVRIWKHORQJLWXGLQDO
)LJXUHGLVSOD\VWKHDPSOLWXGHRIWKHZKHHODQJXODUYHORFLW\YDULDWLRQDV D IXQFWLRQ RI WKH LPSRVHG IUHTXHQF\ WKDW LV LQYHUVHO\ SURSRUWLRQDO ZLWK WKH ZDYHOHQJWKRIWKHXQGXODWLRQ,WLVVHHQWKDWWKHUHVRQDQFHSHDNKHLJKWDQGWKH UHVRQDQFHIUHTXHQF\VWURQJO\GHSHQGRQWKHIRUHDQGDIWVXVSHQVLRQVWLIIQHVV
$QWL/RFN%UDNLQJ$OJRULWKPDQG6LPXODWLRQ
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
DQG RI
0 ,W LV DVVXPHG WKDW WKH EUDNH
WRUTXHUDWHFDQEHFRQWUROOHGDWWKUHHGLIIHUHQWOHYHOV8SRQDSSOLFDWLRQRIWKH EUDNHWKHWRUTXHULVHVDWDFRQVWDQWUDWHXQWLOSRLQW%LVUHDFKHG$IWHUWKDWWKH WRUTXHLVUHGXFHGOLQHDUO\XQWLOSRLQW&7KHEUDNHWRUTXHUHPDLQVFRQVWDQWLQWKH LQWHUYDOEHWZHHQSRLQWV&DQG(&RPSRXQGFULWHULDDUHFRQVLGHUHGDWSRLQWV$
$33/,&$7,2162)75$16,(177<5(02'(/
DQG%IRUSUHGLFWLQJZKHQEUDNLQJVKRXOGEHUHGXFHGWRSUHYHQWWKHZKHHOIURP ORFNLQJ7KXVZHKDYHIURPWKHVWDUWLQJRIEUDNHDSSOLFDWLRQ
W W W
,QFUHDVHRIWRUTXHIRU
%
7KHSUHOLPLQDU\SUHGLFWLRQWLPH
UHR 0 W$ ' 7 J
ZKHUH
U
HR
DFFRUGLQJWR G
$
W
%
ZHKDYH
'HFUHDVHRIWRUTXHIRU
W W W %
&
DFFRUGLQJWR G
7KHLQVWDQWRISUHGLFWLRQDQGDFWLRQ
UHR ^
7KHWLPH
W
W &
W
$
% `
W
%
7
0 W 5 %
G
IRU D FRQVWDQW
LVGHILQHGE\
& 7 J W% & W$ ' 9%
PDUNVWKHEHJLQQLQJRIWKHFRQVWDQWWRUTXHSKDVH&(LH &
&RQVWDQWWRUTXHIRU &
G
0
ZKLFKVHWVWKHWUHVKROGIRU
W
%
VWDQGV IRU WKH QRPLQDO HIIHFWLYH UROOLQJ UDGLXV DQG
)URPWLPH
7LPH
0 W 5
LVJLYHQE\
W W W &
(
G
0 W %
G
LVIRXQGIURP
0 W &
'
6LPLODUO\WKHFULWHULRQIRULQFUHDVLQJWKHWRUTXHDW(
W W
,QFUHDVHRIWRUTXHIRU !
(
DFFRUGLQJWR G
0 W 5 %
G
LV JLYHQE\WKHFRPSRXQGUHVHOHFWLRQFRQGLWLRQVJHQHUDWHGDWSRLQWV'DQG( ZLWK
W
W W ' '
'
DQG
DQG
UHR> ^
(
EHLQJIRXQGIURP
W &
W
'
( `
%
J
W' @ W( & W' @ ' 9(
)RUWKHVLPXODWLRQWKHIROORZLQJYDOXHVKDYHEHHQXVHG
7
9
%
U
HR
W
%
DQG
9
(
PV
7KHEUDNHWRUTXHUDWHVKDYHEHHQVHWWR
5
5
1PV
,QRUGHUWRVLPXODWHWKHEUDNLQJPDQRHXYUHRIDTXDUWHUYHKLFOHHTXLSSHGZLWK WKH DQWLORFN EUDNLQJ V\VWHP WKH GHFHOHUDWLRQ RI WKH YHKLFOH LV WDNHQ WR EH SURSRUWLRQDO WR
)
[
7KH UHVXOWV RI WKH VLPXODWLRQV SHUIRUPHG DUH SUHVHQWHG LQ
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
$33/,&$7,2162)75$16,(177<5(02'(/
7KHUHVXOWVRIWKHVLPXODWLRQVKRZWKDWEUDNHVOLSYDULDWLRQVRFFXUERWKRQD IODWDVZHOODVRQXQGXODWHGURDGVXUIDFHV+RZHYHUWKHYHU\ODUJHYDULDWLRQVRI
NLQWKHODWWHUFDVHOHDGWRDIXUWKHUGHWHULRUDWLRQRIWKH
WKHWUDQVLHQWVOLSYDOXH
EUDNLQJSHUIRUPDQFH7KHDYHUDJHYHKLFOHGHFHOHUDWLRQGURSVGRZQIURPPV
WRDSSUR[LPDWHO\PV $OWKRXJKODUJHIOXFWXDWLRQVLQWKHYHUWLFDOW\UHIRUFHDUH PDLQO\UHVSRQVLEOHIRUWKLVUHGXFWLRQLWLVHTXDOO\FOHDUWKDWVHYHUHSHUWXUEDWLRQ RFFXUULQJERWKLQ
DQG
J PD\EHDQDGGLWLRQDOVRXUFHRIPLVLQIRUPDWLRQIRU
WKHDQWLORFNFRQWURODOJRULWKP7KHLPSRUWDQWUHGXFWLRQLQEUDNLQJHIIHFWLYHQHVV
&
UHVXOWLQJIURPYHUWLFDOW\UHIRUFHYDULDWLRQVPD\EHDWWULEXWHGWRWKHWHUP
F]
LQ
WKHOLQHDULVHGFRQVWLWXWYHUHODWLRQ RIWKHUROOLQJDQGVOLSSLQJW\UHDQGLQ SDUWLFXODUWR WKHFRQWULEXWLRQRIWKHYDULDWLRQRIWKHORQJLWXGLQDOVOLSVWLIIQHVV ZLWKZKHHOORDGG
&
)
G
)
]
%RWKYHUWLFDODQGKRUL]RQWDOYLEUDWLRQVRIWKHD[OHDQG
WKHYHUWLFDOORDGIOXFWXDWLRQVRQZDY\URDGVDSSHDUWRDGYHUVHO\LQIOXHQFHWKH EUDNLQJSHUIRUPDQFHRIWKHW\UHDVZHOODVWKDWRIWKHDQWLORFNV\VWHP ,Q 7DQJX\PDGHDSUHOLPLQDU\VWXG\RIVXFKHIIHFWVXVLQJDGLIIHUHQW FRQWURODOJRULWKP+HSRLQWHGRXWWKDWZKHHOYLEUDWLRQVRQXQHYHQURDGVFDQSRVH VHULRXVSUREOHPVRIPLVLQIRUPDWLRQIRUWKHFRQWUROORJLFRIWKHDQWLORFNV\VWHP 7KH UHVXOWV RI WKH VLPXODWLRQV GLVFXVVHG DERYH DQG UHSRUWHG E\ 9DQ GHU -DJW FRQILUP7DQJX\¶VILQGLQJV
9
T2 A
HRΩ
∆9B
U
∆9E E D
B C
(wheel)
T1
B E
C 0
W
.
HRΩ
D
0
7J
HRΩ
U
W
0% U
9 (vehicle)
C A
W
(time)
)LJ([DPSOHRIDQ$%6DOJRULWKPXVHGWRPRGXODWHWKHDSSOLHGEUDNHWRUTXH
1000
1000
0%
0%
[Nm]
[Nm]
0
0
0.04
0.04
[
[
[m]
[m]
0
0
4000
5000
)[
)]
[N]
[N]
0
1000
0.5
4000
κ
)
[
[N]
0
0
1000
0.5
.
Ω
κ
0
2
[rad/s ] -1000
0
20
1000
.
Ω
0
9 10
Ω
[m/s]
2
[rad/s ] -1000
U
20
9
0 0
1
W
[s]
2
10
ΩU
[m/s] )LJ /HIW 6LPXODWLRQRIEUDNLQJ ZLWK$%6RQDQLGHDOO\IODWURDG
5LJKW 6LPXODWLRQRIEUDNLQJ ZLWK$%6RQDZDY\URDG
0 0
1
W
[s]
2
$33/,&$7,2162)75$16,(177<5(02'(/
6WDUWLQJIURP6WDQGVWLOO
,Q WKLV VHFWLRQ WKH DELOLW\ RI WKH WUDQVLHQW PRGHOV WR RSHUDWH DW DQG QHDU ]HUR VSHHGFRQGLWLRQVZLOOEHGHPRQVWUDWHG7KHIRXUPRGHOVWUHDWHGLQ6HFV DQGZLOOEHHPSOR\HGLQWKHVLPXODWLRQRIWKHORQJLWXGLQDOPRWLRQRID TXDUWHUYHKLFOHPRGHORQDQXSZDUGVORSHRIFI)LJ 7KHIROORZLQJWKUHHGLIIHUHQWPDQRHXYUHVDUHFRQVLGHUHG
6WDQGLQJVWLOORQVORSHVWHSZLVHDSSOLFDWLRQRIGULYHWRUTXHDQGVXEVHTXHQWO\
)URPVWDQGVWLOORQVORSH IUHHO\UROOLQJEDFNZDUGVWKHQSRZHUIXOSURSXOVLRQ
UROOLQJDWFRQVWDQWVSHHG
IROORZHGE\IUHHUROOLQJ
)URPVWDQGVWLOO UROOLQJEDFNZDUGVWKHQEUDNLQJWRZKHHOORFNIROORZHGE\ IUHHUROOLQJDJDLQDIWHUZKLFKDVKRUWSKDVHRIGULYHWRUTXHLVDSSOLHG)LQDOO\ WKHTXDUWHUYHKLFOHVORZVGRZQRQWKHVORSH
7KHVWHDG\VWDWHORQJLWXGLQDOIRUFHIXQFWLRQRIWKHWUDQVLHQWORQJLWXGLQDOVOLS ¶ LVGHVFULEHGE\WKHIRUPXOD )
['
'
[ VLQ>& [ DUFWDQ ^%[
& [ [ & DUFWDQ [ (
%
%
`@
)RUWKHVXFFHVVLYHWUDQVLHQWW\UHPRGHOVWKHHTXDWLRQVZLOOEHUHSHDWHGEHORZ 6HPLQRQOLQHDUWUDQVLHQWPRGHO )LUVWRUGHUGLIIHUHQWLDOHTXDWLRQIRUORQJLWXGLQDOW\UHGHIOHFWLRQXDFFRUGLQJWR (T
GX GW
ZKHUH
%
_9[_ X
' & V[
9
RQO\GHSHQGVRQDSRVVLEO\YDU\LQJYHUWLFDOORDG)]7KHWUDQVLHQWVOLS
Ω
9[
0' )[
PJ
UH
β )LJ2QWKHSUREOHPRIVWDUWLQJIURPVWDQGVWLOORQDVORSH
$33/,&$7,2162)75$16,(177<5(02'(/
UHDGV
'
X
$WORZYDOXHVRIVSHHG_9[_ 9ORZWKHGHIOHFWLRQXPD\KDYHWREHUHVWULFWHGDV ZDVIRUPXODWHGLQJHQHUDOE\(T )RURXUSUHVHQWSUREOHPZHKDYH LI _
_
!
GX GW
VO DQG _9[ _ 9ORZ DQG 9V[ % _9 [_X
$
'
X
HOVH
GX
' & V[ & 9
GW
_9 [_ X
7KHVOLSZKHUHWKHSHDNIRUFHRFFXUVLVURXJKO\
VO '
' [ &)
ZLWKWKHVOLSVWLIIQHVV &
' [ [ [
)
%
& '
)RUWKHIDFWRU$WKH YDOXHLVVXJJHVWHGEXWDKLJKHUYDOXHPD\LPSURYHWKH SHUIRUPDQFHHVSHFLDOO\ZKHQWKHIRUFHFKDUDFWHULVWLFH[KLELWVDSURQRXQFHGSHDN )XOO\QRQOLQHDUWUDQVLHQWPRGHO 7KHVDPHHTXDWLRQV KROGEXWZLWK
UHSODFHGE\WKH
NGHSHQGHQW
DQGGRZQZDUGVOLPLWHGTXDQWLW\
'
(
ZKHUH
PD[
R )
&
R
@
% ) ) _% )
_)[_ _
&
PLQ
UHSUHVHQWVWKHYDOXHRIWKHUHOD[DWLRQ OHQJWKDW
N
:HKDYHWKH
HTXDWLRQ
GX GW
%
(
_9[_ X
' & V[ 9
7KHWUDQVLHQWVOLSQRZUHDGV
'
X
(
$GHIOHFWLRQOLPLWDWLRQLVQRWQHFHVVDU\IRUWKLVPRGHO+RZHYHUEHFDXVHRIWKH DOJHEUDLF ORRS WKDW DULVHV WKH TXDQWLW\ PXVW EH REWDLQHG IURP WKH SUHYLRXVWLPHVWHS
$33/,&$7,2162)75$16,(177<5(02'(/
5HVWULFWHGIXOO\QRQOLQHDUPRGHO $QDORJRXVWR(T ZHKDYHWKHHTXDWLRQIRU
G
9
GW
9
NGHSHQGHQWUHOD[DWLRQOHQJWK
ZLWKWKH
R
'
N
% _ [_ ' & V[
M)
[
M
)
&
7KH PRGHO LV QRW VHQVLWLYH WR ZKHHO ORDG YDULDWLRQV ZKLFK FRQVWLWXWHV WKH UHVWULFWLRQRIWKHPRGHO)RUWKHSUREOHPDWKDQGWKLVUHVWULFWLRQLVQRWUHOHYDQW DQGWKHPRGHOFDQEHXVHG7KHJUHDWDGYDQWDJHRIWKHPRGHOLVWKHIDFWWKDWDQ DOJHEUDLFORRSGRHVQRWRFFXU$QGDJDLQDXOLPLWDWLRQLVQRWQHHGHG$VWUDLJKW IRUZDUG VLPXODWLRQ FDQ EH FRQGXFWHG )RU WKH UHODWLRQ WKH IROORZLQJ DSSUR[LPDWHIXQFWLRQLVXVHG
' PD[ R & ) &
'
ZKHUH
PLQ
[
PLQ
UHSUHVHQWV WKH PLQLPXP YDOXH RI WKH UHOD[DWLRQ OHQJWK WKDW LV
LQWURGXFHGWRDYRLGQXPHULFDOGLIILFXOWLHV (QKDQFHGQRQOLQHDUWUDQVLHQWPRGHO )ROORZLQJ 6HF DQG )LJ ZH KDYH IRU WKH GLIIHUHQWLDO HTXDWLRQ IRU WKH ORQJLWXGLQDOPRWLRQRIWKHFRQWDFWSDWFKPDVVPF
0 (
F V[ % N F[ X0 % FF[ X '
P 9
)
[
ZKHUH)[GHQRWHVWKHFRQWDFWIRUFHJRYHUQHGE\(T 7KHWUDQVLHQWVOLSYDOXHLVREWDLQHGIURPWKHGLIIHUHQWLDOHTXDWLRQ
F
G
GW
% _ [_ ' & V[(
9
9
DQHTXDWLRQVLPLODUWR EXWKHUHZLWKDFRQVWDQW VPDOO FRQWDFWUHOD[DWLRQ OHQJWK7KHORQJLWXGLQDOFDUFDVVVWLIIQHVVFF[LQWKHFRQWDFW]RQHVKRXOGEHIRXQG E\VDWLVI\LQJWKHHTXDWLRQ
R'
)
& &
)[
'
)
& F
F[
% F
7KH GHIOHFWLRQ UDWH QHHGHG LQ (TV LV HTXDO WR WKH GLIIHUHQFH LQ VOLS YHORFLWLHVRIFRQWDFWSDWFKDQGZKHHOULP X 0
' V[( & V[ 9
9
7KHORQJLWXGLQDOIRUFHDFWLQJRQWKHZKHHOULPUHVXOWVIURP
$33/,&$7,2162)75$16,(177<5(02'(/
)
[D ' NF[ X0 % FF[ X
/RZVSHHGDGGLWLRQDOGDPSLQJ $W ]HUR IRUZDUG VSHHG ZLWK HDFK RI WKH ILUVW WKUHH W\UH PRGHOV D YLUWXDOO\ XQGDPSHG YLEUDWLRQ LV H[SHFWHG WR RFFXU $Q DUWLILFLDO GDPSLQJ PD\ EH LQWURGXFHGDWORZVSHHGE\UHSODFLQJ
NLQ E\
& 9 ORZ V[ N
9
)
&
ZKLFKFRUUHVSRQGVZLWKWKHVXJJHVWHG(T 7KHJUDGXDOUHGXFWLRQWR]HUR DW9[
9ORZLVUHDOLVHGE\XVLQJWKHIRUPXOD
9 ORZ '
N
N
N
9 ORZ '
9 ORZ
% FRV
_9 [_ 9
ORZ
LI_9[_
#9
ORZ
LI_9[_!9ORZ
$OVRZLWKWKHHQKDQFHGPRGHOWKHDGGLWLRQDOGDPSLQJZLOOEHLQWURGXFHGZKLOH NF[LVWDNHQHTXDOWR]HURWREHWWHUFRPSDUHWKHUHVXOWV
9HKLFOHDQGZKHHOPRWLRQ )RUWKHDERYHW\UHPRGHOVWKHZKHHOVOLSVSHHGIROORZVIURP 9
V[ '
9
[ & UH
7KH YHKLFOH YHORFLW\ DQG WKH ZKHHO VSHHG RI UHYROXWLRQ DUH JRYHUQHG E\ WKH GLIIHUHQWLDOHTXDWLRQV
0
['
P9
,
Z
0
)
[D & PJ WDQ
' ' & H [D 0
U
)
ZKHUH UH LV WKH HIIHFWLYH PRPHQW DUP ZKLFK WXUQV RXW WR FRUUHVSRQG ZLWK H[SHULPHQWDO HYLGHQFH FI &KDS )LJ )RU WKH ILUVW WKUHH PRGHOV ZLWK FRQWDFWSDWFKPDVVQRWFRQVLGHUHGZHKDYH )
[D '
)
[
:KHQDWEUDNLQJWKHZKHHOEHFRPHVORFNHGZHKDYH
5HVXOWV 7KHWKUHHGLIIHUHQWORQJLWXGLQDOPDQRHXYUHVOLVWHGDERYHKDYHEHHQVLPXODWHG ZLWK HDFK RI WKH IRXU WUDQVLHQW W\UH PRGHOV ,Q WKH )LJV VRPH RI WKH UHVXOWVKDYHEHHQSUHVHQWHG7KHSDUDPHWHUYDOXHVKDYHEHHQOLVWHGLQ7DEOH )LJXUHVDQGGHSLFWWKHSURFHVVRIVWDQGLQJVWLOORQDXSZDUGVORSH DSSURSULDWHEUDNHRUGULYHWRUTXH IROORZHGE\DVWHSGULYHWRUTXHLQSXWDQG VXEVHTXHQWO\UROOLQJDWFRQVWDQWVSHHG EDFNWRHTXLOLEULXPGULYHWRUTXH
$33/,&$7,2162)75$16,(177<5(02'(/
7DEOH3DUDPHWHUYDOXHV
P
NJ
%[
WDQ
,Z
NJP
&[
N9ORZ
1VP
PLQ
P
J
PV
'[
1
NF[
F
P
UH
P
([
PF
NJ
R
)
P
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
FRQVWDQW DSSHDUWRSHUIRUPOHVVJRRGXQGHUWKHVH
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µFRQWDFWSDWFK¶&DOFXODWLRQV KDYH EHHQ FRQGXFWHG ZLWK WKH VLPSOH VHPL QRQOLQHDU WUDQVLHQW W\UH PRGHO 3UDFWLFDOO\HTXDOUHVXOWVKDYHEHHQREWDLQHGZKHQXVLQJRQHRIWKHRWKHUWKUHH WUDQVLHQWW\UHPRGHOV
$33/,&$7,2162)75$16,(177<5(02'(/
[
[
)'
HΩ
U
2
semi non-linear model upward slope 5%
9[
' ( H [)
0 U'
HΩ
U
ORZ
N
1
[ [m/s]
9
)[
V[
9
'
0
)[
ORZ ORZ0
NN
0
V[
9
V[
9
-1
0
0.5
1
time [s]
1.5
2
2.5
)LJ 6WDQGLQJ VWLOO RQ VORSH DFFHOHUDWLRQ DQG VXEVHTXHQWO\ UROOLQJ DW FRQVWDQWVSHHG 7\UHZKHHO ZLQGXS RVFLOODWLRQV RFFXU ERWK DW VWDUW DQG DW HQG RI VWHS FKDQJH LQ
0'
SURSXOVLRQWRUTXH
&RPSXWDWLRQVKDYHEHHQSHUIRUPHGZLWKWKHVHPLQRQOLQHDU
9
WUDQVLHQWW\UHPRGHOZLWKFRQVWDQWUHOD[DWLRQOHQJWK
[
[
)'
HΩ
2
PV
enhanced non-linear model upward slope 5%
[
U
ORZ
9
' ( H [)
0 U'
HΩ
U
ORZ
N
[ [m/s] 1
9
V[ ORZ ORZ0
'
0
[
)
9
[
)
NN
0
V[
9
V[
9
-1
0
0.5
1
time [s]
1.5
2
2.5
)LJ 6DPHVLPXODWLRQDVLQ)LJEXWQRZXVLQJWKHHQKDQFHGQRQOLQHDUWUDQVLHQWW\UH
9
PRGHO ZLWK FDUFDVV FRPSOLDQFH
ORZ
PV 2QO\ DW ORZ VSHHG D IUHTXHQF\
GLIIHUHQFHFDQEHREVHUYHGWRRFFXU9HU\VLPLODUUHVXOWVDUHREWDLQHGZLWKWKHIXOO\ QRQOLQHDUWUDQVLHQWPRGHOVZLWKFKDQJLQJUHOD[DWLRQOHQJWK
$33/,&$7,2162)75$16,(177<5(02'(/
4
[
[
enhanced non-linear model upward slope 5%
)'
3
HΩ
' ( H [)
0 U'
HΩ
U
9[
2
U
ORZ
N
[ [m/s] 1 9V[ 9
'
)[
0
ORZ ORZ0
NN
HΩ
U
0
[ <0
9
-1
propulsion
wheel
wheel spins up
slows down
V[
V[
9
9
-2 0
0.5
1
time [s]
1.5
2
2.5
)LJ )URP VWDQGVWLOO RQ VORSH IUHHO\ UROOLQJ EDFNZDUGV WKHQSRZHUIXOSURSXOVLRQ IROORZHGE\IUHHUROOLQJ7KHW\UHORQJLWXGLQDOIRUFHSDVVHVLWVSHDNDQGUHDFKHVWKH ORZHUHQGRILWVFKDUDFWHULVWLFZKLFKHQDEOHVWKHZKHHOWRVSLQXS1H[WZKHQWKH ZKHHO VORZV GRZQ UDSLGO\ WKH SHDN LV UHDFKHG DJDLQ DQG D GDPSHG ZKHHOW\UH
9
ZLQGXSRVFLOODWLRQRFFXUV&DOFXODWLRQVXVLQJHQKDQFHGWUDQVLHQWW\UHPRGHO
ORZ
PV ZKLFK WRJHWKHU ZLWK WKH UHVWULFWHG IXOO\ QRQOLQHDU PRGHO WXUQV RXW WR SHUIRUPEHVW
[
semi non-linear model upward slope 5%
[
)'
2
' ( H [)
0 U'
HΩ
propulsion
ORZ
N
[ [m/s] 1 9V[ 9
braking
ORZ ORZ0
wheel lock Ω =0
[
NN
)
[
)
'
0
0
HΩ
0
1
[
)
V[
9
[
9
U
)LJ
U Ω [ H
9
U
2
time [s]
3
4
)URPVWDQGVWLOOUROOLQJEDFNZDUGVWKHQ EUDNLQJ WRZKHHOORFNIROORZHGE\IUHH UROOLQJDJDLQDIWHUZKLFKDVKRUWSKDVHRIDFFHOHUDWLRQRFFXUV)LQDOO\WKHTXDUWHU YHKLFOHVORZVGRZQRQWKHVORSH'XULQJZKHHOORFNWKHYHKLFOHYLEUDWHV DW ORZ IUHTXHQF\ZLWKUHVSHFWWRWKH µFRQWDFWSDWFK¶&DOFXODWLRQVZLWKWKHVLPSOHVHPL QRQOLQHDUWUDQVLHQWW\UHPRGHO3UDFWLFDOO\VDPHUHVXOWVDUHREWDLQHGZKHQXVLQJ RQHRIWKHRWKHUWKUHHWUDQVLHQWW\UHPRGHOV
$33/,&$7,2162)75$16,(177<5(02'(/
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
&KDSWHU 6+257:$9(/(1*7+,17(50(',$7( )5(48(1&<7<5(02'(/
,QWURGXFWLRQ
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
PRGHOOLQJ URDG REVWDFOH HQYHORSLQJ SURSHUWLHV DQG SDUNLQJ SKHQRPHQD WKH IUHTXHQF\ LV UHODWLYHO\ KLJK FD +] ZKLOH WKH OHYHO RI VOLS FDQ EH KLJK
6LWXDWLRQVLQZKLFKFRPELQHGVOLSRFFXUVFDQEHKDQGOHGDQGWKH0DJLF)RUPXOD PRGHOFDQEHXVHGDVWKHEDVLVIRUWKHQRQOLQHDUIRUFHDQGPRPHQWGHVFULSWLRQ $VDUHVXOWDFRQWLQXRXVWUDQVLWLRQIURPWLPHYDU\LQJVOLSVLWXDWLRQVWRVWHDG\ VWDWHFRQGLWLRQVLVUHDOLVHG7KHRULJLQDOPRGHOGHYHORSPHQWZDVUHVWULFWHGWRWKH PRUHLPSRUWDQWUHVSRQVHVWRYDULDWLRQVRIORQJLWXGLQDODQGVLGHVOLS$OVRWKH SRVVLELOLW\WRWUDYHUVHGLVWLQFWURDGLUUHJXODULWLHV FOHDWV ZDVLQFOXGHGLQWKHW\UH PRGHO7KHPRGHOLVEDVHGRQWKHZRUNRI=HJHODDU DQG0DXULFH DQGLVUHIHUUHGWRDVWKH
6:,)7PRGHO FRUUHVSRQGLQJWRWKHWLWOHRIWKHSUHVHQW
FKDSWHU 6XEVHTXHQW GHYHORSPHQWV RI WKH PRGHO GLG PDNH LW SRVVLEOH WR DOVR LQFOXGHYDULDWLRQVRIFDPEHUDQGWXUQVOLS 7KHFUXFLDOVWHSWKDWZDVWDNHQWRUHDFKIXUWKHUWKDQRQHFDQE\XVLQJWKH VWULQJPRGHOLVWKHVHSDUDWLRQRIPRGHOOLQJWKHFDUFDVVDQGWKHFRQWDFWSDWFK,Q WKLVZD\DPXFKPRUHYHUVDWLOHPRGHOFDQEHHVWDEOLVKHGWKDWFRUUHFWO\GHVFULEHV VOLS SURSHUWLHV DW VKRUW ZDYHOHQJWKV DQG DW KLJK OHYHOV RI VOLS 7KH PRGHO DFKLHYHGFDQEHVHHQDVDIXUWKHUGHYHORSPHQWRIWKHHQKDQFHGVLQJOHFRQWDFW
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
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
ϕ . -ψ
belt
γ wheel plane
wheel rim
Ω
]
ZHII
9
B
UHII
βHII
belt
9F
)[ )]
eff. road plane
B
α 9
&
α
ϕ
residual springs
)\
0]
actual road surface
)LJ *HQHUDO FRQILJXUDWLRQ RI WKH 6:,)7 PRGHO IHDWXULQJ ULJLG EHOWULQJUHVLGXDO VWLIIQHVVHVFRQWDFWSDWFKVOLSPRGHODQGHIIHFWLYHURDGLQSXWV
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
GLVWULEXWLRQVDQGGHPRQVWUDWHVWKHXVHRIWKHPRGHOZLWKVLPXODWLRQVRIWKHW\UH WUDYHUVLQJDVLQXVRLGDOURDGVXUIDFHDOVRDWDVOLSDQJOHDQGDWEUDNLQJFI*LSVHU HWDO ,QWKHHQVXLQJWKHRUHWLFDOWUHDWPHQWILUVWLQ6HFWLRQ WKHVOLSPRGHORIWKH FRQWDFW SDWFK FRYHULQJ VPDOO ZDYHOHQJWKV DQG ODUJH VOLS ZLOO EH GHDOW ZLWK 6HFRQGO\WKHPRGHOIRUWKHGHVFULSWLRQRIWKHG\QDPLFEHKDYLRXURIWKHULJLGEHOW ULQJZLOOEHDGGHG6HF 7KLUGO\WKHIHDWXUHRIWKHPRGHOWKDWWDNHVFDUHRI UXQQLQJRYHUXQHYHQURDGVZLOOEHDGGUHVVHG6HF )XOOVFDOHW\UHWHVWUHVXOWV ZLOOGHPRQVWUDWHWKHYDOLGLW\RIWKHPRGHO
7KH&RQWDFW3DWFK6OLS0RGHO
,QWKLVVHFWLRQZHZLOOILUVWUHSUHVHQWWKHFRQWDFWSDWFKZLWKWUHDGHOHPHQWVE\ WKHEUXVKPRGHO%HFDXVHRILWVUHODWLYHFRPSOH[LW\WKHDQDO\WLFDOPRGHOWKDW GHVFULEHVWKHQRQVWHDG\VWDWHUHVSRQVHWRVOLSYDULDWLRQVLVDSSUR[LPDWHGE\D VHWRIILUVWRUGHUGLIIHUHQWLDOHTXDWLRQV7KLVFRQWDFWPRGHOLVWHVWHGE\DWWDFKLQJ WKHEDVHOLQHRIWKHEUXVKPRGHOWRWKHZKHHOSODQHWKURXJKDFRPSOLDQWFDUFDVV )RUUHDVRQVRISUDFWLFDOXVHZHILQDOO\LQWURGXFHWKH0DJLF)RUPXODWRKDQGOH WKHQRQOLQHDUEHKDYLRXURIWKHPRGHO
%UXVK0RGHO1RQ6WHDG\6WDWH%HKDYLRXU
7KHVWHDG\VWDWHFKDUDFWHULVWLFVRIWKHPRGHOKDYHEHHQGLVFXVVHGLQ&KDSWHU $VDILUVWVWHSZHZLOOGHULYHWKHHTXDWLRQVWKDWJRYHUQWKHUHVSRQVHRIWKHIRUFHV DQGPRPHQWWRVPDOOYDULDWLRQVRIWKHZKHHOVOLSZLWKUHVSHFWWRDJLYHQOHYHORI ZKHHOVOLSLQGLFDWHGZLWKVXEVFULSWR
/RQJLWXGLQDO6OLS )RUWKHFDVHRISXUHORQJLWXGLQDOVOLS
6FRRIWKHFRQWDFWSDWFK WKHSRLQWRI
WUDQVLWLRQ IURP DGKHVLRQ WR VOLGLQJ LV ORFDWHG DFFRUGLQJ WR (T ZLWK
[
DWDGLVWDQFH WIURPWKHFRQWDFWFHQWUH [
W
'
_ D
%
_ FR
&
FR
2
7KHFRPSRVLWHSDUDPHWHU KDVEHHQGHILQHGLQ&KDSWHU(T 7KHOHQJWK RIWKHDGKHVLRQUDQJH E\ WKH IUDFWLRQ
DP D[WWKDWEHJLQVDWWKHOHDGLQJHGJHLVFKDUDFWHULVHG
P WKDW LQ WKH SUHVHQW FKDSWHU UHSODFHV WKH V\PERO 8 WR DYRLG
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
FRQIXVLRQZLWKWKHQRWDWLRQIRUWKHZDYHOHQJWK:HKDYH
_
' &
P
_
FR
_ LI
%
%
FR
_
FR
'
HOVH P
FR
$QLPSRUWDQWREVHUYDWLRQLVWKDWDWVPDOOYDULDWLRQVRIVOLSZHPD\DVVXPHWKDW RQO\LQWKHDGKHVLRQUDQJHFKDQJHVLQGHIOHFWLRQRFFXU )RUWKHGHYHORSPHQWRIWKHWUDQVLHQWPRGHOZHVWDUWRXWIURPWKHEDVLFUROOLQJ FRQWDFWGLIIHUHQWLDOHTXDWLRQV ,QWKHDGKHVLRQUDQJHZHILQGIRUWKH
X
ORQJLWXGLQDOWUHDGHOHPHQWGHIOHFWLRQ FLQWKHFDVHWKDWRQO\ORQJLWXGLQDOVOLSLV FRQVLGHUHG
MX
MX
a
a
&
F
MW
9
'
F
M[
UFR
'&
a
9
V[F
a
9
[FR
F
:KHUHWKHWLOGHGHVLJQDWHVWKHYDULDWLRQZLWKUHVSHFWWRWKHVWHDG\VWDWHOHYHO7KH DYHUDJHOLQHDUVSHHGRIUROOLQJRIWKHFRQWDFWSDWFKLVHTXDOWRWKDWRIWKHZKHHO ULPDQGUHDGV 9
'
UFR
9
UR
'
9
&
[
9
'
V[R
9
[
%
FR
8DQG.GHQRWLQJWKH
7KH)RXULHUWUDQVIRUPRIWKHGHIOHFWLRQEHFRPHVZLWK
X
WUDQVIRUPHGTXDQWLWLHVRI DQG UHVSHFWLYHO\DQG a
8
F
'
L
&
&
L
H
&
V D
[ 9
VWKHSDWKIUHTXHQF\
a
[ 9 UFR
.
F
V
+HUH WKH ERXQGDU\ FRQGLWLRQ ZKLFK VD\V WKDW WKH GHIOHFWLRQ YDQLVKHV DW WKH OHDGLQJ HGJH LV VDWLVILHG %\ LQWHJUDWLQJ RYHU WKH UDQJH RI DGKHVLRQ DQG
F
PXOWLSO\LQJZLWKWKHWUHDGHOHPHQWVWLIIQHVV SWKHIUHTXHQF\UHVSRQVHIXQFWLRQ RI WKH YDULDWLRQ RI
)[ WR WKH YDULDWLRQ RI
F LV REWDLQHG :LWK WKH UHGXFHG
IUHTXHQF\
V
P
'
V
%
FR
DQGWKHORFDOGHULYDWLYHRI
&
)
F
'
FSP
%
D
)[WR6F
_ LI
FR
FR
%
_
HOVH &
)
F
'
FR
WKHH[SUHVVLRQIRUWKHUHVSRQVHIXQFWLRQEHFRPHVZKHQVWLOODGKHVLRQRFFXUVLQ WKHFRQWDFWSDWFK ILUVWFRQGLWLRQRI
+
)
F
L
V
'
&
)
L
F
D V
&
L
D
&
&
L
H
D
V
V
,Q)LJ WKHUHVXOWLQJDPSOLWXGHDQGSKDVHFKDUDFWHULVWLFVKDYHEHHQVKRZQ
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
~
10
&)κFκ~F
10
)[
0
-1
analytic solution approx. 1st order
10
-2
0
φ)κ
o
-45
o
-90 0 10 )LJ
10
1
1/D
10
2
ωV
[rad/m]
10
3
)UHTXHQF\UHVSRQVHIXQFWLRQRIWKHORQJLWXGLQDOIRUFHYDULDWLRQWRORQJLWXGLQDOVOLS YDULDWLRQRIWKHEUXVKPRGHOYHUVXVUHGXFHGSDWKIUHTXHQF\DFFRUGLQJWR WKHH[DFW DQDO\WLFDOVROXWLRQDQGWKDWRIDQDSSUR[LPDWHVXEVWLWXWHILUVWRUGHUV\VWHPDWDJLYHQ OHYHORIORQJLWXGLQDOVOLS
WRJHWKHUZLWKWKRVHRIDQDSSUR[LPDWHILUVWRUGHUV\VWHP(VSHFLDOO\WKHSKDVH FXUYH DSSHDUVWRH[KLELWDZDY\SDWWHUQLQWKHKLJKHUIUHTXHQF\UDQJH7KHVH ZDYHVDUHFRQVLGHUDEO\DWWHQXDWHGZKHQWKHFRQWDFWPRGHOLVLQFRUSRUDWHGLQD PRUHFRPSOHWHW\UHPRGHOLQFOXGLQJFDUFDVVFRPSOLDQFHDVZLOOEHVKRZQEHORZ 7KHILUVWRUGHUVXEVWLWXWHPRGHOKDVDIUHTXHQF\UHVSRQVHIXQFWLRQWKDWUHDGV
+
)
F
L
V
'
&
%
)
L
F
D
V
'
&
)
%
L
F
V
F
7KHDSSUR[LPDWLRQVKRZVWKHVDPHKLJKIUHTXHQF\DV\PSWRWHDQGVWHDG\VWDWH OHYHODVWKHH[DFWPRGHO$SSDUHQWO\WKHDFWXDOFXWRIISDWKIUHTXHQF\UHDGV
&
VF
R
'
%
FR
PD
D 6FR YDQLVKHV DQG DV D P EHFRPHVHTXDOWRXQLW\:KHQWKHDYHUDJHVOLSLVFKRVHQ
WKDW REYLRXVO\ UHGXFHV WR ZKHQ WKH DYHUDJH VOLS FRQVHTXHQFH
ODUJHU WKH OHQJWK RI WKH DGKHVLRQ UDQJH GHFUHDVHV DQG WKH FXWRII IUHTXHQF\ EHFRPHVKLJKHU7KHUHOD[DWLRQOHQJWKRIWKHDSSUR[LPDWHFRQWDFWPRGHOUHDGV DFFRUGLQJWR(T ZLWK
F
'
PD
%
FR
ZKLFKUHGXFHVWR]HURZKHQWRWDOVOLGLQJRFFXUV 7KH SHUIRUPDQFH RI WKH ILUVWRUGHU V\VWHP LV UHDVRQDEOH EXW DSSHDUV WR
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
~
)[
&)κ
10
0
10
-1
10
-2
κ~
analytic solution approx. 1st order
0
φ)κ
o
-45
o
1/σ
-90 0 10
1
10
1/D
10
2
ωV
[rad/m]
10
3
)LJ )UHTXHQF\UHVSRQVHIXQFWLRQRIWKHORQJLWXGLQDOIRUFHYDULDWLRQWRORQJLWXGLQDOVOLS YDULDWLRQ RI WKH EUXVK PRGHO DWWDFKHG WR D IOH[LEOH FDUFDVV YHUVXV SDWK IUHTXHQF\ DFFRUGLQJ WR WKH H[DFW DQDO\WLFDO VROXWLRQ DQG WKDW RI WKH DSSUR[LPDWH ILUVWRUGHU V\VWHPDW]HURORQJLWXGLQDOVOLS
LPSURYH ZKHQ WKH ILOWHULQJ DFWLRQ RI WKH FDUFDVV FRPSOLDQFH LV WDNHQ LQWR DFFRXQW,QWKDWFRQILJXUDWLRQZHKDYHWKHZKHHOVOLSYHORFLW\9V[WKDWDFWVDVWKH LQSXWTXDQWLW\$GGLQJWKHWLPHUDWHRIFKDQJHRIWKHFDUFDVVGHIOHFWLRQSURGXFHV WKHVOLSYHORFLW\RIWKHEUXVKPRGHO7KHFDUFDVVGHIOHFWLRQXHTXDOV ZLWKIRUH DQGDIWFDUFDVVVWLIIQHVVF[LQWURGXFHG
'
X
) F
[
[
7KHIHHGEDFNORRSLQWKHDXJPHQWHGV\VWHPDSSDUHQWO\FRQWDLQVDJDLQHTXDOWR
T
L
F[7KHUHVXOWLQJFRPSOHWHIUHTXHQF\UHVSRQVHIXQFWLRQUHDGV
V
) L
+
V
) F L
V
+
'
%
) F L
V L
+
V F[
:KHQWKHDSSUR[LPDWHILUVWRUGHUV\VWHPLVHPSOR\HGIRUWKHFRQWDFWPRGHOWKH IUHTXHQF\UHVSRQVHIXQFWLRQIRUWKHFRPSOHWHPRGHOEHFRPHVXVLQJ ZLWK
) L
+
V
&
'
%
) F
L
V
ZKHUHWKHWRWDOUHOD[DWLRQOHQJWKKDVEHHQLQWURGXFHGZKLFKDSSDUHQWO\UHDGV
'
F
%
&
) F
F
[
'
DP
%
&
)
F
[
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
7KHORFDOORQJLWXGLQDOVOLSVWLIIQHVVRIWKHFRQWDFWSDWFKLVKHUHHTXDOWRWKHORFDO ORQJLWXGLQDOVOLSVWLIIQHVVRIWKHFRPSOHWHPRGHOZKLFKREYLRXVO\LVGXHWRWKH DVVXPHG LQH[WHQVLELOLW\ RI WKH EDVH OLQH RI WKH EUXVK PRGHO 6R ZH KDYH DW VWHDG\VWDWH
M)[
'
M
'
&
)
&
) F
$WYDQLVKLQJVOLSZHZLOODGGDVXEVFULSW :KHQWRWDOVOLGLQJRFFXUVERWK DQG&) UHGXFHWR]HURDQGWKHWRWDOUHOD[DWLRQOHQJWK
FYDQLVKHV
F
)LJXUH SUHVHQWV WKH IUHTXHQF\ UHVSRQVH IXQFWLRQ IRU WKH WRWDO PRGHO DW YDQLVKLQJVOLS
6
R
,WLVQRWHGWKDWFRPSDUHGZLWKWKHFXUYHVRI)LJ WKH
ZDY\SDWWHUQLVFRQVLGHUDEO\UHGXFHG7KHDSSUR[LPDWHV\VWHPSHUIRUPVYHU\ ZHOO$ZDYHOHQJWKRIFPRFFXUVDWWKHSDWKIUHTXHQF\
T
V
FDUDGP
7KH GLIIHUHQWLDO HTXDWLRQ WKDW JRYHUQV WKH WUDQVLHQW VOLS UHVSRQVH RI WKH FRQWDFWSDWFKDQGWKURXJKWKDWWKHORQJLWXGLQDOIRUFHUHVSRQVHEHFRPHVIRUWKH DSSUR[LPDWHV\VWHP a
F
G F
GW
%_
9
a
[_ F
'
a
_9[_ F
'&
a
9
V[F
7KHYDULDWLRQRIWKHIRUFHEHFRPHV a
)
[
'
a
&
) F F
7KHVWUXFWXUHRI(T FRUUHVSRQGVZLWKWKDWRI(T RI&KDSWHU7KH UHVSRQVHKRZHYHULVLQVHQVLWLYHWRORDGYDULDWLRQVEXWVKRZVDQLFHEHKDYLRXU 7KHWUDQVLHQWUHVSRQVHWRORDGYDULDWLRQVLVVXIILFLHQWO\WDNHQFDUHRIWKURXJKWKH HIIHFWRIFDUFDVVFRPSOLDQFH$VDUHVXOWWKHUHOD[DWLRQOHQJWKIRUWKHUHVSRQVH WRORDGYDULDWLRQVEHFRPHVHTXDOWR&)
F
F [ZKLFKLVVRPHZKDWVPDOOHUWKDQ
WKDWKROGVIRUWKHUHVSRQVHWRVOLSYDULDWLRQV :KHQWKHDYHUDJHVWHDG\VWDWHUHODWLRQIRUWKHORQJLWXGLQDOVOLS
_9[_ FR
'&
9
V[FR
LVDGGHGWR(T WKHHTXDWLRQIRUWKHWRWDOWUDQVLHQWVOLSLVREWDLQHG
F
G F
GW
%_
9
[_ F
'
_9[_ F
'&
9
V[F
WKDW FRPSOHWHO\FRUUHVSRQGVWR(T RIWKHHQKDQFHGQRQOLQHDUWUDQVLHQW W\UHPRGHO7KHWUDQVLHQWVOLS
NLVVXEVHTXHQWO\XVHGDVLQSXWLQWRWKHVWHDG\ F
VWDWH ORQJLWXGLQDO IRUFH FKDUDFWHULVWLF DV ZLOO EH H[SODLQHG ODWHU RQ DQG KDV DOUHDG\EHHQLQGLFDWHGLQ&KDS6HF
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
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
MYF a
MW
MY F a
&
9
UFR
M[
'
a
9
V\F
'&
a
9
[FR F
1RWHWKDWIRUWKHVDNHRIVLPSOLI\LQJLQWKHSUHVHQWFKDSWHUWKHQRWDWLRQWDQ
LV
ZLWK RU ZLWKRXW VXEVFULSW 6LPLODU DV IRU WKH ORQJLWXGLQDO
UHSODFHG E\
GHIOHFWLRQ ZH ILQG IRU WKH )RXULHU WUDQVIRUP 9F RI WKH ODWHUDO GHIOHFWLRQ UHVSRQGLQJ WR VOLS DQJOH YDULDWLRQV $F GHQRWLQJ WKH EUXVK PRGHO VOLS DQJOH¶V WUDQVIRUP RUDFWXDOO\RIWDQ a
9
F
'
L
V
&
&L
H
F
&[ 9[ 9UF
V D
a
$
F
$JDLQ WKH UHVSRQVHV WR YDULDWLRQV RI WKH VLGH VOLS RQO\ RFFXU LQ WKH UDQJH RI DGKHVLRQ7KHWUDQVLWLRQSRLQWIURPDGKHVLRQWRVOLGLQJQRZRFFXUVDW
[
'
W
D
_ FR _
&
DQGWKHFRUUHVSRQGLQJDGKHVLRQIUDFWLRQEHFRPHVFI(T
' &
P
_ FR _ LI _ FR _
HOVH P
'
$VWKHVOLSDQJOHRIWKHFRQWDFWSDWFKUHPDLQVVPDOOLQWKHUDQJHZKHUHDGKHVLRQ VWLOORFFXUVFRV
FR
KDVEHHQUHSODFHGE\XQLW\%\LQWHJUDWLRQRIWKHWUDQVIRUPHG
GHIOHFWLRQ RYHU WKH UDQJH RI DGKHVLRQ WKH IUHTXHQF\ UHVSRQVH IXQFWLRQV RI WKH IRUFHDQGWKHPRPHQWYDULDWLRQVDUHHVWDEOLVKHG7KH\UHDGDWSXUHODWHUDOVOLS 9UF
9[
) F L
+
DQG
V
'
&
) F
L
DP
V
&
L
DP
V
&
& L VDP
H
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
0 F L
F
'
V
+
L
&
&
S
V
&
& L VDP
H
%
V
D L
% & P
& L VDP
H
%
D
P
&
P
V
7KHORFDOVORSHRIWKHVWHDG\VWDWHVLGHIRUFHFKDUDFWHULVWLFRIWKHEUXVKPRGHO LVJLYHQE\ &
) F
'
F P
S
D
LI _
FR _
HOVH &
) F
'
7KH QRUPDOLVHG UHVSRQVH IXQFWLRQ RI WKH VLGH IRUFH LV LGHQWLFDO WR WKDW RI WKH ORQJLWXGLQDOIRUFHLIWKHIDFWRU
FR
LVRPLWWHGLQ 7KHDSSUR[LPDWHILUVW
RUGHUGHVFULSWLRQZLWKUHVSRQVHIXQFWLRQ
) F L
V
+
ZKHUH
F
'
&
'
%
) F
L
V F
DP
VKRZVWKHVDPHYHU\JRRGDJUHHPHQWZLWKWKHH[DFWUHVXOWDVDVVHVVHGZLWKWKH ORQJLWXGLQDOIRUFHUHVSRQVH6LPLODUO\ZHKDYHWKHGLIIHUHQWLDOHTXDWLRQIRUWKH YDULDWLRQRIWKHWUDQVLHQWVLGHVOLS a
F
G F
GW
%_
9
a
[_ F
'
a
_9[_ F
'&
a
9
V\F
7KHYDULDWLRQRIWKHVLGHIRUFHEHFRPHV a
)
\
'
a
&
) F F
$IWHUDGGLQJWKHVWHDG\VWDWHHTXDWLRQ
_9[_ FR
'&
9
V\FR
WR(T WKHHTXDWLRQIRUWKHWRWDOWUDQVLHQWVLGHVOLSLVREWDLQHG
F
G F
GW
%_
9
[_ F
'
_9[_ F
$V EHIRUH WKH UHVXOWLQJ
'&
9
V\F
N LV XVHG DV WKH LQSXW RI WKH VWHDG\VWDWH VLGH IRUFH F
IXQFWLRQ :HPD\IROORZWKHVDPHSURFHGXUHWRDVVHVVWKHDOLJQLQJWRUTXHHTXDWLRQVDV ZDV GRQH LQ 6HF 7KH UHVXOWLQJ ILUVWRUGHU UHVSRQVH KRZHYHU GRHV QRW DOZD\VDJUHHZLWKWKHDQDO\WLFDOO\IRXQGWHQGHQF\7KHSKDVHODJDQGWKHVORSH RI WKH KLJK IUHTXHQF\ DPSOLWXGH DV\PSWRWH LQGLFDWH WKDW D VHFRQGRUGHU
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
EHKDYLRXU SUHYDLOV DW ]HUR DYHUDJH VOLS DQJOH ZKLOH DW ODUJHU VOLS DQJOHV WKH UHVSRQVHJUDGXDOO\FKDQJHVLQWRDILUVWRUGHUQDWXUH$SSDUHQWO\WKHPHFKDQLVP LV PRUH FRPSOH[ DQG ZH VKRXOG DFFRXQW IRU WKH WUDQVLHQW UHVSRQVH RI WKH SQHXPDWLFWUDLO7KHYDULDWLRQRIWKHDOLJQLQJWRUTXHPD\EHZULWWHQDVIROORZV a
0
]
'&
&
a
W
FR )\
a
W )
F \R
7KHDQDO\VLVFRQGXFWHGE\0DXULFH VKRZVWKDWWKHDQDO\WLFDOO\DVVHVVHG UHVSRQVHIXQFWLRQRIWKHSQHXPDWLFWUDLOYDULDWLRQWRVOLSDQJOHYDULDWLRQVFDQEH DSSUR[LPDWHG E\ WKH ILUVWRUGHU V\VWHP DQG D VRFDOOHG SKDVH OHDGLQJ QHWZRUNLQVHULHV7KHIUHTXHQF\UHVSRQVHIXQFWLRQRIWKHODWWHUUHDGV
S L V
+
% W F %
'
L
V
L
V
&
7KH IDFWRU LQ WKLV IRUPXOD UHSUHVHQWV WKH ORFDO VORSH RI SQHXPDWLF WUDLO FKDUDFWHULVWLF
MWF
W F'
&
M_
F_
ZKLFKDSSDUHQWO\LVDQHJDWLYHTXDQWLW\$FFRUGLQJWR0DXULFHDGHTXDWHYDOXHV IRUWKHSDUDPHWHUV
'
&
D
DQG
FDQEHREWDLQHGWKURXJKWKHIRUPXODH
_ FR_
RUDOWHUQDWLYHO\
'
W
F
DQG
'
&
P
7KHEORFNGLDJUDPRIWKHFXUUHQWV\VWHPJRYHUQHGE\WKHHTXDWLRQV LV SUHVHQWHG LQ WKH XSSHU GLDJUDP RI )LJ 7KH ORZHU GLDJUDP VKRZV DQ DOWHUQDWLYHVWUXFWXUHRIWKHVDPHV\VWHPWKHUHE\GLVSOD\LQJWKHH[WUDPRPHQW 0] ZKLFK LV JRYHUQHG E\ WKH UDWLR RI SDUDPHWHUV 7KLV UDWLR WHQGV WR LQILQLW\ZKHQIXOODGKHVLRQRFFXUV7KHQP DQG
FR
DVEHFRPHVFOHDUIURP
(T 7KHVLQJXODULW\LQYROYHGKDVEHHQFLUFXPYHQWHGE\0DXULFHWKURXJK WKHLQWURGXFWLRQRIDIXQFWLRQWKDWOLPLWVWKHYDOXHRI DURXQG]HURODWHUDO VOLS
FR
7KLVKRZHYHUZLOOVOLJKWO\GLVWXUEWKHSURSHUUHVSRQVHDW]HURVOLS
DQJOHDQGWKXVGHJUDGHVWKHOLQHDUDQDO\VLVDURXQG]HURVOLS$QDOWHUQDWLYHZD\ RI DYRLGLQJ WKH VLQJXODULW\ ZKLFK REYLRXVO\ LV FDXVHG E\ WKH IDFW WKDW ZH DFWXDOO\PD\KDYHDPRPHQWZLWKRXWWKHVLPXOWDQHRXVSUHVHQFHRIDIRUFHLVWKH
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
a )
)αF
\
&
αF a
1 1+ ωVσF
αF a
L
FR
W
phase leading network
1+ ωVσ1 WαF 1+ ω σ V 2 L
&
a W
F
-
a 0
]
\R
)
L
a )
\
)αF
&
αF a
1 1+ ωVσF
αF a
L
FR
W
αW
a
1 1+ ωVσ2 L
+ )LJ
WαF)\R
&
σ1 σ2 WαF \R &
)
--
a 0
]
∆ ] a
0
%ORFN GLDJUDP RI WKH FRQWDFW SDWFK PRGHO WR JHQHUDWH VKRUW ZDYHOHQJWKWUDQVLHQW UHVSRQVHVRIWKHVLGHIRUFHDQGWKHDOLJQLQJWRUTXHWRVPDOOVOLSDQJOHYDULDWLRQV7KH RULJLQDO XSSHU GLDJUDP FDQ EH UHSODFHG E\ WKH ORZHU GLDJUDP WKHUHE\ DYRLGLQJ VLQJXODULW\DW]HURVOLS
FRQVLGHUDWLRQRIWKHH[WUDWUDQVLHQWPRPHQW
0]7KLVPRPHQWLVREWDLQHGE\
PXOWLSOLFDWLRQRIWKHGLIIHUHQFHRIWKHWUDQVLHQWVOLSTXDQWLWLHVIRUWKHIRUFHDQG IRU WKH SQHXPDWLF WUDLO ZLWK WKUHH IDFWRUV WKH FRPELQDWLRQ RI ZKLFK PD\ EH GHVLJQDWHGZLWK&
&
0
'&
0
&
W F )\R
,WWXUQVRXWWKDWQRZWKHVLQJXODULW\GRHVQRWVKRZXSEHFDXVHERWK)\RDQG
EHFRPH]HURDWWKHVDPHWLPH7KLVLQGLFDWHVWKDWLQGHHGDPRPHQWPD\DULVH DOWKRXJK DW WKDW LQVWDQW RI WLPH WKH VLGH IRUFH LV ]HUR $IWHU ZULWLQJ RXW WKH IDFWRUVLQ E\XVLQJH[SUHVVLRQVIRUWKHVLGHIRUFH WKHSQHXPDWLF
2
WUDLO DQGIXUWKHU ZKLOHLQ LVUHSODFHGE\PDQG FR
]ZHILQGIRU&
0
\
\
E\
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
&
0
'
ZKHUH&0
F
'
D&
) F
&
0 F
GHQRWHVWKHDOLJQLQJVWLIIQHVVRIWKHEUXVKPRGHODW]HURVOLSDQJOH
DQGWKHQRQGLPHQVLRQDOIDFWRU
& &
' ZLWK
]
'
]
]
&
&
LVLQWURGXFHG
&
]
& %
]
]
]
LI ] HOVH
'
_ FR_
(YDOXDWLRQ RI UHYHDOV WKDW IRU ] WKH IDFWRU
PD\ SRVVLEO\ EH
DSSUR[LPDWHGE\
. &
]
7KLV DSSUR[LPDWLRQ ZLOO EH LQWURGXFHG ODWHU RQ ZKHQ ZH ZLOO GHDO ZLWK WKH DSSOLFDWLRQRIWKH0DJLF)RUPXOD 7KHGLIIHUHQWLDOHTXDWLRQVWKDWDSSO\IRUWKHFRQWDFWSDWFKPRGHOVXEMHFWHGWR VPDOOVLGHVOLSYDULDWLRQVZLWKUHVSHFWWRDJLYHQVLGHVOLSOHYHOQRZUHDG a
G F
F
GW
%_
9
[_ F
%_
9
[_ W
a
'
_9[_ F
'
_9[_ F
a
'&
a
9
V\F
a
G W
GW
a
a
IURPZKLFKWKHYDULDWLRQRIWKHIRUFHDQGPRPHQWUHVXOW a
)
\
a
0
]
'
a
&
) F F
'&
a
W
FR ) \
&
&
a
W F W ) \R
%
&
&
a
0 F
a
W
,WLVRILPSRUWDQFHWRFKHFNWKHVWHSUHVSRQVHVWRVLGHVOLSVWDUWLQJIURP]HURVOLS DQJOH HVSHFLDOO\ ULJKW DIWHU WKH VWDUW RI WKH VWHS FKDQJH ,QLWLDOO\ ZH KDYH DOO YDULDEOHVHTXDOWR]HURZKLOHWKHVOLSDQJOHLQSXWKDVUHDFKHGWKHQHZYDOXH
F
7KHYDULRXVWLPHGHULYDWLYHVEHFRPH W
'
G F
GW G)\ GW
F
'
'
_9[_
F
F
_9[_ F
F
F
'
&
) F
F
' W '
G W
GW
)
'
\
0
]
'
'
G0] GW
'&
W
F
G) \ GW
%
&
0
G F
GW
'
7KHVH UHVXOWV DUH FRUUHFW 7KH ODVW HTXDWLRQ KROGV LQGHHG EHFDXVH ZH KDYH DW
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
wheel plane base line
Y
sliding
α
F\
Fψ
β
9
αF
9V\F
YF
2DP
D
9V\
9[
D
0]
)\
)LJ 7KHEUXVKPRGHODWWDFKHGWRDFDUFDVVSRVVHVVLQJODWHUDODQGWRUVLRQDOFRPSOLDQFH
YDQLVKLQJVOLSDQJOHDFFRUGLQJWR(T ZKHUHWF
DQGFRQVHTXHQWO\&
0
W
F
&)
F
D(YLGHQWO\WKHH[WUDPRPHQWLVUHVSRQVLEOHIRUWKHSURSHUVWDUW
RIWKHFRXUVHRIWKHDOLJQLQJWRUTXHVKRZLQJ]HURVORSH7KLVSURSHUW\KDVEHHQ DVFHUWDLQHGWRRFFXUERWKLQUHDOLW\DQGZLWKPRGHOVFI)LJVDQG$OVR WKHUHVSRQVHWR D ODWHUDOZKHHOGLVSODFHPHQW\
I9
V\
GWDW]HURIRUZDUGVSHHG
GHYHORSVFRUUHFWO\:HILQGZLWK(T IRUWKHIRUFH )\ PRPHQW 0]
&)\ \DQGIRUWKH
7KHHTXDWLRQVPD\QRZIXUWKHUEHDSSUDLVHGE\ILUVWLQWURGXFLQJW\UHFDUFDVV ODWHUDODQGWRUVLRQDOFRPSOLDQFHDVGHSLFWHGLQ)LJ DQGLQWKHFRUUHVSRQGLQJ EORFNGLDJUDPRI)LJ DQGVXEVHTXHQWO\FRPSDULQJWKHUHVXOWVZLWKDQDO\WLFDO VROXWLRQVREWDLQHGE\XVLQJ(TV $V D UHIHUHQFH WKH VWHDG\VWDWH FKDUDFWHULVWLFV RI WKH PRGHO KDYH EHHQ SUHVHQWHG LQ )LJ 7KH GLDJUDP FRQWDLQV WKH FXUYHV IRU ERWK WKH FRPSOHWH PRGHODQGIRUWKHEUXVKPRGHODORQH,WLVRILQWHUHVWWRQRWHWKHORZHUFRUQHULQJ VWLIIQHVVGXHWRWKHLQWURGXFWLRQRIFDUFDVVFRPSOLDQFH7KHH[SUHVVLRQIRUWKH ORZHUVLGHVOLSVWLIIQHVVFDQEHIRXQGWRUHDG
&
)
'
F
&
%
F
&
0 F
) F
7KHUHOD[DWLRQOHQJWKRIWKHFRPSOHWHPRGHOIRXQGIURPWKHFXWRIIIUHTXHQF\RI WKHVLGHIRUFHUHVSRQVHIXQFWLRQWXUQVRXWWREHFRPH
'
F F
%
DP &
0 F
%
&
) F
F
\
'
F F
%
DP &
0 F
%
&
)
F
\
7KHZD\WKHUHOD[DWLRQOHQJWKFKDQJHVZLWKVOLSDQJOHLVGHSLFWHGLQWKHULJKW KDQGGLDJUDPRI)LJ $VRQHPLJKWH[SHFWZHILQGWKDWWKLVOHQJWKPXOWLSOLHG ZLWK WKH LQFUHPHQW LQ ZKHHO VOLS DQJOH LV HTXDO WR WKH LQFUHDVH RI WKH VXP RI FDUFDVV ODWHUDO GHIOHFWLRQ DQG WKH DYHUDJH GHIOHFWLRQ RI WKH DGKHULQJ WUHDG
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
tyre α
a
+
\
)
)αF
&
αF a
+ +
a
tread 1 1+ ωVσF
-
L
αF a
FR
W
αW 1 1+ ωVσ2
a
L
-
+
.
Y
a
WαF)\R
&
a
--
]
0
0
&
∆
ωV
L
\
F
β
a
1 ψ
F
)LJ
%ORFN GLDJUDP RI WKH DXJPHQWHG V\VWHP LQFOXGLQJ FDUFDVV ODWHUDO DQGWRUVLRQDO FRPSOLDQFH
HOHPHQWV $WWKUHHGLIIHUHQWOHYHOVRIVLGHVOLS
R
DQGUDGWKHFRPSDULVRQ
RIWKHVLPXODWLRQPRGHOZLWKWKHDQDO\WLFDOPRGHOKDVEHHQFRQGXFWHGLQWHUPV RI WKH SDWK IUHTXHQF\ UHVSRQVH IXQFWLRQV )LJXUH VKRZV WKH UHVXOWV 7KH JHQHUDO FRQFOXVLRQ LV WKDW DW OHDVW IRU ZDYHOHQJWKV ODUJHU WKDQ FP WKH
)\ [kN]
4
[m] 0.4
)\
-0.10]
with without carcass compliance
[Nm] 2
0]
W
W
[cm]
σF
0
0 0 )LJ
σ
0.1 tanα
0.2
0
0.1 tanα
0.2
6WHDG\VWDWHVLGHVOLSIRUFHDQGPRPHQWFKDUDFWHULVWLFVDQGWKHUHOD[DWLRQOHQJWKRI WKHEUXVKPRGHO WKHFRQWDFWSDWFK DQGRIWKHPRGHOLQFOXGLQJWKHIOH[LEOHFDUFDVV WKURXJKZKLFKWKHEUXVKPRGHOLVDWWDFKHGWRWKHZKHHOSODQH
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
FRUUHVSRQGHQFHFDQEHMXGJHGWREHYHU\JRRG7KHXSSHUSDLURIGLDJUDPVWKDW UHIHUWRWKHVLGHIRUFHUHVSRQVHVKRZVWKHVLGHZD\VµVKLIW¶RIWKHSKDVHODJFXUYHV ZKLFK LV FDXVHG E\ WKH GURS LQ UHOD[DWLRQ OHQJWK ZLWK LQFUHDVLQJ DYHUDJH VOLS DQJOH7KHGLDJUDPIRUWKHDOLJQLQJWRUTXHUHVSRQVHFOHDUO\VKRZVWKHWUDQVLWLRQ IURPVHFRQGWRILUVWRUGHUEHKDYLRXUZKHQWKHDYHUDJHVOLSDQJOHFKDQJHVIURP ]HURWRODUJHUYDOXHV
5
10
)\
4
10
3
10
N rad
analytic solution simulation model
2
10
0
o
-30
φ)α
λ =0.2m
o
o
-60
αo = 0.00 rad 0.08 rad 0.16 rad
o
-90 -1 10
10
0
ωV
1
10
2
[rad/m] 10
3
10
3
0]
10
2
10
1
10
Nm rad 100 -1
αo = 0.00 rad 0.08 rad 0.16 rad
10
λ =0.2m
o
180
φ α 0
o
90
analytic solution simulation model
o
0
o
-90 -1 10
10
0
ωV
1
10
[rad/m] 10
2
3
10
)LJ )UHTXHQF\UHVSRQVHIXQFWLRQVRIOLQHDULVHG V\VWHPLQFOXGLQJFDUFDVVIOH[LELOLW\DW GLIIHUHQW DYHUDJH VOLS DQJOH OHYHOV DFFRUGLQJ WR WKH DQDO\WLFDO VROXWLRQ DQG WR WKH DSSUR[LPDWH VLPXODWLRQ PRGHO 7KH SDWK IUHTXHQF\ DW D ZDYHOHQJWK RI WKH LQSXW ZKHHOSODQHPRWLRQHTXDOWRFPKDVEHHQLQGLFDWHG
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
V\F
9
αF =
9[
αF
αF 9V\F σF
αW =
\
)
σF
σ
[ (α W αF ) σ
9
WF
-
0
&
]
0
-
&∆0
∆
-
pneumatic trail
+
)LJ
x
αW
αF
side force
∆0] extra transient moment
x
%ORFNGLDJUDPRIWKHQRQOLQHDUPRGHORIWKHFRQWDFWSDWFK WUHDG DOVRYDOLGIRU ODUJHUVOLSDQJOHYDULDWLRQV
7KH(TV FDQEHPDGHDSSOLFDEOHWRWKHJHQHUDOFDVHRIODUJHVOLS DQJOH YDULDWLRQV E\ DGGLQJ WR WKH VWHDG\VWDWH UHODWLRQV DQG E\ UHZULWLQJ WKH (TV LQ FRPSOHWH QRQOLQHDU IRUP VR WKDW ZKHQ FRQVLGHULQJDVPDOOYDULDWLRQWKHOLQHDULVHGHTXDWLRQVDUHUHFRYHUHG:HREWDLQ IRUWKHWUDQVLHQWVOLSTXDQWLWLHV
F
G
F % _9
[_ F '
_9[_
F ' & 9V\F
W % _9 _ ' [ W GW
_9[_
F
GW G
DQGIRUWKHIRUFHDQGPRPHQW
\ ' )\ F 0 ' & W ) % & ] F W \ 0 F & W
)
7KHODVWWHUPUHSUHVHQWLQJWKHH[WUDPRPHQWLVOHIWLQOLQHDULVHGIRUP7KLVWHUP PD\EHUHSODFHGE\WKHGLIIHUHQFHRIWZRHTXDOIXQFWLRQVWKHGHULYDWLYHRIZKLFK HTXDOV&
0
RQHZLWKDUJXPHQW
NDQGWKHRWKHUZLWK N%HFDXVHRIWKHIDFWWKDW F
W
WKHGLIIHUHQFHRIWKHVHWZRDUJXPHQWVUHPDLQVVPDOOWKHOLQHDULVHG YHUVLRQLV H[SHFWHGWREHVXIILFLHQWO\DFFXUDWH7KHEORFNGLDJUDPRIWKHQRQOLQHDUV\VWHP GLVSOD\HGLQ)LJ PD\IXUWKHUFODULI\WKHVWUXFWXUHRIWKHPRGHO 7RHQVXUHWKDWWKHDERYHHTXDWLRQVFRUUHFWO\GHVFULEHWKHUHVSRQVHWRODUJH VOLSDQJOHYDULDWLRQVWKHVLPXODWLRQPRGHOUHVXOWVKDYHEHHQFRPSDUHGZLWKWKH UHVSRQVH RI D SK\VLFDO PRGHO 7KDW PRGHO IHDWXUHV D ILQLWH QXPEHU RI WUHDG
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
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
DQGPWZRDYHUDJHVOLSDQJOHOHYHOV
DQGRQHDPSOLWXGH Ö
R
DQGUDG
UDG
7DEOH 3DUDPHWHUYDOXHVXVHGIRUEUXVKPRGHOZLWKIOH[LEOHFDUFDVV
YHUWLFDOORDG
)]
1
IULFWLRQFRHIILFLHQW
KDOIFRQWDFWOHQJWK
D
P
ORQJLWXGLQDOFDUFDVVVWLIIQHVV
F[
î
ODWHUDOFDUFDVVVWLIIQHVV
F\
î
WRUVLRQDOFDUFDVVVWLIIQHVV
F
WUHDGHOHPHQWVWLIIQHVVP
FS
FRPSRVLWHW\UHSDUDPHWHU
θ
1P
1P
1PUDG
1P
)LJXUH SUHVHQWVWKHUHVXOWVIRUWKHWZRPRGHOV7KHUDQJHRIWKHDEVFLVVD KDV EHHQ FKRVHQ VXFK WKDW SUHFLVHO\ WZR ZDYHOHQJWKV DUH FRYHUHG DQG WKH GLVWDQFH UROOHG LV ODUJH HQRXJK WR KDYH D VLWXDWLRQ FORVH WR WKH SHULRGLF VWDWH $JDLQWKHDJUHHPHQWLVTXLWHJRRGDQGZHPD\KDYHFRQILGHQFHLQWKHPRGHO 7KH ERWWRP GLDJUDP SUHVHQWV WKH YDULDWLRQ RI WKH VOLS DQJOH 7KH WRS SDLU RI GLDJUDPVVKRZVWKHUHVSRQVHVDWWKHUHODWLYHO\ORQJZDYHOHQJWKRIPVRWKDWD FRQGLWLRQFORVHUWRVWHDG\VWDWHKDVEHHQUHDFKHG)URP)LJ LWFDQEHVHHQ WKDWDWWKHPD[LPXP
E
RI UDGRUDOPRVW WKHDOLJQLQJWRUTXHSHDNKDV
EHHQ VXUSDVVHG E\ IDU 7KLV H[SODLQV WKH WZR GLSV SHU ZDYHOHQJWK $W D PD[LPXPRIUDGWKHSHDNKDVMXVWEHHQVXUSDVVHG 7KH OHVV GHHS GLSV RFFXUULQJ LQ WKH XSSHU FXUYHV RI GLDJUDPV E DQG G DUH GHOD\HGZLWKUHVSHFWWRWKHVOLSDQJOHZKHQWKLVKDVUHDFKHGLWVPLQLPXPYDOXH HTXDOWR]HUR$WVWHDG\VWDWHZKHQ
6 4WKHPRPHQW DQGWKHIRUFH ZRXOG
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
4
30
)\
0] [Nm]
[kN]
0
0
λ =5m
λ =5m
-4
-30
D
E
4
30
)\
0] [Nm]
[kN]
0
0
λ =1m
λ =1m
physical model
-4
G
4
30
)\
0] [Nm]
λ =0.2m
0
[o]
λ =0.2m
0
10
α
physical model
-30 F
[kN]
10
α
0
[o]
-10 8
9
V
H
λ
10
0
-10 8
9
λ
V
10
I
)LJ 6LGHIRUFHDQGDOLJQLQJWRUTXHUHVSRQVHVRIWKHQRQOLQHDUEUXVKVLPXODWLRQPRGHO ZLWKIOH[LEOH FDUFDVV WR VLQXVRLGDOVOLSDQJOHLQSXWRIWKHZKHHOSODQHZLWKDVOLS DQJOHDPSOLWXGHRIUDGDQGDYHUDJHOHYHOVRI WKLQ FXUYHV DQGUDG IDW FXUYHV FRPSDUHGZLWKUHVXOWVRIWKHSK\VLFDOPRGHO EURNHQFXUYHV
KDYHEHFRPHHTXDOWR]HURDWWKDWLQVWDQW7KHGHHSHUGLSVEHORQJWRWKHPD[LPD RI WKH VOLS DQJOH YDULDWLRQ 7KH FRQVLGHUDEOH UHGXFWLRQ LQ DPSOLWXGH RI WKH UHVSRQVHDWWKHVKRUWHUZDYHOHQJWKVQLFHO\DJUHHVZLWKWKHILQGLQJVRI)LJ 7KH IRUFH UHVSRQVHV FRUUHVSRQG DOPRVW SHUIHFWO\ ZLWK WKH RXWFRPH RI WKH SK\VLFDOPRGHO(YLGHQWO\DVLPLODUFRUUHVSRQGHQFHLVH[SHFWHGWRRFFXUZLWKWKH
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
UHVSRQVHRIWKHORQJLWXGLQDOIRUFHWRORQJLWXGLQDOVOLSYDULDWLRQV
7XUQ6OLS 7KH ODVW LWHP WR EH VWXGLHG LQ WKH GHYHORSPHQW RI WKH FRQWDFW PRGHO LV WKH UHVSRQVHWRYDULDWLRQVLQSDWKFXUYDWXUHZKLOHWKHVOLSDQJOHUHPDLQV]HUR:H ZLOO QRW DWWHPSW WR GHYHORS D EDFNJURXQG DQDO\WLFDO PRGHO EXW WDNH D PRUH KHXULVWLF URXWH 7KH UHVXOWV ZLOO EH FKHFNHG E\ FRPSDULQJ WKHVH ZLWK WKH FRPSXWHGUHVSRQVHVRIWKHSK\VLFDOPRGHO7KHUHVSRQVHVGHULYHGIRUWKHVWULQJ PRGHO ZLWK WUHDG HOHPHQWV DV SUHVHQWHG LQ &KDSWHU HVSHFLDOO\ WKH VWHS UHVSRQVHVWR
n DVGHSLFWHGLQ)LJFDQGG H[WUHO DQG)LJPD\EH
KHOSIXO :H REVHUYH WKDW WKH IRUFH UHVSRQVH LV VLPLODU WR WKDW RI WKH DOLJQLQJ WRUTXHWRDVWHSFKDQJHLQVOLSDQJOH,QERWKFDVHVWKHVORSHDWWKHVWDUWLV]HUR 7KHIXUWKHUDSSURDFKRIWKHIRUFHWRLWVVWHDG\VWDWHOHYHOLVDVVXPHGWRRFFXU DFFRUGLQJWRWKHILUVWRUGHUHTXDWLRQZLWK
F
DVUHOD[DWLRQOHQJWK7KH]HURVORSH
DWWKHVWDUWPD\EHPRGHOOHGE\VXEWUDFWLQJDUHVSRQVHFXUYHWKDWVWDUWVZLWKWKH VDPHVORSHEXWGLHVRXWDIWHUKDYLQJUHDFKHGLWVSHDN6XFKDVKRUWWHUPUHVSRQVH PD\EHREWDLQHGE\WDNLQJWKHGLIIHUHQFHRIWZRUHVSRQVHVHDFKOHDGLQJWRWKH VDPH OHYHO EXW VWDUWLQJ DW GLIIHUHQW VORSHV WKH GLIIHUHQFH RI ZKLFK VKRXOG FRUUHVSRQG WR WKH LQLWLDO VORSH RI WKH XQFRUUHFWHG IRUFH UHVSRQVH FXUYH )RU VLPSOLFLW\ZHWDNHIRURQHRIWKHWZRUHVSRQVHVWKHXQFRUUHFWHGIRUFHUHVSRQVH VR WKDW
)
HTXDOWR YHORFLW\
F
F
F
)
0
GW
7KH UHOD[DWLRQ OHQJWK RI WKH VHFRQG UHVSRQVH VKRXOG WKHQ EH
7KHUHVXOWLQJHTXDWLRQVIRUWKHIRUFHUHVSRQVHWRWKHWXUQVOLS
UHDG
F
GnF
)
%_
9
Gn)
GW
[_ nF '
&0 F
[_ n) '
%_
9
&0 F
7KHWUDQVLHQWWXUQVOLSIRUWKHIRUFHILQDOO\EHFRPHV
n) ' nF & n)
& n) ' nF & n)
DQGWKHVLGHIRUFHDWSXUHSDWKFXUYDWXUHLVREWDLQHGIURPWKHQRQOLQHDUVWHDG\ VWDWHUHVSRQVHIXQFWLRQ
\'
)
)
\ n)
)RUWKHUDQJHRIWXUQVOLSD_ )
\'
n_
)nFn) '&0 Fn) '
&
WKHUHODWLRQUHPDLQVOLQHDUDQGHTXDOV F D
S
n)
7KHPRPHQWUHVSRQVHPD\EHGLYLGHGLQWRWKHUHVSRQVHGXHWRWKHFRQWDFWSDWFK OHQJWKWKDWLQYROYHVODWHUDOWUHDGGHIOHFWLRQVDQGWKHUHVSRQVHGXHWRWUHDGZLGWK
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
JLYLQJ ULVH WR ORQJLWXGLQDO GHIOHFWLRQV )LUVW ZH ZLOO DGGUHVV WKH PRPHQW JHQHUDWHGE\WKHEUXVKPRGHOZLWK]HURZLGWK)LJXUHV DQGLQGLFDWH WKDWZHDUHGHDOLQJKHUHZLWKDUHVSRQVHWKDWDIWHUKDYLQJUHDFKHGDSHDNWHQGV WR ]HUR $JDLQ ZH PD\ PRGHO WKLV EHKDYLRXU E\ VXEWUDFWLQJ WZR ILUVWRUGHU UHVSRQVHV )RU WKLV ZH LQWURGXFH WZR WUDQVLHQW WXUQ VOLS TXDQWLWLHV ZLWKUHVSHFWLYHUHOD[DWLRQOHQJWKV
n
n
DQG
n¶ DQG n¶
7KHWZRGLIIHUHQWLDO HTXDWLRQV
EHFRPH
Gn
n
GW Gn
n
GW
%_
[_ n '
&0 F
%_
[_ n '
&0 F
9
9
$W]HURVSHHGWKHUHVSRQVHRIWKHGLIIHUHQFHZRXOGEHFRPH
n & n ' &
n
&
F
n
7KHGHIOHFWLRQDQJOHRIWKHWUHDGGXHWRWUDQVLHQWVSLQLVGHILQHGDV
0 ' & D n & n
ZKLFKDW]HURVSHHGEHFRPHV
0 ' D
&
n
F
n
7KHFRQGLWLRQWREHVDWLVILHGLVWKDWWKHGHIOHFWLRQDQJOHLVHTXDOWRWKH\DZDQJOH +HQFHZHKDYH
n
&
n
'
D
,QWKHFDVHRIVPDOODQJOHVZHPD\ZULWHIRUWKHPRPHQW
] ' & &0 F 0
0
)URP(TV LWFDQEHDVVHVVHGWKDWWKHLQLWLDOVORSHRIWKHUHVSRQVHRIWKH PRPHQWWRDVWHSFKDQJHLQVSLQ
G0] GV
' &
D&
0 F
n
nF '& 0 F 9[ IURP]HURWRn
&
n
nF
'&
F
WXUQVRXWWREH
0 F nF
&
)RUWKHVHFRQGFRQGLWLRQWRDVVHVVWKHUDWLR¶VRIWKH
n¶VWRKDOIWKHFRQWDFWOHQJWK
DWKHEHVWILWRIWKHUHPDLQLQJFRXUVHRIWKHVWHSUHVSRQVHPD\VHUYH :KHQWKHDQJOHRIURWDWLRQ
FRQWLQXRXVWRJURZWKHVWDWHRIWRWDOVOLGLQJ
ZLOOEHDWWDLQHGDQGWKHPRPHQWFDQEHFDOFXODWHGWREHFRPH
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
]n 4 ' &
0
D )
]
,WKDVQRWEHHQWULHGWRGHULYHWKHIXQFWLRQDOUHODWLRQVKLSEHWZHHQPRPHQWDQG LQFUHDVLQJVWHHUDQJOHIRUWKHVWDQGLQJ QRQUROOLQJ EUXVKPRGHO7KHIROORZLQJ QRQOLQHDUIXQFWLRQWRGHVFULEHWKHPRPHQWUHVSRQVHLQEHWZHHQWKHWZRH[WUHPHV KDVEHHQFKRVHQ
] ' & 0]n 4 VLQ DUFWDQ
0
7KH IDFWRU
&
0 F ]n 4
0
gn 0
gn KDV EHHQ LQWURGXFHG WR KDYH D SDUDPHWHU DYDLODEOH WR EHWWHU
DSSURDFKWKHUHVSRQVHVKRZQE\WKHSK\VLFDOPRGHO7KHYDOXHDSSHDUHG WREHDSSURSULDWH6LPLODUWRWKHUHOD[DWLRQOHQJWK
n
(TV WKHOHQJWKV
F
DUH UHGXFHG LQ SURSRUWLRQ ZLWK WKH PDJQLWXGHV RI WUDQVLHQW WXUQ VOLS
TXDQWLWLHV
n¶DQGn¶
)RUWKHHYDOXDWLRQRIWKHPRGHODFRPSDULVRQZLWKWKHSK\VLFDOEUXVKPRGHO KDVEHHQH[HFXWHG)LUVWWKHIOH[LEOHFDUFDVVLVDWWDFKHGWRWKHWUHDGPRGHOFI )LJ 7RVLPXODWHWKLVPRUHFRPSOH[VLWXDWLRQWKHDSSURDFKRIWKHHQKDQFHG WUDQVLHQWPRGHORI6HFKDVEHHQDGRSWHG7KHDGGLWLRQDOG\QDPLFHTXDWLRQV IRU WKH FRQWDFW SDWFK ZLWK PDVV
PF
DQG PRPHQW RI LQHUWLD
,F
DQG FDUFDVV
VWLIIQHVVHVF\ DQGGDPSLQJFRHIILFLHQWVN\ UHDG
F 9\F % 9 [ 0 F % N \ Y0 % F\ Y ' 0
P
Y 0
,
'
9
\F & 9V\
0 F F % N % F
0
'
9
V\F '
0
\
'
F& 0 9
)
V\ % Y0 & 9[
0
]
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
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
DQG DV D UHVXOW WKH EDVH OLQH LV \DZHG DQG WKXV LQGXFHV VLGH VOLS 7KLV HIIHFW YDQLVKHVDWVWHDG\VWDWHWXUQLQJ+RZHYHULIZHZRXOGDGGWKHHIIHFWRIWUHDG ZLGWKWKHVSLQWRUTXHDOVRDFWVDWVWHDG\VWDWHDQGWKHUHE\FRQWULEXWHVODUJHO\WR WKHVLGHIRUFHUHVSRQVHWRVSLQRIWKHFRPSOHWHPRGHO 'XHWRWKHFRPSOH[LW\WKDWDULVHVZKHQDGGLQJWUHDGZLGWKWRWKHEUXVKPRGHO
4
30
)\
0]
λ =5m
[Nm]
[kN]
0
0
λ =5m
-10 D
E
4
30
)\
0] [Nm]
[kN]
0
0
λ =1m
λ =1m
-4
-30
F
G
4
30
)\
0] [Nm]
[kN]
0
0
simulation model
λ =0.2m
λ =0.2m
physical model
-4 0.2
Dϕ
[ -]
-30 0.2
Dϕ
0
[- ]
-0.2 8
9
H
λ
V
0
-0.2
10
8
10
9 I
)LJ 6LGHIRUFHDQGDOLJQLQJWRUTXHUHVSRQVHVRIWKHQRQOLQHDUEUXVKVLPXODWLRQPRGHO ZLWKIOH[LEOHFDUFDVVWRVLQXVRLGDOSDWKFXUYDWXUHYDULDWLRQV WXUQVOLSDW
RIWKH
n RIDQGDYHUDJHOHYHOVRI WKLQFXUYHV
ZKHHOSODQHZLWKDQDPSOLWXGHRID
DQG IDWFXUYHV FRPSDUHGZLWKUHVXOWVRIWKHSK\VLFDOPRGHO EURNHQFXUYHV
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
)\
with without carcass compliance
[kN] 4
100] [kNm]
)\
2
0]
0 0
0.1
Dϕ
0.2
0.3
)LJ 6WHDG\VWDWHWXUQVOLSIRUFHDQGPRPHQWFKDUDFWHULVWLFVRIWKHEUXVKPRGHOERWKZLWK DQGZLWKRXWIOH[LEOHFDUFDVV7KHHIIHFWRIWUHDGZLGWKKDVQRWEHHQLQFOXGHG
FI &KDS LW KDV EHHQ GHFLGHG WR FRQVLGHU WKLV DVSHFW ZKHQ GHDOLQJ ZLWK WKH XOWLPDWHPRGHODGDSWHGWRWKHXVHRIWKH0DJLF)RUPXODLQWKHQH[WVHFWLRQ 7KHGLDJUDPVRI)LJ SUHVHQWWKHFRPSXWHGUHVSRQVHVWRYDU\LQJWXUQVOLS ,W LV VHHQ WKDW WKH FRUUHVSRQGHQFH ZLWK WKH SK\VLFDO PRGHO LV TXLWH JRRG 7KH GHIRUPDWLRQRIWKHPRPHQWUHVSRQVHFXUYHRFFXUULQJDWVKRUWHUZDYHOHQJWKVLV FDXVHG E\ WKH H[WUD PRPHQW
0] JHQHUDWHG WKURXJK WKH EDVH OLQH VOLS DQJOH
YDULDWLRQ
$V D UHIHUHQFH WKH VWHDG\VWDWH FKDUDFWHULVWLFV RI WKH IRUFH DQGPRPHQW
UHVSRQVH WR WXUQ VOLS DV FRPSXWHG IRU WKH VLQJOH URZ EUXVK PRGHO ZLWK DQG
n
ZLWKRXWFDUFDVVFRPSOLDQFHKDYHEHHQVKRZQLQ)LJ 8SWRD
WKH
DOLJQLQJ WRUTXH UHPDLQV ]HUR ZKLFK FDXVHV WKH FKDUDFWHULVWLFV IRU WKH FDVHV ZLWKRXWDQGZLWKIOH[LEOHFDUFDVVWREHFRPHLGHQWLFDO7KHUHPDLQLQJFRXUVHRI WKHFXUYHVIRUWKHV\VWHPLQFOXGLQJFDUFDVVFRPSOLDQFHKDVQRWEHHQFRPSXWHG DVWKDWSDUWOLHVRXWVLGHWKHUDQJHRIHYDOXDWLRQ
&RPELQHG6OLS 7RFRYHUWKHFDVHRIFRPELQHGVOLSDOVRLQFOXGLQJORQJLWXGLQDOVOLSWKHVWHDG\ VWDWHEUXVKPRGHOFKDUDFWHULVWLFVDUHWREHDGDSWHGDVIRUPXODWHGLQ&KDSWHU 6HF ,Q DGGLWLRQ WKH IDFWRU P WKDW LQGLFDWHV WKH IUDFWLRQ RI WKH FRQWDFW OHQJWKZKHUHDGKHVLRQRFFXUVDQGLVXVHGWRUHGXFHWKHUHOD[DWLRQOHQJWK
F
LVWR
EHDGDSWHGE\XVLQJWKHFRPSRVLWHPDJQLWXGHRIVOLSRIWKHFRQWDFWSDWFK
F
'
% F
% F
F
7KLV H[SUHVVLRQ KROGV EHFDXVH ZH KDYH DVVXPHG WKDW WKH EUXVK PRGHO LV LVRWURSLF8VLQJWKHPDJQLWXGHRIFRPELQHGVOLSDFFRUGLQJWR WKHIDFWRU
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
PFDQEHDVVHVVHG
P
' &
F LI F
HOVH P
'
0DXULFH IRXQG H[FHOOHQW DJUHHPHQW ZLWK WKH SK\VLFDO PRGHO IRU WKH FRPELQHGVOLSFDVHV
R
UDGDQG Ö
E
UDGDQG
R
DQG Ö
ZLWKDSKDVHGLIIHUHQFHRI DQGZDYHOHQJWKVRIDQGP $GGLQJWXUQVOLSZLOOLQIOXHQFHWKHFRPELQHGVOLSUHVSRQVHIXUWKHU7KHQH[W VHFWLRQDSSURDFKHVWKLVPDWWHULQDSUDJPDWLFZD\
7KH0RGHO$GDSWHGWRWKH8VHRIWKH0DJLF)RUPXOD
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µVHHV¶ D VOLS DQJOH WKDW GLIIHUV LV VPDOOHU IURPWKDWRIWKHZKHHOSODQH$VHWRIDGDSWHG0)SDUDPHWHUVPD\EH HVWDEOLVKHGRIIOLQHIRUWKHFRQWDFWSDWFKRUDQLWHUDWLRQORRSPD\EHLQFOXGHGWR DFKLHYHWKHFRUUHFWVWHDGVWDWHEHKDYLRXURIWKHWRWDOPRGHO$SUDFWLFDOZD\KDV EHHQIRXQGZKLFKHPSOR\VDILUVWRUGHUIHHGEDFNORRSZLWKVPDOOWLPHFRQVWDQW ,QVWHDGRILQWURGXFLQJDQDGGLWLRQDOILUVWRUGHUGLIIHUHQWLDOHTXDWLRQWKHDOUHDG\ SUHVHQW ILUVWRUGHU HTXDWLRQ IRU WKH WUDQVLHQW VLGH VOLS
N KDV EHHQ XVHG 7KH
GLDJUDP RI )LJ LOOXVWUDWHV WKH VHWXS $ VLPLODU DSSURDFK LV IROORZHG LQ 6HF WR DFFRXQW IRU WKH FDPEHU DQJOH RI WKH EHOW EHLQJ GLIIHUHQW IURP WKH
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
contact patch dynamics
wheel velocities
contact patch slip velocities
’static’ deflection tyre carcass angle compliance β st slip forces and moments
0DJLF )RUPXOD
first-order contact transient slip equations
quasi wheel slip input quantities
)LJ 'LDJUDPH[SODLQLQJWKHPRGHOVWUXFWXUHXVLQJWKH0DJLF)RUPXOD
FDPEHUDQJOHRIWKHZKHHOSODQH 7KHWUDQVLHQWVOLSILUVWRUGHUGLIIHUHQWLDOHTXDWLRQVOLVWHGEHORZDUHLGHQWLFDO WRWKRVHGHULYHGLQWKHSUHYLRXVVHFWLRQH[FHSWIRUWKHILUVWHTXDWLRQIRUWKHODWHUDO WUDQVLHQWVOLS WUDQVLHQWVOLSHTXDWLRQVIRUVLGHVOLSWXUQVOLSDQGFDPEHU
G
F
GW G
% _9 [_ ' & 9V\F & _9[_
W
GW
% _9[_ W '
nF
G
F
n)
G
GW
n
G
n
GW
n
G
n
[
% _9[_ nF ' & 0 F
GW
)
_9 _
VW
GW
% _9 [_ n) ' & 0 F
% _9 [_ n ' & 0 F
% _9 [_ n ' & 0 F
ZKHUHLQ WKHFDOFXODWHGGHIOHFWLRQDQJOHKDVEHHQXVHG
VW
' 0 ] F
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
FRPSRVLWHWUDQVLHQWVOLSTXDQWLWLHV
' nF & n)
( n0
' gn n % g F n n)
n & n
G\QDPLFFRQWDFWSDWFKHTXDWLRQV
0
P 9
F
\F
% 9 [ 0 F % N\ Y0 % F \ Y ' )\
' 9 \F & 9V\ , % N 0% F F F Y 0
' 0]
0
'
0
9
V\F
' 9V\ % Y0 & 9[
F
&0
VLPSOLILHGVLGHIRUFHDQGDOLJQLQJWRUTXH0DJLF)RUPXODH 0)
' '\ VLQ^& \ DUFWDQ %\ \ ` 0 ' &W) % 0 % 0 ] \ ] ]U & ' S \ &\ ' ' ) @ \ ] % ' . & ' \ \ \ \ . ' & @ \ ) \ ' % 6 +\ 6
` +\ ' '+\n VLQ ^&+\nDUFWDQ %+\n 5Rn) W ' ' FRV^& DUFWDQ % W W W W ` W ' W % 6 +W 6 +W ' ' ' T W '] 5R@ '
` Un ' ''UnVLQ^&'UnDUFWDQ %'Un 5Rn0 0 ' 0 . ' ]U ]n Un 0 ' & ] 0 & W ' FRV^DUFWDQ %\n 5Rn) ` ' FRV^DUFWDQ S.\n 5R n) ` ' FRV^DUFWDQ T'Wn 5Rn) ` )
\
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
IDFWRUVUHGXFHGZLWKVOLS
&
0
' &)
W
@ PD[ &
F
' D @ PD[ &
'
n
n
OLP
W
F
D
' E) ' En ' En
)
F
F
F
ZLWKW\UHFRPSRVLWHSDUDPHWHU
&
'
)
)
]
DQGWKHWRWDOPDJQLWXGHRIHTXLYDOHQWVLGHVOLS
'
% D gn
_ _
%
n & n _
_
ZKHUHLQWKHSUHVHQWDSSOLFDWLRQ ¶
%
&
)
)
&
_ _
%
nF _
E_
RWKHUSDUDPHWHUUHODWLRQV
&)
&)
F
WF
&0n F &0
F
&0
&0nF W
F
F
F
&0n FWF&)nF
&0D
&)
W&) F
W&)
&\
S&\
%W
T%]&)
&W
T&]
&)n &0n
F F
&)
'+\n
S+\n
%+\n
&)n 5R&+\n'+\n&)
&'Un
T'Un
''Un
0]n4VLQ &'Un
.]5nU &'Un''Un
0] n 4
T&Un5R)]
N\FULW
&0nF F &0 F
F &0
F
F
&0nW&)n 5R
%'Un
N
&)nF&) &0nFF
F
S+\n
.]5nU
WF
S'\n
&+\n
E &) WF&)
&)nF
&)
%\ n
F
T']5R
FULW
N\ N
% P F % , F
F
F
N N
N\FULW
N
FULW
\
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
7DEOH
3DUDPHWHU YDOXHV IRU W\UH PRGHO ZLWK 0DJLF )RUPXOD LQFOXGLQJTXDQWLWLHV LQWURGXFHGODWHURQ
) F F P ,
1
]
î 1P
\
î 1PUDG NJ
F
F
NJP
N
E
)
) F
)
(
En 9
]
P
gn
OLP
S T T T S
R
& D E 5
ORZ
D
P
&\
%]
&]
']
'\
n
gn
En P
PV
TF
S S S T T T
.\
n
+\
n
+\
n
&U
n
'U
'W
n
n
NJ
F
R
6WHDG\6WDWH6WHS5HVSRQVHDQG)UHTXHQF\5HVSRQVH&KDUDFWHULVWLFV 7RGHPRQVWUDWHWKHSHUIRUPDQFHRIWKHPRGHODQXPEHUW\SLFDOFKDUDFWHULVWLFV ZLOOEHSUHVHQWHG7KHK\SRWKHWLFDOVWHDG\VWDWHSXUHVLGHVOLSDQGSXUHWXUQVOLS RIWKHPRGHODUHVKRZQLQ)LJV 7KHVWHSUHVSRQVHJUDSKVRI)LJ VKRZWKHSURSHUVKDSHVRIWKHYDULRXV FXUYHVQRWDEO\WKHLQLWLDOKRUL]RQWDOWDQJHQWRIWKHUHVSRQVHFXUYHVRIWKHVLGH IRUFHWRWXUQVOLSDQGRIWKHPRPHQWWRVLGHVOLSDOVRVKRZQLQ)LJ QRWUHDG ZLGWK $OVRWKHSHDNRIWKHPRPHQWUHVSRQVHFXUYHWRVSLQDQGWKHGLSRIWKH
)\ [kN] 4
)\
VLGHVOLS
0
-0.1 ]
[kN] 4
0
-0.1 ]
)\
[Nm]
2
[Nm] 2
0]
W
WXUQVOLS
)\ 0]
W
[cm]
0
0 0
0.1
tanα
0.2
0.3
0
0.1
Dϕ
0.2
0.3
)LJ 6WHDG\VWDWHVLGHVOLSDQGWXUQVOLSIRUFHDQGPRPHQWFKDUDFWHULVWLFVRIWKHRYHUDOO W\UHPRGHODV GHILQHGE\WKH0DJLF)RUPXOD7KHHIIHFWRIWUHDGZLGWKKDVEHHQ LQFOXGHG
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
FXUYHRIWKHPRPHQWUHVSRQVHWRVWHHUDQJOHDUHH[KLELWHGDVH[SHFWHG )LJXUH SUHVHQWVWKHIUHTXHQF\UHVSRQVHIXQFWLRQVRIWKHOLQHDULVHGV\VWHP DW]HURDYHUDJHVLGHDQGWXUQVOLSZLWKWUHDGZLGWKHIIHFWLQFOXGHG7KHFXUYHV PD\ EH FRPSDUHG ZLWK WKRVH RI WKH VWULQJ PRGHO ZLWK WUHDG HOHPHQWV FI )LJV ZLWK]HURWUHDGZLGWK ,Q)LJ WKH1\TXLVWSORWVRIWKHPRPHQW UHVSRQVH WR VWHHU DQJOH KDYH EHHQ GHSLFWHG 7KH XSSHU GLDJUDP VKRZV WKH LQIOXHQFHRIWUHDGZLGWKE\FKDQJLQJWKHSDUDPHWHU
(
gn ,IHTXDOWR]HURWKHWKLQ
W\UHLVUHSUHVHQWHG7KHYDOXHFRUUHVSRQGVZLWKWKHEDVHOLQHFRQILJXUDWLRQ7KH ORZHUJUDSKSURYLGHVLQVLJKWLQWKHLQIOXHQFHRIWKHSDUDPHWHU
gn
WKHPDJQLWXGHRIWKHHIIHFWRIWKHWUDQVLHQW\DZGHIOHFWLRQDQJOH
WKDWJRYHUQV
0
7KH
YDOXHLVXVHGLQWKHEDVHOLQHFRQILJXUDWLRQ:KHQFRPSDUHGZLWKWKHSORWVRI )LJVDQGLWPD\EHFRQFOXGHGWKDWWKHPRGHOLVFDSDEOHRIDSSURDFKLQJ WKHUHVSRQVHVRIPRUHFRPSOH[LQILQLWHRUGHUPRGHOVDQGRIDFWXDOW\UHV
1
0.5
0
0
)\ψ &)α
)\α &)α )\ϕ &)ϕ 0.2
0.4
0.6
V
0.8
1
0.8
1
[m]
1
]ϕ 0ϕ
0 &
]α 0α
0
0.5
&
-1
]ψ
0
2 & α 0
0
0
0.2
0.4
V
0.6
[m]
)LJ 6WHSUHVSRQVHFXUYHVRIWKHVLGHIRUFHDQGRIWKHDOLJQLQJWRUTXHWRVLGHVOLS WXUQ
nDQGVWHHUDQJOH
VOLS SDWKFXUYDWXUH
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
)\ & αα )\ & ϕϕ )\ & αψ
1.0
)
)
0.5
)α )ϕ )ψ
0 0
φ)α φ) ϕ φ)ψ
0] &
)ψ
)α
)
-90
)ϕ
[o] -180 0
0.05
0.1
Dλ
0.15
0.2
0.25
0.15
0.2
0.25
1.5
α
0ϕ
α
0
1.0
0] &
ϕ
ϕ
0
0.5
&
ψ
α
0
φ0α φ0ϕ φ0ψ
0 α
0 ψ
0]
0 90
0 ψ 0
0ϕ -90
0 α -180 0
)LJ
0.05
0.1
Dλ
)UHTXHQF\UHVSRQVHIXQFWLRQRIV\VWHPZLWKWUHDGZLGWK&XUYHVIRUWKHIRUFHDQG
n DQGVWHHUDQJOH
PRPHQWUHVSRQVHVWRVLGHVOLS WXUQVOLS SDWKFXUYDWXUH EHHQLQGLFDWHG
KDYH
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
εϕ12 = 4
1
Im
0]
ψ
εϕ* = 0
0α
&
0.5 1
0.5
1
1.5
1
1
1
0.06
0
0
0.04 0.02
λ
D
-0.5
Re
0.01 0
0.5
1
0]
ψ
0α
&
1.5
2
1
Im
0]
εϕ12 = 4
ψ
0α
&
6
0.5
1
2 1
εϕ* =1
0 0.06 0
0 0.04 0.02
-0.5 0 )LJ
λ
D
Re
0.01 0.5
1
1.5
0]
0α
&
ψ 2
1\TXLVW IUHTXHQF\ UHVSRQVH SORWV RI WKH DOLJQLQJ WRUTXH WR VWHHU DQJOH8SSHU GLDJUDP LQIOXHQFHRIWUHDGZLGWK LVEDVHOLQHYDOXH ORZHUGLDJUDP LQIOXHQFHRI WKHWUDQVLHQW\DZGHIOHFWLRQUHVSRQVHWRWXUQVOLS LVEDVHOLQHYDOXH
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
/DUJH6OLS$QJOHDQG7XUQ6OLS5HVSRQVH6LPXODWLRQV
7KHVDPHVLQXVRLGDOPDQRHXYUHVKDVEHHQVLPXODWHGDVZDVGRQHEHIRUHZLWKWKH EUXVKEDVHGFRQWDFWPRGHO7KHFRPSOHWHQRQOLQHDUVHWRIHTXDWLRQV WR KDYHEHHQXVHGZLWKSDUDPHWHUVOLVWHGLQ7DEOH
30
4
)\
0] [Nm]
[kN]
0
0
λ =5m
λ =5m
-4
-30
D
4
E
30
)\
0] [Nm]
[kN]
0
0
λ =1m
λ =1m
-4
-30 F
G
4
30
)\ [kN]
0] [Nm]
λ =0.2m
0
0
10
α
[o]
10
α
0
-10 8
λ =0.2m
[o]
9
λ
V
10
H
0
-10 8
9
λ
V
10
I
)LJ 6LGHIRUFHDQGDOLJQLQJWRUTXHUHVSRQVHVRIWKH0DJLF)RUPXODEDVHGVLPXODWLRQ PRGHOZLWKIOH[LEOHFDUFDVVDQGILQLWHWUHDGZLGWKWRVLQXVRLGDOVOLSDQJOHLQSXWRI WKHZKHHOSODQHZLWKDVOLSDQJOHDPSOLWXGHRIUDGDQGDYHUDJHOHYHOVRI WKLQ FXUYHV DQGUDG IDWFXUYHV
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
7KH VOLS DQJOH YDULDWLRQ LV VLQXVRLGDO DURXQG DYHUDJH OHYHOV UDGZLWKRQHDPSOLWXGH Ö
UDGDWWKUHHZDYHOHQJWKV
6LPLODUO\ WKH WXUQ VOLS KDV WZR DYHUDJH OHYHOV DPSOLWXGH
Dn Ö
DOVRDWZDYHOHQJWKV
Dn
R
DQG
R
DQGP
DQG DQG RQH
DQGP
7KHGLDJUDPVRI)LJV SUHVHQWWKHUHVXOWV:HREVHUYHWKDWWKHFXUYHV
4
150
)\
0] [Nm]
[kN]
0
0
λ =5m
λ =5m
-4
-150 D
E
150
4
)\
0] [Nm]
[kN]
0
0
λ =1m
λ =1m
-4
-150 F
4
)\
G
150
λ =0.2m
[kN]
0
0
0.2
Dϕ
[-]
0.2
Dϕ
0
-0.2 8
λ =0.2m
0] [Nm]
[ -]
9
λ
10
V
0
-0.2 8
9
λ
10
V
H
I
)LJ 6LGHIRUFHDQGDOLJQLQJWRUTXHUHVSRQVHVRIWKH0DJLF)RUPXODEDVHGVLPXODWLRQ PRGHO ZLWK IOH[LEOH FDUFDVV DQG ILQLWH WUHDG ZLGWK WR VLQXVRLGDO SDWK FXUYDWXUH YDULDWLRQV WXUQVOLSDW
nRI
RIWKHZKHHOSODQHZLWKDQDPSOLWXGHRID
DQGDYHUDJHOHYHOVRI WKLQFXUYHV DQG IDWFXUYHV
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
DUHTXLWHVLPLODUWRWKRVHGHSLFWHGLQWKH)LJV DQG2QO\DVH[SHFWHGWKH PRPHQWUHVSRQVHWRWXUQVOLSLVYHU\PXFKDIIHFWHGE\WKHQRZLQWURGXFHGHIIHFW RIWKHW\UHWUHDGZLGWK 0DXULFHKDVFRQGXFWHGH[WHQVLYHH[SHULPHQWVZLWKD5 9W\UHDW EDULQIODWLRQSUHVVXUHRQDPGUXPWHVWULJ7KHGLDJUDPVRI)LJ D SUHVHQWWKHFRPSXWHGUHVXOWVFRPSDUHGZLWKH[SHULPHQWDOGDWDIRUWKHFDVHRI SXUHVLGHVOLS7KHH[SHULPHQWVKDYHEHHQFDUULHGRXWDWYHU\ORZVSHHGWRDYRLG LQHUWLDHIIHFWVRIWKHPRYLQJW\UH7KHZDYHOHQJWKUDQJHVIURPWRP7KH
E
XSSHUGLDJUDPUHIHUVWRWKHFDVHRI]HURDYHUDJHVOLSDQJOHDQG DPSOLWXGH7KH
E
ORZHUGLDJUDPVKRZVWKHUHVSRQVHVDURXQGDQDYHUDJHVOLSDQJOHRI ZKLFK FDXVHVWKHFXUYHVWRGHYLDWHFRQVLGHUDEO\IURPWKHLQSXWVLQXVRLGDOVKDSH7KH FXUYHVFOHDUO\VKRZWKDWDVKRUWHUZDYHOHQJWKFDXVHVWKHUHVSRQVHDPSOLWXGHWR
)\
)] = 4000 N
test
3
α o= 0o
model
α o= 0o
α$ = 4o
9 = 0.6 m/s
2 1
[kN] 0 -1
λ= 0.3 0.6 1.2 2.4m
-2 -3
0
0.2
0.4
λ
0.6
0.8
1
0
0.2
V
)\
λ
0.6
0.8
1
V
)] = 4000 N
test
5
0.4
α o= 4o
model
α o= 4o
α$ = 4o
9 = 0.6 m/s
4 3
[kN] 2
λ= 0.3 0.6 1.2 2.4m
1 0
0
0.2
0.4
λ
0.6
0.8
1
V
0
0.2
0.4
λ
0.6
0.8
1
V
)LJ D &RPSDULVRQRIWKHRUHWLFDOPRGHOFDOFXODWLRQVDQGH[SHULPHQWDOUHVXOWVSHUIRUPHG DWNPKRQD PGUXP 0DXULFH 7KHFXUYHVFRYHURQHZDYHOHQJWKRIWKH SHULRGLFUHVSRQVHVWRVLQXVRLGDOVOLSDQJOHYDULDWLRQV)RUFHUHVSRQVHWRVOLSDQJOH YDULDWLRQVDWWZROHYHOVRIVLGHVOLS
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
λ=
2.0
)]o = 4000 N )$] = 2000 N 9 = 0.6 m/s
0.3 0.6 1.2 2.4m
)\
80
model
α = 1o
0] 60
1.5
[Nm]
[kN]
40
1.0 20
0.5
0
2.0
80
)\
test
α = 1o
0] 60
1.5
[Nm]
[kN]
40
1.0 20
0.5 0
0.2
0.4
λ
0.6
0.8
0
1
0
0.2
0.4
V
6
λ=
5
0.3 0.6 1.2 2.4m
)\
V
)]o = 4000 N )$] = 2000 N 9 = 0.6 m/s
λ
0.6
150
1
0.8
model
α
0]
=5
o
100
[kN]
[Nm]
4
50 3
)\
2
0
6
150
test
α
0]
5
=5
o
100
[kN]
[Nm]
4
50 3
2
0
0.2
0.4
λ
0.6
0.8
1
0 0
0.2
V
0.4
λ
0.6
0.8
1
V
)LJ E &RPSDULVRQRIWKHRUHWLFDOPRGHOFDOFXODWLRQVDQGH[SHULPHQWDOUHVXOWVSHUIRUPHG DWNPKRQD PGUXP 0DXULFH 7KHFXUYHVFRYHURQHZDYHOHQJWKRIWKH SHULRGLF UHVSRQVHV WR VLQXVRLGDO ORDG YDULDWLRQV )RUFH DQG PRPHQW UHVSRQVH WR YHUWLFDOORDGYDULDWLRQVDWWZRYDOXHVRIVOLSDQJOH
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
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
E
DJUHHPHQWKDVEHHQHVWDEOLVKHG6LQFHDWWKHPD[LPXPVOLSDQJOHRI WKHSHDN RI WKH PRPHQW FKDUDFWHULVWLF KDV EHHQ VXUSDVVHG WKH UHVXOW EHFRPHV TXLWH VHQVLWLYH
WR
VPDOO
GLIIHUHQFHV
EHWZHHQ
DFWXDO
DQG
PRGHO
VWHDG\VWDWH
FKDUDFWHULVWLFVRIWKHDOLJQLQJWRUTXH
3DUNLQJ0DQRHXYUHV
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
0
(T ZDVXVHG7KHHTXDWLRQJRYHUQVWKHYDULDWLRQRIWKHDOLJQLQJWRUTXH WKDWDULVHVZKHQWKHQRQUROOLQJW\UHLVVWHHUHGDQGWKHVWHHUDQJOH
]
LVLQFUHDVHG
IURP]HURWRDQGEH\RQGWKHVWDWHRIIXOOVOLGLQJ8OWLPDWHO\WKHWRUTXHUHDFKHV WKH PDJQLWXGH WKDW ZRXOG DOVR DULVH ZKHQ WKH UROOLQJ W\UH LV VXEMHFWHG WR D FRQVWDQWUDWHRIWXUQLQJG VSHHG
9
W
G ZKLOHWKHVOLSDQJOHUHPDLQV]HUR DWDIRUZDUG
[WKDWGHFUHDVHVWR]HUR7KHQWKHUDGLXVRIWXUQ
5
UHGXFHVWR]HURDQG
WKXVWKHVSLQDSSURDFKHVLQILQLW\)LJXUH LOOXVWUDWHVWKHVLWXDWLRQ 7KH PLVVLQJ FKDUDFWHULVWLF ZLOO EH PRGHOOHG E\ XVLQJ D IRU WKLV SXUSRVH
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
ϕ= 1/5
5
Eqs.(9.105,104)
]ϕ
0 9
[
]ϕ
0
'Uϕ
'
Eqs.(9.115,116)
]ϕ 4
0
0
]ψ
0
ϕ ψ
4
ψ
(9[= 0)
0
]ψ
0
)LJ $SSURDFKLQJWKHPD[LPXPWRUTXHDWVWDQGVWLOOLQWZRZD\V E\GHFUHDVLQJWKH
5WR]HURE\LQFUHDVLQJWKHVWHHUDQJOH
WXUQUDGLXV
ZKLOHVWDQGLQJVWLOO
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¶V PRGHO LQ WKH H[LVWLQJ PRGHO VWUXFWXUH $ JUDGXDO WUDQVLWLRQ IURP WKH QHZ W\SH RI PRGHO WR WKH H[LVWLQJ RQH ZKHQ WKH VSHHG DSSURDFKHVDQGVXUSDVVHVDORZVSHHGWKUHVKROGKDVEHHQWDNHQFDUHRI 7KHSULQFLSOHRI9DQGHU-DJW¶VDSSURDFKLVWKDWDWDJLYHQUDWHRIFKDQJHRI WKH VWHHU LQSXW WKH JURZWK UDWH RI WKH W\UH DQJXODU GHIOHFWLRQ GHFUHDVHV LQ SURSRUWLRQZLWKDIXQFWLRQRIWKHUHPDLQLQJGLIIHUHQFHEHWZHHQWKHPD[LPXP DFKLHYDEOH GHIOHFWLRQ DQG WKH FXUUHQW GHIOHFWLRQ 7KH WRUVLRQDO VWLIIQHVV LV DVVXPHGWREHDFRQVWDQWDQGWKHUHVXOWLQJFKDUDFWHULVWLFRIWKHWRUTXHEHFRPHV VLPLODU WR D ILUVWRUGHU UHVSRQVH IXQFWLRQ 7KH FDOFXODWHG PRPHQW JUDGXDOO\ DSSURDFKHV LWV PD[LPXP YDOXH :KHQ WKH GLUHFWLRQ RI URWDWLRQ RI WKH ZKHHO DERXWWKHYHUWLFDOD[LVLVFKDQJHGWKHGLVWDQFHWRWKHQHZRSSRVLWHSHDNWRUTXH LVODUJHDQGDFFRUGLQJO\WKHUDWHRIUHGXFWLRQRIWKHPRPHQWLVODUJHDVZHOO,W LVWKLVIHDWXUHRIWKHPRGHOWKDWLVDWWUDFWLYHVLQFHDVLPLODUEHKDYLRXUKDVEHHQ IRXQG WR RFFXU ZLWK WKH DFWXDO W\UH VXEMHFWHG WR DQ DOWHUQDWLQJ OHIW DQG ULJKW
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
3.0
model experiment
2.0
0]
100 1.0 0
[Nm] -1.0 -2.0 -3.0 -20
-10
0
ψ
10 o [ ]
20
)LJ &DOFXODWHGDQGH[SHULPHQWDOO\DVVHVVHGYDULDWLRQRIWKHPRPHQWYHUVXVVWHHUDQJOH
)]
IRUDQRQUROOLQJW\UHSUHVVHGDJDLQVWDIODWSODWHDWDORDG
1
VHTXHQFHRIWXUQLQJ7KHHTXDWLRQVWKDWJRYHUQWKHPRPHQWJHQHUDWLRQDWVWDQG VWLOODUHDVIROORZV
0
] FR
0
' & & S / 0
]
0
S
' 0
'
]n4
0
0
/ 0
&
LI
VJQ
û& VJQ 0
F
)RUWKHSDUDPHWHUYDOXH
R
HOVH S
'
)LJ SUHVHQWVWKHFDOFXODWHGYDULDWLRQRIWKH
WRUTXHYHUVXVWKHVWHHUDQJOHFRPSDUHGZLWKH[SHULPHQWDOO\REWDLQHGUHVXOWVDV UHSRUWHGE\9DQGHU-DJW7KHQRQUROOLQJW\UH VL]H35 LVORDGHGWR 1RQDIODWSODWHDQGVXEVHTXHQWO\VWHHUHGDWDUDWHRIDQGGHJV7KH FRUUHVSRQGHQFHLVTXLWHJRRGSHUKDSVH[FHSWIRUWKHLQLWLDOSKDVHZKHUHWKHZKHHO VWDUWV WR EH VWHHUHG IURP WKH FRQGLWLRQ ZKHUH
0
]
7R LPSURYH WKH PRGHO
SHUIRUPDQFH 9DQ GHU -DJW VXJJHVWV WR XVH DQ H[SRQHQW GHSHQGV RQ WKH ODVW H[WUHPH RI WKH GHIOHFWLRQ DQJOH
F
R
WKH YDOXH RI ZKLFK
)RU SRVVLEOH IXUWKHU
UHILQHPHQWVRIWKHPRGHOZHUHIHUWRWKHRULJLQDOZRUN :KHQLQVWHDGRIWKHQHZDSSURDFKWKH0DJLF)RUPXODZRXOGEHXVHGZLWK WKH LQWHJUDWLRQ OLPLWDWLRQ DV VXJJHVWHG DFFRUGLQJ WR (TV D VKDUS SHDNZRXOG DULVHLQWKHFXUYHZKHUHWKHGLUHFWLRQRIWXUQLQJLVFKDQJHG$VD UHVXOWWKHPRPHQWGHFUHDVHVDWDPXFKVORZHUUDWHWKDQVKRZQE\WKHWHVWUHVXOW 7KHSUREOHPLVQRZKRZWRLQWHJUDWHWKHQHZPRGHOIHDWXUHLQWKHRULJLQDO PRGHOVWUXFWXUH7KHWUDQVLHQWVOLSTXDQWLW\
n0
(T PD\EHUHFRJQLVHG
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
WR EH SURSRUWLRQDO ZLWK WKH GHIOHFWLRQ DQJOH $V FDQ EH VHHQ IURP (TV WKLVTXDQWLW\LVREWDLQHGIURPLQWHJUDWLRQRI
(
0 '&g n n 0
F
% [nF & gn
0 F
9
n
% [n &
0 F
9
n
% [n
0 F
9
,QWKHQHZFRQILJXUDWLRQWKHLQWHJUDWLRQLVFRQGXFWHGDWDJUDGXDOO\GHFUHDVLQJ UDWHZKLOHDSSURDFKLQJWKHPD[LPXPWRUTXHYDOXH:HKDYH
n0
'
&
]n
0 S / 0
]n4
0
'
LI
$W]HURVSHHG
]n '
0
Z
VJQ n0
9ORZ
0
ZKHUH S
0 GW n 0
/
û VJQ n0 0
HOVH S
' 9ORZ
Z
7KHPRPHQWLVFDOFXODWHGZLWKWKHOLQHDUIXQFWLRQ
'Un&'Un %'Un 5Rn0
'
9
)RUWKHVWDQGLQJW\UHZLWKVSHHG
[
HTXDOWR]HURWKHUHVSRQVHWRDOWHUQDWLQJVWHHU
DQJOHYDULDWLRQVZLOOIROORZDFRXUVHVLPLODUWRWKDWRI)LJ ,WLVQRZGHVLUHGWRJUDGXDOO\FKDQJHWRWKHRULJLQDOHTXDWLRQVZKHQWKHW\UH VWDUWVUROOLQJ)RUWKLVWKHIROORZLQJVSHHGZLQGRZLVXVHG
9ORZ '
Z
% FRV
9 9
[
LI _9
[_ 9
ORZ
9ORZ '
HOVH Z
ORZ
DOUHDG\ DSSHDULQJ LQ (T ,Q DGGLWLRQ ZH QHHG D GRZQZDUGV OLPLWHG ZLQGRZIDFWRUWREHXVHGLQ
9ORZ % ' 9OO % Z
Z
OLP
OLP
ZLWK
OLP
OLNHLQ UHSUHVHQWLQJDVPDOOSRVLWLYHTXDQWLW\VD\
7KHDGDSWHGPRPHQWHTXDWLRQ LVIRUPXODWHGDVIROORZV
]n' ''Un VLQ
0
9OO
Z
DUFVLQ
9OO &'UnDUFWDQ
Z
Z
9OO
WDQ
9OO %'Un 5R n0
Z
ZKLFK PD\ EH DVFHUWDLQHG WR UHGXFH WR WKH OLQHDU IXQFWLRQ ZKHQ
9
EHFRPHVHTXDOWRXQLW\:KHQWKHVSHHGH[FHHGVWKHYDOXH
Z
9OO
WKHHTXDWLRQJHWV ORZ
YHU\FORVHWRWKHRULJLQDOHTXDWLRQ 7KHDGDSWHGPRGHOZLOOEHDSSOLHGQRZWRWKHVLPXODWLRQRIWKHPRWLRQRID ULJLGTXDUWHUFDUPRGHOZLWKPDVV
P
TF
WKDWZKLOHDVLQXVRLGDOVWHHULQJLQSXWLV
DSSOLHGVWDUWVUROOLQJDIWHUVHFZLWKDOLQHDUO\LQFUHDVLQJVSHHG7KHODWHUDO
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
)\
[N]
1000 0]
100
)\
ψ
[Nm]
ψ
[rad] 0
9[
[m/s]
\TF
2 \TF
[m]
due to lateral vibrations of quarter car body on tyre
-1
0 )LJ
9[
0]
1
1
2
t
[s]
3
4
5
6LPXODWLRQUHVXOWVRIDSDUNLQJPDQRHXYUH FDUOHDYLQJWKHSDUNLQJORWZKLOHVWHHULQJ VLQXVRLGDOO\
1.5 0] 1.0
100
0.5 0
[Nm] -0.5 -1.0 -1.5 -0.3
-0.2
-0.1
0
ψ
0.1
0.2
0.3
[rad]
)LJ 7KHVWHHUWRUTXHSORWWHGYHUVXVVWHHUDQJOHGXULQJWKHPDQRHXYUHRI)LJ
DFFHOHUDWLRQRIWKHTXDUWHUFDUD[OHUHVXOWVIURPWKHDFWLRQRIWKHVLGHIRUFHWKDW EHJLQVWRGHYHORSDIWHUWKHZKHHOKDVVWDUWHGWRUROO
TF '
)
\
P
\
TF
7KH ODWHUDO ZKHHO VOLS YHORFLW\ LV QRZ QRW RQO\ D UHVXOW RI WKH \DZ DQJOH DW D IRUZDUGVSHHGRIWKHYHKLFOH D[OH
\
9
0
TF
[
0TF EXWDOVRGXHWRWKHODWHUDOYHORFLW\RIWKHZKHHO
:HKDYH
V\ ' & [0TF VLQ
% 0TF FRV \
ZKLFKVHUYHVDVDQLQSXWLQWRWKHHTXDWLRQV 7KHDGGLWLRQDOSDUDPHWHU YDOXHVKDYHEHHQDSSHQGHGLQ7DEOH
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
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
7\UH'\QDPLFV
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
'\QDPLF(TXDWLRQV
7KHZKHHOD[OHSRVLWLRQLVGHILQHGE\WKHORFDWLRQRIZKHHOFHQWUH ;<= DQG
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
γE γEU
γ
wheel body accelerations
wheel rim
$ .
Ω θ EU ; =
<
Ω
belt
Ω
U U U E
ξ
η
ζ
ωξ $ . γ wheel rim &
belt deflections
[E \] E
ψ
U E
. ψ ω ζ
ψE
[
contact patch
ψ
ψ &
U F
U [F \ U U contact patch F ]F ] residual deflections
belt
.F 7F )F\ )F] 0F\ 0F]
\
]
[
9ξ 9η
)F[ 0F[
residual spring forces and moments
)LJ 0RGHO VWUXFWXUH IHDWXULQJ FRQWDFW SDWFK UHVLGXDO FRPSOLDQFH ULJLG EHOWFDUFDVV FRPSOLDQFHDQGZKHHOULP
RULHQWDWLRQ
7KHZKHHOVSHHGRIUHYROXWLRQLVGHQRWHGE\
$PRYLQJ
D[HVV\VWHP &[\] KDVEHHQGHILQHGRIZKLFKWKH[D[LVSRLQWVIRUZDUGVDQG UXQVDORQJWKHOLQHRILQWHUVHFWLRQRIZKHHOSODQHDQGURDGSODQH7KH\D[LVOLHV LQWKHSODQHQRUPDOWRWKHURDGSODQHDQGSDVVLQJWKURXJKWKHZKHHOVSLQD[LV 7KH]D[LVIRUPVWKHQRUPDOWRWKHURDGVXUIDFH7KHRULJLQRIWKHPRYLQJWULDG LVWKHFRQWDFWFHQWUH&$QRWKHUPRYLQJV\VWHPRID[HV $ LVLQWURGXFHG RIZKLFKWKHRULJLQ$LVORFDWHGLQWKHZKHHOFHQWUHWKH D[LVLVKRUL]RQWDODQG WKH D[LVUXQVDORQJWKHZKHHOVSLQGOHD[LV:LWKUHVSHFWWRWKHZKHHOULPWKH U
U
U
EHOWVKRZVUHODWLYHGLVSODFHPHQWV [E \E ]E DQG
U E
U E
U E
7KHFRQWDFWSDWFK
LVGLVSODFHGKRUL]RQWDOO\ZLWKUHVSHFWWRWKHEHOWFRUUHVSRQGLQJWRWKHGHIOHFWLRQV U
U
U
RIWKHUHVLGXDOVSULQJV [F \F ]F DQG
U F
U
7KHVXSHUVFULSW GHVLJQDWHVDUHODWLYH
GLVSODFHPHQW ZLWKRXWWKHVXSHUVFULSWZHKDYHWKHGLVSODFHPHQWZLWKUHVSHFWWR WKHLQHUWLDOV\VWHP ;<= $OOUHODWLYHGLVSODFHPHQWVDUHFRQVLGHUHGVPDOODQG WKHG\QDPLFHTXDWLRQVPD\EHOLQHDULVHG 7KHZKHHOPRWLRQIRUPVWKHLQSXWWRWKHW\UHV\VWHP SRVVLEO\WRJHWKHUZLWK WKH URDG SURILOH )URP WKHVH WKH ZKHHO YHORFLWLHV 9 9 9 DQG
GHILQHGZLWKUHVSHFWWRWKHD[OHWULDG $ RUDOWHUQDWLYHO\ 0
GHILQHG
0
ZLWKUHVSHFWWRWKHPRYLQJFRQWDFWWULDG &[\] DQGWKHFDPEHUDQJOH DQGWKH UDGLDOW\UHGHIOHFWLRQ
]
DUHDYDLODEOH7KHFDPEHUDQJOH ZLOOEHWUHDWHGKHUHDV
DVPDOOTXDQWLW\ 7KHEHOWFRQVLGHUHGDVDULJLGFLUFXODUERG\KDVDPDVVPEDQGPRPHQWVRI LQHUWLD ,E[\] 7KH FDUFDVV VLGH ZDOOV SRVVHVVHV VWLIIQHVVHV FE[\] DQG GDPSLQJ FRHIILFLHQWVNE[\],QWKHILJXUHWKHIRUFHDQGPRPHQWYHFWRUV
.
FDQG
7
FGHILQHG
WRDFWIURPFRQWDFWSDWFKWRDQGDERXWWKHFHQWUHRIWKHEHOWKDYHEHHQLQGLFDWHG
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
7KHLUFRPSRQHQWVDUHGHILQHGZLWKUHVSHFWWRWKH $ WULDG7KHILUVWRIWKH
WZRVHWVRIILUVWRUGHUGLIIHUHQWLDOHTXDWLRQVIRUWKHVL[GHJUHHVRIIUHHGRPUHDGV G\QDPLFEHOWHTXDWLRQV
% NE[ [0EU % FE[ [EU % NE] ]EU ' .F
PE 90E & 9E
PE 90 E % N E] ]0EU % FE] ] EU & NE[ [EU ' .F
,E\ 0 E % NE
PE 90 E % 9E
E
,E[ 0 E % ,E\
E
,E] 0 E &,E\
E
E
0U E
% FE
U E
' 7F
% NE\ \0EU % FE\ \ EU ' .F % NE 0 EU % FE EU % NE
%NE
0
U E
%FE
U E
&NE
U E U E
' 7F
' 7F
6HYHUDOFRXSOLQJWHUPVVKRZXS7KHVHDUHGXHWRWKHJ\URVFRSLFHIIHFWDQGGXH WRWKHDFWLRQRIWKHURWDWLQJUDGLDOGDPSHUVZLWKUHVXOWLQJFRHIILFLHQWNE[ NE]DQG WKHODWHUDOGDPSHUVZLWKUHVXOWLQJDQJXODUGDPSLQJFRHIILFLHQWVNE
NE )LJXUH
LOOXVWUDWHV WKH PHFKDQLVP WKDW JLYHV ULVH WR WKH LQWHUDFWLRQ WHUPV 7KH H[DPSOHFRQFHUQVWKHWHUP N
E]
U E
] LQ 7KHIROORZLQJUHODWLRQVEHWZHHQ
WKHWZRVHWVRIZKHHODQGD[OHDQJXODUYHORFLWLHVKROG
'0 ' & % 0 VLQ . & % 0 ' 0 FRV . 0
)RUWKHUHODWLYHGLVSODFHPHQWVEHWZHHQEHOWDQGZKHHOULPZHKDYHWKHVHFRQG VHWRIVL[ILUVWRUGHUGLIIHUHQWLDOHTXDWLRQV
DQG
[0EU ' 9E & 9 \0EU ' 9E & 9 ]0EU ' 9E & 9 U
0E
' ' '
0U E U 0E
&0 0 E % & % 0 E &
E
U E
7KHIRUFHVDQGPRPHQWVDSSHDULQJLQWKHULJKWKDQGPHPEHUVRI(TV
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
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
.F ' )F[ .F ' )F] & )F\ 7F ' UO )F[ % 0F\ .F ' )F\ % )F] 7F ' &UO )F\& UO E&\FU )F]%0F[ 7F ' 0F]&\FU )F[
2EYLRXVO\DSURSHUD[HVWUDQVIRUPDWLRQLVWREHSHUIRUPHGLIWKHURDGSODQHLV QRWKRUL]RQWDO$VDUHVXOWWUDQVYHUVHDQGIRUZDUGVORSHVZLOODIIHFWWKHWHUPV DSSHDULQJLQWKHULJKWKDQGPHPEHUVRI(TV ,WLVOHIWWRWKHXVHUWR LQWURGXFHWKHVHWUDQVIRUPDWLRQV 7KHFRQWDFWSDWFKERG\LVVXEMHFWHGWRIRUFHVDFWLQJLQWKHUHVLGXDOVSULQJV VXEVFULSW F DQG H[WHUQDO IRUFHV DFWLQJ IURP URDG VXUIDFH WR FRQWDFW SDWFK
Ω
Ω
belt
U E
[
belt
rim
]
U E
U
E
]
rim
radial dampers NE]
)U
rolling resistance force
)1
)LJ 7KHURWDWLQJUDGLDOGDPSHUVRIWKHYHUWLFDOO\GHIOHFWHGW\UHJLYHVULVHWRDUHVXOWLQJ IRUHDQGDIWIRUFHDFWLQJEHWZHHQEHOWDQGULP7KHUHVXOWLQJORQJLWXGLQDOGHIOHFWLRQ
) WKURXJKWKHDFWLRQRIWKHQRUPDOORDG)1
SURGXFHVDUROOLQJUHVLVWDQFHIRUFH
U
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
rim belt
YHUWLFDO
ODWHUDO
] U
rim
0V\ FF[
U F
[
)1
)V\ 0V]
belt
F
DQG\DZ
FF]
)V[ 0V[ WDQJHQWLDO
wheel plane
belt
ψFU
FF\
FFψ
U F
\
)LJ 'HIOHFWLRQVRIWKHUHVLGXDOVSULQJVDWWDFKLQJFRQWDFWSDWFKWREHOW
VXEVFULSWV )LJXUH LOOXVWUDWHVWKHVLWXDWLRQ7KHVHFRQGRUGHUGLIIHUHQWLDO HTXDWLRQVWKDWJRYHUQWKHKRUL]RQWDOG\QDPLFVRIWKHFRQWDFWSDWFKERG\UHDG G\QDPLFFRQWDFWSDWFKHTXDWLRQV
PF 90 F[& 9F\ 0 F % NF[ [0FU % F F[ [FU ' )V[ PF 90 F\% 9F[ 0 F % NF\ \0FU % FF\ \FU ' )V\ ,F F % NF 0 FU % FF FU ' 0V]
DQGLQDGGLWLRQGLIIHUHQWLDOHTXDWLRQVIRUWKHUHVLGXDOGHIOHFWLRQV
[0FU ' 9F[ & 9E % UH & 0 EU \0FU ' 9F\ & 9E % U O 0 % 0 EU
U 0F ' 0 &0 F
& 0 EU
$OVRKHUHLQWKHULJKWKDQGPHPEHUVDURDGVORSHZLOOKDYHDQHIIHFW$QD[HV WUDQVIRUPDWLRQLVQHHGHGWRSURSHUO\LQWURGXFHWKHEHOWYHORFLWLHVZLWKUHVSHFWWR WKHORFDWLRQRIWKHFRQWDFWSDWFK)URPWKHGHIOHFWLRQVDQGGHIOHFWLRQUDWHVWKH UHVLGXDO VSULQJ DQG GDPSHU IRUFHV DSSHDULQJ LQ (TV FDQ EH GHWHUPLQHG:HKDYH
)F[ ' NF[ [0FU % FF[ [FU )F\ ' NF\ \0FU % F F\ \FU 0F] ' NF 0 FU % FF FU
ZKLOH
0F[ ' 0V[
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
residual spring forces and moments
belt
U F
U F
[
\
FF[ FF\
)F\
0F\ 0F[
)F[
) F]
U F
0F\
]
FF]
ψFU
)F\
0F]
PF,F
FFψ
0F]
)F]
0F[
)F[
)1
)V[ )V\
0V]
0V[
0V\
ground contact forces and moments )LJ ,QWHUQDODQGH[WHUQDOIRUFHVDQGPRPHQWVDFWLQJRQWKHFRQWDFWSDWFKERG\
0F\ ' 0V\ )F] ' )V]
7KHFRQWDFW IRUFHVDQGPRPHQWVUHVXOWIURPWKHFRQWDFWVOLSPRGHOHTXDWLRQV GHYHORSHGLQWKHSUHFHGLQJVHFWLRQ7KHFRPSXWHGIRUFHVDQGPRPHQWVKDYHEHHQ GHILQHGDFFRUGLQJWRWKH0DJLF)RUPXODPRGHOZKLFKDWVWHDG\VWDWHDFWZLWK UHVSHFWWRWKHPRYLQJD[HV V\VWHP &[\] 7KHVHIRUFHVDQGPRPHQWVKHUH SURYLGHGZLWKVXEVFULSW&DUHWREHWUDQVIRUPHGWRDUULYHDWWKHVHWRIIRUFHVDQG PRPHQWVGHILQHGDFFRUGLQJWRWKHV\VWHPZLWKOLQHVRIDFWLRQVKLIWHGVLGHZD\V U RYHU WKH FDOFXODWHG µVWDWLF¶ ODWHUDO GLVSODFHPHQW \ RI WKH FRQWDFW SDWFK ZLWK VW UHVSHFWWRWKHZKHHOSODQH7KHVHFRUUHFWHGTXDQWLWLHVFRUUHVSRQGWRWKHIRUFHV
V
DQG PRPHQWV SURYLGHG ZLWK VXEVFULSW RFFXUULQJ LQ WKH (TV DQG
)V[ ' )[ & )V\ ' )\ & 0V] ' 0] & % \VWU )[ & 0V[ ' 0[ & & \VWU )] & 0V\ ' 0\ & )V] ' )] &
@
@
ZKHUHWKHVWDWLFODWHUDOGHIOHFWLRQLVFRPSXWHGIURPWKHVLGHIRUFHDQGWKHRYHUDOO ODWHUDOFRPSOLDQFHRIWKHVWDQGLQJW\UHDWJURXQGOHYHO
\VWU '
)V\ U ' )V\ % % O &)\ FF\ F E\ FE
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
9[ 9\ψ
axle
γ EUψEU
U E
U E
U [ \ ] E
Ω road surface
belt
contact patch
ψFU
U
6
U θE U
ψF
[ \ U F
F
9V[F 9V\F
)LJ 6OLSYHORFLWLHVRIFRQWDFWSDWFKIRUPLQJLQSXWWRWUDQVLHQWVOLSHTXDWLRQV
:LWKWKHWUDQVLHQWUHVSRQVHYDULDEOHVFRPSXWHGZLWKWKHDLGRI(TV SOXV (T WKH FRQWDFW IRUFHV DQG PRPHQWV PD\ EH IRXQG E\ XVLQJ WKHVH YDULDEOHVDVDUJXPHQWLQWKHVWHDG\VWDWHHTXDWLRQVSUHVHQWHGLQ6HFWLRQV DQG:HVKRXOGKRZHYHUSURSHUO\DFFRXQWIRUWKHUHVSRQVHWRDYDU\LQJ FDPEHUDQJOHRIWKHEHOWSODQH
)RUWKLVSXUSRVHZHLQWURGXFHWKHW\UHWRWDOVSLQYHORFLW\
0
FI(T
ZKLFKKDVEHHQFRUUHFWHGIRUWKHVWDWLFEHOWFDPEHUGHIOHFWLRQWRHQDEOHWKHGLUHFW XVHRIWKHUHOHYDQWPDJLFIRUPXODH DQDORJRXVWRWKHXVHRI
'
0
0
F & & g
VLQ
%
U U E& E
VW
VW
LQ(T
ZLWKWKHµVWDWLF¶EHOWGHIOHFWLRQDQJOH
U E
VW
'&
O V\
U )
E
F
)XUWKHUZHZULWHLQVWHDGRI
U VW
'
0
V]
F
F
%
VW
F
E
7KHWUDQVLHQWVOLSILUVWRUGHUGLIIHUHQWLDOHTXDWLRQVDUHUHSHDWHGEHORZ7KH\ DUHLGHQWLFDOWR(TV SOXV(T H[FHSWIRUWKHQRZDGGHGHIIHFWRI WKHFDPEHUDQJOHLQWKHULJKWKDQGPHPEHUV,Q
F PD\EHUHSODFHGE\
DVDWVWHDG\VWDWHWKHVHDUHHTXDOIRUFRQWDFWSDWFKDQGRYHUDOOW\UHPRGHO7KH VDPHKROGVIRUWKHVSLQYDULDEOHV
nF DQG n
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
WUDQVLHQWVOLSHTXDWLRQVIRUORQJLWXGLQDOVLGHVOLSWXUQVOLSDQGFDPEHU
G
F
GW G
F
GW G
%_
9
[_
%_
9
[_
W
%_
Gn
F
GW
GW Gn
n
&0
GW
VW
F
[_ n)
' &0
%_
[_ n '
&0
%_
[_ n '
&0
GW Gn
n
U
V\F & _9[_
[_ n '
9
9
_9[_
%_
Gn)
)
'&
V[F
9
[_ W '
9
GW
'&
%_
9
9
9
,QDGGLWLRQZHQHHGWKHFRPSRVLWHWUDQVLHQWVOLSTXDQWLWLHV
' n & n) ( n0 ' gn n % gn n)
n
&n
7KH VOLS YDULDEOHV HPSOR\HG LQ WKH PDJLF IRUPXODH 0) DUH UHSODFHG E\ WKH WUDQVLHQW VOLS YDULDEOHV DV LQGLFDWHG LQ WKH DUJXPHQWV RI WKH IROORZLQJ H[SUHVVLRQV RXWSXWIRUFHVDQGPRPHQWV
[& '
)
)
)[
n) )1
)\
n)
0)
\& '
0)
)1
] & ' & W & )\ & % 0]U & % 0] W ' 0)
) & W W n0 1 0
]U & '
0
0
[& '
0
0
\&'
&
0
0]U
0)
]'
0
&
W
n0
)1
[
\
0
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
]&
)
'&
1
)
(TXDWLRQ KDV EHHQ DGGHG ZKLFK LV LQ DJUHHPHQW ZLWK WKH VKRUW ZDYHOHQJWKWUDQVLHQWVOLSWKHRU\FI(TV 7KHLQSXWWRWKHWUDQVLHQW VOLS HTXDWLRQV LV FRQVWLWXWHG E\ WKH YHORFLWLHV RI WKH FRQWDFW SDWFKFI)LJ
V[F ' 9F[
V\F '
9
9
U U F\ & 9[ E % F
9
F' 0F
0
7KHRYHUWXUQLQJFRXSOH0[FDQEHPRGHOOHGZLWKWKHIXQFWLRQ ( ZKHUHWKH )\SDUWPD\EHUHSODFHGE\WKHH[SUHVVLRQ ZLWKWUDQVLHQWVOLSDQJOH DVDUJXPHQWLIWKHDFWXDOPRPHQWDU\ORDGHGUDGLXVUO GLVWDQFHEHWZHHQSRLQWV $DQG& KDVEHHQSURSHUO\DFFRXQWHGIRULQWKH VWHDG\VWDWH PHDVXUHPHQWVDQG IXUWKHUSURFHVVLQJ7KHZKHHOORDG)1
_)]_ DQGWKHUROOLQJUHVLVWDQFHPRPHQW
0\ GHSHQG RQ WKH UDGLDO GHIOHFWLRQ DQG RQ D QXPEHU RI RWKHU YDULDEOHV 7KH VXEVHTXHQW VHFWLRQ SURYLGHV LQIRUPDWLRQ RQ WKH H[SHULPHQWDOO\ DVVHVVHG IXQFWLRQDOUHODWLRQVKLSV )LQDOO\ZHQHHGWRHVWDEOLVKWKHRXWSXWIRUFHVDQGPRPHQWVWKDWDFWIURPWKH W\UH XSRQ DQG DERXW WKH ZKHHO FHQWUH 7KHVH TXDQWLWLHV DUH GHQRWHG ZLWK WKH V\PEROV. DQG 7 DQGDUHSURYLGHGZLWKWKHVXEVFULSW D 7KHFRPSRQHQWVDUH GHILQHGWRDFWDORQJDQGDERXWWKHD[HVRIWKHD[OHWULDG $
D
.
D
.
D
.
7
7
7
D D
D
' ' ' ' ' '
:HILQG
E
E
E
.
. .
U U 7 E % \ E .E & ]E .E U U 7 E % ] E .E & [E .E U U 7 E % [E .E & \E .E
ZKHUHWKHIRUFHVDQGPRPHQWVDFWLQJIURPEHOWFHQWUHWRULPDUHUHWULHYHGIURP (TV
E
'
E
'
E
'
.
.
.
U U E[ [0E % FE[ [E % NE] U U N E\ \0E % FE\ \E U U N E] ]0E % FE] ]E & NE[ N
]
U E
U [ E
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
E
7
7
7
E E
' ' '
U U E 0 E % FE E % NE U 0 U N E E % FE E U U N E 0 E % FE E & NE N
U E
U E
9DOXHVRILQHUWLDSDUDPHWHUVQRUPDOLVHGZLWKW\UHPDVVPRDQGUHIHUHQFHPRPHQW
RILQHUWLDPRURZLWKURWKHXQORDGHGW\UHUDGLXVKDYHEHHQOLVWHGLQ$SS
&RQVWLWXWLYH5HODWLRQV
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
_)]_ DQG WKH IROORZLQJ
IRUPXODHKDYHEHHQIRXQGWRJLYHDJRRGDSSUR[LPDWLRQ D
'
D
T
1 )1R % TD
)
1 )1R UR
)
0.12
0.08
0.04
0
2.5 m drum
flat plate
[m]
0
2
E
E
D
D 4
)1 6
[kN]
8
10
0
D
2
4
E
)1 6
[kN]
8
10
)LJ 0HDVXUHGDQGFDOFXODWHGKDOIOHQJWK DQGKDOIZLGWK RIWKHFRQWDFWSDWFKYVZKHHO ORDGIRUWKHFDVHV ORDGLQJRQDIODWSODWHDQGRQDGUXPVXUIDFH
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
DQG
E
'
T
E
1 )1R % TE
)
1 )1R UR
)
ZLWKURGHQRWLQJWKHIUHHW\UHUDGLXV)LJXUH SUHVHQWVWKHFXUYHVFRPSDUHG ZLWKWKHPHDVXUHGHIIHFWLYHTXDQWLWLHVIRUWKHW\UHSUHVVHGRQDIODWVXUIDFHDQG RQ D FXUYHG GUXP VXUIDFH ZLWK P GLDPHWHU 2EYLRXVO\ WKH UHVXOWV DUH VDWLVIDFWRU\7KHGLPHQVLRQVRIWKHW\UHZHUHDJDLQ 5 9DWEDU LQIODWLRQSUHVVXUH7KHQRQGLPHQVLRQDOSDUDPHWHUYDOXHVFDQEHIRXQGLQ7DEOH ,Q$SSDQDOWHUQDWLYHH[SUHVVLRQIRUDLVSUHVHQWHGEDVHGRQWKHUDGLDO GHIOHFWLRQ
]
LQVWHDGRIRQWKHQRUPDOORDG)17KHUHVXOWLQJYDOXHLVPXFKOHVV
GHSHQGHQWRQWKHSRVVLEO\FKDQJHGLQIODWLRQSUHVVXUHFI)LJ$ 7DEOH 3DUDPHWHUYDOXHVIRUFRQWDFWSDWFKGLPHQVLRQV 5 9DWEDU
)1R
_)]R_
TDIODW TDGUXP
1UR
TDIODW
5R
TDGUXP
PGUXPGLDPHWHU TEIODW TEGUXP
P
TEIODW TEGUXP
7KH6LGHZDOO6WLIIQHVVHVDQG'DPSLQJ 7KHULJLGULQJPRGHORIWKHW\UHIUHHO\UROOLQJDQGORDGHGRQWKHURDGVKRZVWKUHH LQSODQHPRGHVRIYLEUDWLRQ WKHYHUWLFDOPRGHDQGWZRDQJXODUPRGHV2QHRI WKHVHURWDWLRQDOPRGHVYLEUDWHV LQ SKDVHZLWKULPDQJXODUYLEUDWLRQZKLOHWKH RWKHUPRYHVLQDQWLSKDVH7KHQDWXUDOIUHTXHQFLHVKDYHEHHQHVWLPDWHGZLWKWKH DLGRIH[SHULPHQWVFRQGXFWHGRQWKHGUXPWHVWVWDQGZKHUHWKHZKHHODWIL[HG D[OHSRVLWLRQUROOVRYHUDVKRUWFOHDWRULVH[FLWHGE\EUDNHWRUTXHIOXFWXDWLRQV FI6HF 7KHH[SHULPHQWVLQGLFDWHWKDWWKHQDWXUDOIUHTXHQFLHVO\LQJLQWKHUDQJHRI +] GHFUHDVH ZLWK YHORFLW\ 2WKHU UHVHDUFKHUV IRXQG WKH VDPH WHQGHQF\ QRWDEO\%UXQL&KHOLDQG5HVWD 6LQFHWKHVLGHZDOOVWLIIQHVVHVDUHPXFK ODUJHUWKDQWKHUHVLGXDOVWLIIQHVVHVLWLVGHFLGHG WR PDNH WKHLQSODQHVLGHZDOO VWLIIQHVVHVGHSHQGHQWRQWKHVSHHGRIUROOLQJ$VWRWKHRXWRISODQHYLEUDWLRQV 0DXULFH GLG QRW DVFHUWDLQ WKH QHFHVVLW\ WR PDNH WKH ODWHUDO \DZ DQG FDPEHU VWLIIQHVVHVVSHHGGHSHQGHQW =HJHODDULQWURGXFHVDYDULDEOHTXDQWLW\49WKDWLVDPHDVXUHRIWKHWLPHUDWH RI FKDQJH RI WKH ORDGHG W\UH GHIRUPDWLRQ GXH WR UROOLQJ :H KDYH WKH QRQ GLPHQVLRQDOTXDQWLW\ 9RUHSUHVHQWLQJWKHUHIHUHQFHYHORFLW\FI6HF
4
9'
_
_
R
9
[
U E
%
]
U E
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
7KH IROORZLQJ H[SUHVVLRQV IRU WKH VLGHZDOO VWLIIQHVVHV KDYH EHHQ IRXQG WR EH DSSURSULDWH
F
F
F
E[ ' FE[
E] ' FE]
'
E
E
F
& & &
E9[
4
9
E9]
4
9
E9
4
9
T T
T
7KHDGGLWLRQDOVXEVFULSWGHVLJQDWHVWKHVLWXDWLRQRIWKHORDGHGQRQURWDWLQJW\UH 7KHYHUWLFDODQGORQJLWXGLQDOVWLIIQHVVHVKDYHEHHQDVVXPHGHTXDOWRHDFKRWKHU 7KHSDUDPHWHUVTE9[] JRYHUQWKHVSHHGGHSHQGHQF\RIWKHVWLIIQHVVHV 7KHVLGHZDOOGDPSLQJFRHIILFLHQWVNEL
NE[\]
DUHFRQVLGHUHGWREHFRQVWDQW
TXDQWLWLHV7KHLQWHUDFWLRQ WHUPVDSSHDULQJLQ(TV FRQWDLQLQJWKH FRHIILFLHQWVNEL
DUHRPLWWHGVLQFHWKHVHWHUPVDIIHFWWKHUROOLQJUHVLVWDQFHDQG
WKHDOLJQLQJWRUTXH DOVRLQVWHDG\VWDWH DQGZRXOGPDNHWKHVHVSHHGGHSHQGHQW 7KH LQWURGXFWLRQ RI PDWHULDO GDPSLQJ EHLQJ LQYHUVHO\ SURSRUWLRQDO ZLWK IUHTXHQF\ZRXOGEHFORVHUWRUHDOLW\)XUWKHURQWKHUROOLQJUHVLVWDQFHZLOOEH LQWURGXFHGLQDQDOWHUQDWLYHEHWWHUFRQWUROOHGZD\ )RUWKHFRQVWDQWVWLIIQHVVHVQRQGLPHQVLRQDOSDUDPHWHUVPD\EHLQWURGXFHG :HGHILQHZLWK)1RUR
5R DQGPR WKHUHIHUHQFHORDGIUHHW\UHUDGLXVDQGW\UH
PDVV WKHQRQGLPHQVLRQDOSDUDPHWHUVT
' ' E E[ \ ] ' E[ \ ]
F F
N
E
N
FE[ \ ] )1RUR
FE
T
T
T
'
NE[ \ ]
NE
T
)
1R UR
P
R )1RUR
P
R )1R U R
7R SURYLGH PRUH GDPSLQJ ZKHQ WKH ZKHHO VSHHG JHWV FORVH WR ]HUR ZH PD\ IROORZWKHWKHRU\RI&KDSWHUDQGLQWURGXFHN9ORZDVGHPRQVWUDWHGLQ&KDS (TV ZKHUHWKHVOLSVSHHG9V[PD\EHUHSODFHG9V[F,QDVLPLODUZD\ WKHUHVLGXDOVWLIIQHVVDQGGDPSLQJSDUDPHWHUVFFDQGNFKDYHEHHQQRUPDOLVHG
7KH1RUPDO)RUFH 7KHVSULQJZLWKUHVLGXDOVWLIIQHVVFF]LQGLFDWHGLQ)LJ KLGHVDVWUXFWXUHWKDW LVDORWPRUHFRPSOH[WKDQDVSULQJZLWKFRQVWDQWVWLIIQHVV([SHULPHQWVUHYHDO WKDWWKHIRUFHGHIOHFWLRQFKDUDFWHULVWLFVDUHQRQOLQHDU WKHIRUFHGHYHORSVDIWHU FRQWDFWKDVEHHQPDGHDQGLQFUHDVHVVOLJKWO\PRUHWKDQSURSRUWLRQDOO\ZLWKWKH RYHUDOOUDGLDOGHIOHFWLRQ
]
$OVRWKHW\UHJURZVZLWKVSHHGGXHWRWKHFHQWULIXJDO
DFWLRQ)LJXUH LOOXVWUDWHVERWKSKHQRPHQD)XUWKHUPRUHLWKDVEHHQIRXQG XVHIXO WR LQWURGXFH )[ DQG )\ LQWHUDFWLRQ WHUPV LQ WKH YHUWLFDO VWLIIQHVV FI
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
5HLPSHOO HW DO 7KH IROORZLQJ IRUPXODH LV SURSRVHG IRU WKH RYHUDOO VWLIIQHVVDQGJURZWKIXQFWLRQV )
1 ' _)]_ '
%
_
9
T
U
_
R R
9
)
&
)F[
T
)
[
1R
)
&
)F\
T
)
\
]
)]
T
1R
U
R
%
T
]
)]
U
R
1R
)
ZLWKGHIOHFWLRQ
] ' U R& UO%
U
DQGJURZWK U
'
9 UR
T
R 9R
U
+HUHURGHQRWHVWKHUDGLXVRIWKHIUHHQRQURWDWLQJW\UHUOWKHORDGHGUDGLXV D[OH KHLJKW DQG
UWKHLQFUHDVHLQIUHHW\UHUDGLXVGXHWRZKHHOURWDWLRQYHORFLW\7KH
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
1 ' _)]_ ' D ]U% D
]U% D
]U
ZLWKWKH) 1UHODWHGUHVLGXDOVSULQJGHIOHFWLRQ WDNLQJLQWRDFFRXQWJHRPHWULFDO
9[ =
10
)1
1.5
∆U
140
6 [kN]
1.0 [mm]
92 59 25 km/h
2
0.5
0
0 0 )LJ
ρ]
40
20 [mm]
0
50
9[ 100[km/h]150
9HUWLFDOORDGYVUDGLDOGHIOHFWLRQFKDUDFWHULVWLFVDWYDULRXVIRUZDUGYHORFLWLHVDQGW\UH UDGLXVJURZWKZLWKVSHHG
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
LQWHUDFWLRQWHUPVZLWKKRUL]RQWDOGHIOHFWLRQV
]U ' ]E%
U
&
[UR &T)F\
\ UR & Z
)F[
T
+HUH]ELVWKHYHUWLFDOGLVSODFHPHQWRIWKHFHQWUHRIWKHEHOWULQJ ZLWKUHVSHFWWR LWVORFDWLRQLQWKHMXVWWRXFKLQJXQORDGHGDQGQRQURWDWLQJFRQGLWLRQ RQDIODW Z
KRUL]RQWDOURDG)1
[\
UHSUHVHQWWKHORQJLWXGLQDODQGODWHUDOW\UHFRQWDFW
GHIOHFWLRQVDQGZGHQRWHVWKHYDULDWLRQRIWKHYHUWLFDOSRVLWLRQRIWKHURDGSODQH DWWKHFRQWDFWFHQWUHZLWKUHVSHFWWRLWVLQLWLDOOHYHO7KHDFWXDOORDGHGUDGLXVUO UHDGVZLWKUR U
5R FI)LJIRUPHDVXUHGHYLGHQFH
U O ' UR % ] E %
& ]U&
U
[ % T)F\
)F[
T
\ UR
7KH FRHIILFLHQWV DSSHDULQJ LQ FDQ EH H[SUHVVHG LQ WHUPV RI VLGHZDOO VWLIIQHVVFE]DQGZKHHOVSHHGRIUHYROXWLRQ
D
D
D
ZKHUH $
$
' '
F F
' '
E] $
E]& F
E] $
F
T
)]
$
$
%
E] $
&
$
FE]
)]
&
T
$
FE]
'
9
T
_
_UR9R )1R UR
T)] UR
ZKHUH WKH WZR QRQGLPHQVLRQDO SDUDPHWHUV T)] RI (T DSSHDU 7KH KRUL]RQWDOW\UHGHIOHFWLRQVDWURDGVXUIDFHOHYHOZLWKUHVSHFWWRWKHZKHHOULPDUH
U [ ' [ E % UR U \ ' \ E & UR
U U E % [F U U E % \F
$SSHQGL[ SURYLGHVWKHUHOHYDQWSDUDPHWHUYDOXHVIRUWKHSDVVHQJHUFDUW\UH WKDWKDVEHHQWHVWHG
)UHH5ROOLQJ5HVLVWDQFH ([SHULPHQWVVKRZWKDWWKHUROOLQJUHVLVWDQFHIRUFH)U SRLQWLQJEDFNZDUGV LV SURSRUWLRQDOWRWKHW\UHQRUPDOORDG)1$KLVWRU\RQWKLVVXEMHFWFDQEHIRXQG LQWKHSXEOLFDWLRQRI&ODUN :HKDYH )
U ' IU )1
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
7KHUROOLQJUHVLVWDQFHFRHIILFLHQWIUGHSHQGVRQWKHIRUZDUGVSHHGDQGPD\EH H[SUHVVHGLQWHUPVRISRZHUVRIWKHVSHHGFI0LWVFKNH
U ' TV\
I
%
_9[9R_
V\
T
%
V\
T
9[9R
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
(IIHFWLYH5ROOLQJ5DGLXV%UDNH/HYHU$UP5ROOLQJ5HVLVWDQFH0RPHQW ,Q&KDSWHUWKHQRWLRQRIWKHHIIHFWLYHUROOLQJUDGLXVKDVEHHQLQWURGXFHG)LJXUH VKRZVWKHUHVXOWVRIH[SHULPHQWVRIW\UHVUXQQLQJRYHUDGUXPVXUIDFH,Q 6XEVHFWLRQ WKH WKHRU\ LV UHVWULFWHG WR D OLQHDULVHG UHSUHVHQWDWLRQ RI WKH YDULDWLRQRIWKHHIIHFWLYHUROOLQJUDGLXVZLWKUDGLDOGHIHFWLRQ7KHFRPSOHWHQRQ OLQHDUYDULDWLRQYHUVXVQRUPDOORDG)1PD\EHGHVFULEHGE\WKHH[SUHVVLRQ
U
H ' UR%
ZLWK '
%
U
UH ' TUH
UH '
)
&
UH ] % ' UH DUFWDQ
T
1R &)]
%
UH ]
D
UH 'UH
T
DQGWKHYHUWLFDOVWLIIQHVVRIWKHVWDQGLQJW\UHDWQRPLQDOORDG)1RDVGHULYHGIURP (T
&
)] '
)
1R
U
R
)]
T
%
)]
T
E
7KHORQJLWXGLQDOVOLSYHORFLW\LVGHILQHGZLWKUHLQWURGXFHGDVWKHVOLSUDGLXVFI HJ(T 1RWHWKHVOLJKWO\GHYLDWLQJIRUPXODLQ$SS(T $ ,Q WKH SUHVHQW PRGHO ZLWK WKH EHOW ULQJ DQG FRQWDFW SDWFK PRGHOOHG DV VHSDUDWHERGLHVWKHORQJLWXGLQDOVOLSYHORFLW\9V[FRIWKHFRQWDFWSDWFKLVXVHGDV LQSXW LQ WKH WUDQVLHQW VOLS GLIIHUHQWLDO HTXDWLRQ ,Q (T ZLWK
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
0'
Ω
rim
9[
)' $
UI UH U
O
9V[F
belt ring
9E[ effective rolling circle
)[
)1 0\ )LJ 7KHGULYHQW\UHZKHHOFRPELQDWLRQZLWKGHIOHFWHGEHOWDQGUHVLGXDOVSULQJ
WKH HIIHFWLYH UROOLQJ UDGLXV
UH DFFRPSOLVKHV WKH WUDQVPLVVLRQ RI WKH
URWDWLRQDOVSHHGRIWKHEHOWWRWKHUHVLGXDOGHIOHFWLRQUDWHRIFKDQJH$WVWHDG\ VWDWHFRQGLWLRQWKHGHIOHFWLRQUDWHVYDQLVKDQGZHKDYHWKHIROORZLQJUHODWLRQIRU WKHORQJLWXGLQDOVOLSVSHHG
9V[F ' 9[ & UH
:HPD\FRQVLGHUWKHSRZHUEDODQFHRIDZKHHOVXEMHFWHGWRDSURSXOVLRQWRUTXH
0'DQGDGUDJIRUFH)'DFWLQJEDFNZDUGVRQWKHZKHHOLQLWVFHQWUH)LJXUH GHSLFWVWKHVLWXDWLRQ7KHFRQQHFWHGSRZHUIORZGLDJUDPLVSUHVHQWHGLQ)LJ 7KH6UHSUHVHQWV D SRZHUVRXUFH WKHHQJLQH DQGWKH5¶VDUHUHVLVWRUVZKHUH HQHUJ\LVGLVVLSDWHG7KHEDODQFHRISRZHUUHTXLUHVWKDWWKHHTXDWLRQKROGV
0' ' )' 9[ % 0\ & ) [ 9V[F
RUZLWK
0'& 0\ ' )[ UH
7KLVVXJJHVWVDWOHDVWIRUWKHPRGHOHPSOR\HGWKDWWKHPRPHQWDUPHTXDOVWKH
S
0'Ω 0\Ω R
tyre
)' 9[
R
)[9V[F
R
)LJ 3RZHUIORZGLDJUDP ERQGJUDSK RIGULYHQW\UHZKHHOFRPELQDWLRQLQVWHDG\VWDWH
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
on flat plank
310
on 2.5m drum
310
300
300
[mm]
[mm]
290
290
loaded radius eff. rolling radius brake lever arm
280 270 0
)1
2
280 270
4
6
[kN]
8
0
)1
2
4
[kN]
6
8
)LJ 7\UHUDGLLDVIXQFWLRQRIWKHQRUPDOORDGPHDVXUHGDW]HURRUYHU\ORZVSHHG
HIIHFWLYHUROOLQJUDGLXV GHILQHGDW]HURGULYLQJRUEUDNLQJWRUTXH IUHHUROOLQJ &RQVHTXHQWO\WKHEORFNQDPHGµW\UH¶LQWKHGLDJUDPRI)LJ UHSUHVHQWVZKHQ XQIROGLQJ WKH ERQG JUDSK D MXQFWLRQ VWUXFWXUH FRQWDLQLQJ D WUDQVIRUPHU ZLWK PRGXOXVUHWKDWWUDQVIRUPVWKHDQJXODUVSHHGLQWR DSDUWRI WKHVOLSVSHHGDQG LQRSSRVLWHGLUHFWLRQWKHVOLSIRUFHLQWRWKHGULYHWRUTXH ([SHULPHQWV KDYH EHHQ FDUULHG RXW E\ =HJHODDU RQ ERWK WKH IODW SODQN PDFKLQHDQGWKHGUXPWHVWVWDQGWRHVWDEOLVKWKHHIIHFWLYHUROOLQJUDGLXVDQGWKH PRPHQWDUP,QWKHVHWHVWVDEUDNHWRUTXHZDVDSSOLHGWRWKHZKHHO7KHPRPHQW DUPPD\EHWHUPHGDVWKHEUDNHOHYHUDUP7KHGLDJUDPVRI)LJ KDYHEHHQ REWDLQHGIURPWHVWVSHUIRUPHGDW]HUR EUDNHOHYHUDUP DQGYHU\ORZVSHHGRI WUDYHO HIIHFWLYHUROOLQJUDGLXV (VSHFLDOO\LQFDVHRIWKHIODWVXUIDFHDQH[FHOOHQW DJUHHPHQWKDVEHHQIRXQGWRRFFXU,WLVDVVXPHGWKDWWKHJURZWKRIWKHHIIHFWLYH UROOLQJUDGLXVZLWKVSHHGLVHTXDOWRWKDWRIWKHIUHHW\UHUDGLXVFI(T 7KHGLDJUDPVRI)LJ SUHVHQWWKHLQIOXHQFHRIVSHHGRQWKHWZRUDGLL7KH
310
eff. rolling radius
PRGHO WHVW
308 306
[mm] 304
310
1
) 0=
2000N 4000N 6000N
306
[mm] 304
302
302
300
300
298 0
50
9[
brake lever arm eff. rolling radius
308
298 100
[km/h]
150
0
50
9[
100
[km/h]
150
)LJ 7\UHUDGLLPHDVXUHGRQPGUXPDVDIXQFWLRQRIWKHIRUZDUGVSHHGDWWKUHHD[OH KHLJKWVFRUUHVSRQGLQJWRWKHLQGLFDWHGLQLWLDOORDGV$YHUDJHOHYHODQGDPSOLWXGHRI EUDNHWRUTXHDUHVPDOO
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
312
-d0% /d)[
UR
WHVW
-d0% /d)[
308
)0 =
free radius
2000N 4000N 6000N
312mm
1
R
U
304 300
UH
296 0 )LJ
0.2
0.4
0.6
)[ )1
0.8
1
eff. rolling radius UH PRGHO )10= 2000N 4000N 6000N
loaded radius U O
301mm 290mm 280mm
5DWLRRIEUDNHWRUTXHLQFUHPHQWWREUDNH IRUFH LQFUHPHQWDVIXQFWLRQRIOHYHORI EUDNHIRUFHDWDYHORFLW\RINPKDQGWKUHHGLIIHUHQWD[OHKHLJKWV ORDGHGUDGLL
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
0 ' ) U &U % ) U \
[
H
O
U O
9U 9R
@ DUFWDQ
ZKLFK DOVR FRQWDLQV WKH IUHH UROOLQJ UHVLVWDQFH PRPHQW )UUO DFFRUGLQJ WR (T DQGWKHDUFWDQIXQFWLRQWRWDNHFDUHRIDSRVVLEOHVLJQFKDQJHRIWKH ZKHHOUROOLQJVSHHG ,Q$SSHQGL[DFRPSOHWHVHWRISDUDPHWHUYDOXHVKDVEHHQOLVWHGIRUERWKWKH 0DJLF)RUPXODPRGHODQGWKH6:,)7PRGHO
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
'\QDPLF7\UH0RGHO3HUIRUPDQFH
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
'HGLFDWHG'\QDPLF7HVW)DFLOLWLHV
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
]
DW]HURVSHHG WKH
YDOXHV RI WKH DYHUDJH YHUWLFDO IRUFH GHULYHG IURP PHDVXUHPHQWV DW GLIIHUHQW VSHHGV7KHYDOXHVKDYHEHHQREWDLQHGIURP(TV DIWHUKDYLQJILWWHG WKHSDUDPHWHUVLQYROYHG7KHORDGVVKRZQDSSO\IRUWKHFDVHVRIQRPLQDOORDGV 1LQGLFDWHGLQWKHJUDSKVSUHVHQWHGLQWKHQH[WVHFWLRQ 0XFKFDUHKDVEHHQWDNHQWRPDNHWKHULJVXIILFLHQWO\ULJLG7KLVUHVXOWHGLQ DORZHVWQDWXUDOIUHTXHQF\MXVWRYHU+]ZKLFKDOORZVWKHXVHRIWHVWGDWDXS WRFD+],QDGGLWLRQWRDYRLGIRUFHFURVVWDONWKHEUDNHVKDIWLVFRQQHFWHG
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
frames to constrain axle arm
hinge
cleat drum 2.5 m diameter
flexible couplings piezo force transducers wheel speed sensor
brake torque sensor disc brake
drum
)LJ '\QDPLFEUDNHDQGFOHDWWHVWIDFLOLW\LQVLGHDQGIURQWYLHZ
ZLWKWKHZKHHOVKDIWWKURXJKDQLQWHUPHGLDWHVKDIWHTXLSSHGZLWKWZRIOH[LEOH FRXSOLQJV7KHVHFRXSOLQJVDUHIOH[LEOHLQDOOGLUHFWLRQVH[FHSWDERXWWKHD[LVRI URWDWLRQ7KLVHQVXUHVWKDWRQO\WKHEUDNHWRUTXHLVWUDQVPLWWHGWRWKHZKHHODQG WKDWRWKHUSDUDVLWLFIRUFHVDQGPRPHQWVDUHODUJHO\VXSSUHVVHG7KHEUDNHWRUTXH LVPHDVXUHGZLWKVWUDLQJDXJHVDWWDFKHGWRWKHLQWHUPHGLDWH FRXSOLQJVKDIW$ K\GUDXOLFVHUYRV\VWHPLVXVHGWRFRQWUROWKHEUDNHSUHVVXUH7KHWHVWIDFLOLWLHV ZLWKQXPHURXVH[SHULPHQWDODQGVLPXODWLRQUHVXOWVKDYHEHHQGHVFULEHGLQGHWDLO E\=HJHODDU
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
7DEOH 9HUWLFDOORDGRQPGUXPDWFRQVWDQWD[OHKHLJKWDQGLQFUHDVLQJ VSHHG W\UH 5 9DWEDU
9[>NPK@
LQLWLDOYHUWLFDOGHIOHFWLRQ
]>PP@
YHUWLFDOORDG)1>1@DWFRQVWDQWD[OHKHLJKW
3HQGXOXPDQG
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
drum
hinge
piezo electric measuring hub and steering head
hydraulic actuator
)LJ 7KH WUDLOLQJ DUP µSHQGXOXP¶ WHVW ULJ H[FLWLQJ WKH W\UH DOPRVW SXUHO\ODWHUDOO\ )UHTXHQFLHVXSWRFD+]DGHTXDWHIRUDVVHVVLQJWKHW\UHUHOD[DWLRQOHQJWKDQG J\URVFRSLFFRXSOLQJSDUDPHWHU
drum
piezo electric measuring hub flexible hinge rigid member
hydraulic actuator
)LJ 7KH\DZRVFLOODWLRQWHVWULJIHDWXULQJFHQWUHSRLQWVWHHULQJ)UHTXHQFLHVXSWRFD +]HQDEOLQJWKHDVVHVVPHQWRIW\UHRXWRISODQH LQHUWLDDQGVWLIIQHVVSDUDPHWHUV LQFOXGLQJUHVLGXDOVWLIIQHVVHVDQGULJLGPRGHV
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
WKHWHVWWKHORDGHGUDGLXVUHPDLQVFRQVWDQW 7KH PHDVXULQJ WRZHU SURYLGHG ZLWK D K\GUDXOLF YHUWLFDO D[OH SRVLWLRQLQJ LQVWDOODWLRQLVXVHGWRFRQGXFWSXUHEUDNLQJDQGSXUHVLGHVOLSWHVWVDVZHOODV FRPELQHGVOLSH[SHULPHQWVDWD[OHKHLJKWRVFLOODWLRQVDQGUDGLDOG\QDPLFVWLIIQHVV WHVWVXSWRFD+] $GHWDLOHGGHVFULSWLRQRIWKHYDULRXVVLGHVOLSWHVWIDFLOLWLHVWRJHWKHUZLWKD IXOODFFRXQWRQWKHQXPHURXVH[SHULPHQWVFRQGXFWHGDQGWKHVLPXODWLRQUHVXOWV RIWKHPRGHOKDYHEHHQJLYHQE\0DXULFH DQGIRUPRWRUF\FOHW\UHVE\'H 9ULHV DE
'\QDPLF7\UH6LPXODWLRQDQG([SHULPHQWDO5HVXOWV
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
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
IUHH
rotational 0Hz
vertical 74Hz
torsional 78Hz
longitudinal 74Hz
in-phase 33Hz
vertical 80Hz
anti-phase 76Hz
longitudinal 100Hz
ORDGHG
)LJ &DOFXODWHGLQSODQHYLEUDWLRQDOPRGHVRIW\UHZKHHOV\VWHPZLWKD[OHIL[HGDQGW\UH
)1
IUHHRUORDGHGRQWKHGUXPVXUIDFHZLWKYHUWLFDOORDG
1DQGDW]HURVSHHG
UHDOLVWLFUROOLQJH[SHULPHQWV 7KH FDOFXODWHG YLEUDWLRQDO ULJLG ERG\ PRGHV DW ]HUR VSHHG XVLQJ WKH SDUDPHWHUVDVHVWDEOLVKHGIURPH[SHULPHQWVFDUULHGRXWRQWKHGUXPKDYHEHHQ GHSLFWHGLQ)LJV DQG
IUHH
lateral 42Hz
yaw 46Hz
camber 44Hz
yaw 46Hz
camber 46Hz
ORDGHG
lateral 103Hz
)LJ &DOFXODWHGRXWRISODQHYLEUDWLRQDOPRGHVRIW\UHZKHHOV\VWHPZLWKD[OHIL[HGDQG W\UH IUHH RU ORDGHG RQ WKH GUXP VXUIDFH ZLWK YHUWLFDO ORDG VSHHG
)1
1 DQG ]HUR
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
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¶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
7DEOH 1DWXUDO IUHTXHQFLHV DQG GDPSLQJ RI YLEUDWLRQDO PRGHV RI ULJLG ULQJW\UHPRGHO
)1
FDOFXODWHGDWDYHUWLFDOORDG
IRUZDUGVSHHG>PV@
1IRUWZRYDOXHVRIIRUZDUGVSHHG
Q>+]@
>@
LQSKDVH
YHUWLFDO
DQWLSKDVH
ORQJLWXGLQDO
FDPEHU
\DZ
ODWHUDO
LQSODQHPRGHV
RXWRISODQHPRGHV
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
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
200
30 EHOW
)[
)[
G\QDPLFV
0%
VOLSVWLIIQHVV
κ
[N]
[1/m] 0
0
180 90
180
experiment model
0
0
-90
-90
-180 0
20
Q
40
UHOD[DWLRQOHQJWK
90
60
[Hz]
80
100
-180 0
9 )1
= 25 km/h = 4000 N
20
Q
40
60
[Hz]
80
100
)LJ 0HDVXUHGDQGFDOFXODWHGLQSODQHIUHTXHQF\UHVSRQVHIXQFWLRQVDWDQDYHUDJHEUDNLQJ IRUFHRI1DQGDVVHVVHGDWDEUDNLQJIRUFHVWDQGDUGGHYLDWLRQRI1
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
7KH RXWRISODQH IUHTXHQF\ UHVSRQVH IXQFWLRQV RI WKH VLGH IRUFH DQGWKH
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µSHQGXOXP¶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
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
10
6
10
9 [=
)\ 10
experiment
ψ
α0 = 0
5
6
0]
25 km/h 59 km/h 92 km/h
ψ
10
N rad
4
Nm rad
10
4
10
180
180
90
φ )ψ
90
φ0ψ
0
[o ]
0
[o ]
-90
-90
-180 10
10
0
Q
6
10
1
-180
10
9 [=
)\
ψ
10
0
6
Q
10
Q
10
1
[Hz]
model
0] ψ
α0 = 0
5
10
[Hz]
25 km/h 59 km/h 92 km/h
10
N rad
4
Nm rad
10
4
10
180
2
180
90
φ )ψ
2
90
φ0ψ
0
[ o]
0
[ o]
-90
-90
-180 10
0
Q
10
1
-180 10
[Hz]
0
1
[Hz]
)LJ 0HDVXUHGDQGFDOFXODWHGRXWRISODQHIUHTXHQF\UHVSRQVHIXQFWLRQVRIWKHVLGHIRUFH DQG DOLJQLQJWRUTXHWRVWHHUDQJOHYDULDWLRQVDW]HURDYHUDJHVOLSDQJOHIRUWKUHH
)1
YDOXHVRIIRUZDUGVSHHGDQGQRUPDOORDG
1
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
10
α0 =
6
10
o 0 o 1 o 3o
)\
ψ
10
experiment
0] ψ
5 [ =59 km/h
9
5
6
10
N rad
4
Nm rad
10
4
10
180
180
90
φ)ψ
2
φ0ψ
0
[ o]
90 0
[ o] -90
-90
-180 0 10
10
Q
10
1
-180 0 10
[Hz]
α 0=
6
10
o 0 o 1 o 3o
)\
ψ
10
Q
10
1
[Hz]
model
ψ
9
10
N rad
4
Nm rad
10
4
10
180
2
180
90
φ )ψ
10
0]
5 =59 km/h [
5
6
Q
90
φ0 ψ
0
0
[ o]
[ o]
-90
-90
-180 10
0
Q
10
1
-180 10
[Hz]
0
1
[Hz]
)LJ 0HDVXUHGDQGFDOFXODWHGRXWRISODQHIUHTXHQF\UHVSRQVHIXQFWLRQVRIWKHVLGHIRUFH DQGDOLJQLQJWRUTXHWRVWHHUDQJOHYDULDWLRQVDWRQHYDOXHRIVSHHGDQGDWIRXUOHYHOV
)
RIDYHUDJHVLGHVOLSDQGZKHHOORDG
1
1
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
3000
23
)[
simulation
2000
simulation
Ω
[N] 1000
22
23
[rad/s] 21
22 experiment
20
0
0
21
experiment
0
time [s]
2.5
20
0
20 2.5
time [s]
3000 simulation
)[
experiment
2000
[N] 1000
)1 =4000 N, 9[=25km/h 0
0
2
4
-κ
6
8
[%]
10
12
)LJ %UDNHIRUFHUHVSRQVHWRVXFFHVVLYHVWHSLQFUHPHQWVRIEUDNHSUHVVXUH7KHXSSHU GLDJUDPVGHSLFWWKHYDULDWLRQRIIRUFHDQGZKHHOVSHHGRIUHYROXWLRQZLWKWLPH 7KH ORZHU ILJXUH VKRZV WKH ORRSV LQ WKH IRUFH YHUVXV ZKHHO VOLS GLDJUDP $SSDUHQWO\WKHDFWXDOIULFWLRQFRHIILFLHQWLVORZHUWKDQDVVXPHGLQWKHPRGHO =HJHODDU
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
time
1
0
[s]
2
3
4
)1 =3700 N, 9[=25 km/h
4000
)\
experiment
3000
o
5
o
6
7
o
o
o
8
4000
)\
4 o
2000
3000
3
[N]
2
o
simulation
1000
2000
o
[N]
1
1000
0
ψ =0 1
0
0
1
2
time
time
[s]
2
3
[s]
3
0
4
4
200 0]
experiment
100 0 [Nm] 200
-100
0]
-200
100 0 [Nm] -100 simulation -200 0
1
time
2
[s]
3
4
)LJ 6LGHIRUFHDQGDOLJQLQJWRUTXHUHVSRQVHWRVXFFHVVLYHVWHSFKDQJHVLQVWHHUDQJOH 0DXULFH
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
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
'\QDPLF7\UH5HVSRQVHWR6KRUW5RDG 8QHYHQQHVVHV
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
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
30 mm 10 mm
15 mm
15 mm
50 mm
trapezium
positive step
negative step
)LJ 7KUHHW\SLFDOREVWDFOHVXVHGLQ=HJHODDU¶VUHVHDUFK
7\UH(QYHORSPHQW3URSHUWLHV
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
)
)LJXUH SUHVHQWVWKHPHDVXUHGYHUWLFDOORDG
KRUL]RQWDOORQJLWXGLQDO
9
U 7KHODWWHU TXDQWLW\LVREWDLQHGIURPWKHYDULDWLRQRIWKHZKHHOURWDWLRQUDWHG GVZKLFKLV )
IRUFH
DQGWKHIURPPHDVXUHPHQWVGHULYHGHIIHFWLYHUROOLQJUDGLXV +
H
GHILQHGDV WKHGLIIHUHQFHRIWKHLQFUHPHQWDOZKHHODQJXODUGLVSODFHPHQWDQGWKH FRQVWDQW XQGLVWXUEHG LQFUHPHQWDOURWDWLRQDVDUDWLRWRWKHLQFUHPHQWRIWKH WUDYHOOHGGLVWDQFHRIWKHZKHHOD[OH7KHIROORZLQJHTXDWLRQVDSSO\
UH '
9[
9[ ' V W G
G
'
R&
G G
W
UHR '
9[ R
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
10000 8000
)9 [N]
6000 4000 2000 0 800
)+
400
0
[N] -400
-800
UH
320
300
[mm] 280
260 -100
0
100
-100
V
0
100
-100
0
100
[mm]
)LJ 5ROOLQJRYHUDWUDSH]LXPFOHDWDQXSZDUGDQGDGRZQZDUGVWHS )LJ DWYHU\ ORZVSHHGDQG IRUWKUHHD[OHKHLJKWV'LDJUDPVVKRZPHDVXUHGYDULDWLRQVRIWKH
)
YHUWLFDOIRUFH
)
XSZDUG WKHORQJLWXGLQDOKRUL]RQWDOIRUFH
9
U
HIIHFWLYHUROOLQJUDGLXV
IRUZDUG DQGWKH
+
(T 7\UHGLPHQVLRQV 5
H
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
DQGKHQFH
U H . UHR % UHR
G G
V
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µVKHDUIRUFH¶WKDW DFWVRQWKHHQGSRLQWVRIWKHUDGLDOVSULQJV0RXVVHDXHWDO DQG2HUWHO VLPXODWHWKHW\UHUROOLQJRYHUDSRVLWLYHVWHSE\PHDQVRI GLIIHUHQWW\SHV RI ILQLWHHOHPHQWPRGHOV=HJHODDUXVHVWKHIOH[LEOHULQJ EHOW PRGHORI*RQJ WKDWZDVGHYHORSHGZLWKWKHDLGRIWKHPRGDOH[SDQVLRQPHWKRGDVD UHIHUHQFHPRGHOLQKLVUHVHDUFK7KHDGGLWLRQRIWUHDGHOHPHQWVZLWKUDGLDODQG WDQJHQWLDOFRPSOLDQFHVWR*RQJ¶VPRGHOGLGHQDEOH=HJHODDUWRHPSOR\WKHPRGHO IRU WKH VWXG\ RI WUDYHUVLQJ REVWDFOHV $OVR WKLV PRGHO VKRZV UHVSRQVHV YHU\ VLPLODU WR WKH PHDVXUHG EHKDYLRXU 6FKPHLW] HPSOR\V WKH UDGLDO LQWHUUDGLDOPRGHODVDSRVVLEOHEDVLVIRUWKHSUDJPDWLFPRGHOUXQQLQJRYHUDQ DUELWUDU\URDGVXUIDFH
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
) E\WKH ZLVREWDLQHG
DSSUR[LPDWHO\HTXDOWRWKHYHUWLFDOFRPSRQHQWRIWKHQRUPDOIRUFH UDGLDOVWLIIQHVVRIWKHW\UHWKHHIIHFWLYHURDGKHLJKWYDULDWLRQ
1
7KHHIIHFWLYHVORSHLVIRXQGE\GLYLGLQJWKHORQJLWXGLQDOKRUL]RQWDOIRUFH DIWHU
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
measured vertical and horizontal forces ) +
KD[OH
wheel axle height
YHU\ORZVSHHG
)
radial deflection
ρ
I)
rolling resistance force
U
Z
1
) +
V
distance travelled
]
1
)
effective road plane height - (V) and slope β (V)
H[SHULPHQW
9
)
actual road profile
9
normal load
)LJ 7KH ZKHHOUROOHGRYHUDURDGLUUHJXODULW\DWFRQVWDQWD[OHKHLJKWWRHVWDEOLVKWKH HIIHFWLYHURDGSODQHYDULDWLRQ
KDYLQJ VXEWUDFWHG WKH UHODWLYHO\ VPDOO UROOLQJ UHVLVWDQFH IRUFH E\ WKH YHUWLFDO IRUFH%RWKHIIHFWLYHTXDQWLWLHVDUHIXQFWLRQVRIWKHORQJLWXGLQDOFRRUGLQDWHRIWKH
V
ZKHHOFHQWUH 7KHIROORZLQJIRUPXODHDSSO\FI(T
) &Z ' 9 . &)] a
WDQ
'&
a
)1 a
]
FRV
)+ % IU )1 )9
'
&)]
FRV
FRV
.&
)+ % IU )9 )9
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
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
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
basic curve
shifted basic curves
1 2 KE OE
H -1
OV
wheel centre Z
V
effective road surface height Z
Z
actual road (trapezium)
V
+
OV
β OV
effective road surface slope
OV
β OV
β
+ low load
high load
)LJ 7KHFRQVWUXFWLRQRIWKHFXUYHVUHSUHVHQWLQJWKHHIIHFWLYHKHLJKWHQGHIIHFWLYHVORSH RIWKHORFDOURDGSODQHIURPWKHEDVLFIXQFWLRQDVVRFLDWHGZLWKWKHWUDSH]LXPFOHDW
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
virtual road profile at axle level
effective road plane at axle level V
)1R
IU )1R
slope
)1
9[
β
effective road plane
IU )1
height Z
Z
ρ]R
IU )1R ) 1R
ρ~] ρ]R
ρ] = ρ]R+ ρ]
~
β
IU)1 ~
)1 = )1R+ )1
)LJ 7KHYLUWXDOURDGSURILOHDQGHIIHFWLYHURDGSODQHDVVHQVHGDWZKHHOD[OHOHYHO
DQGVORSH7KHVHWZRHTXLYDOHQWTXDQWLWLHVZLOOODWHUEHFRPSOHWHGZLWKDWKLUG TXDQWLW\ WKHHIIHFWLYHUROOLQJUDGLXVWRIRUPWKHFRPSOHWHVHWRIWKUHHHIIHFWLYH LQSXWVIURPURDGWRW\UHPRGHO )LJXUH GHPRQVWUDWHVWKHXVHRIWKHEDVLFFXUYHIRUWKHHIIHFWLYHURDG SODQH KHLJKW DSSOLFDEOH IRU WKH W\UH UROOLQJ RYHU D VKRUW WUDSH]RLGDO FOHDW DW D FRQVWDQWD[OHKHLJKW7KHYHUWLFDOVFDOHKDVEHHQH[DJJHUDWHG7KHEDVLFFXUYH LVDSSUR[LPDWHGE\DKDOIVLQHZDYH7KHEDVHOHQJWKRIWKHFXUYHLVGHQRWHGZLWK
O LWVKHLJKWZLWKK DQGWKHVKLIWZLWKO 7KHZLWKWUDYHOOHGGLVWDQFHVYDU\LQJ YDOXHVRIZDQG GHILQHWKHORFDOHIIHFWLYHZKHHOSODQH)LJXUH FODULILHVWKH DFWXDOVLWXDWLRQ$WZKHHOSRVLWLRQ VWKHFXUUHQWHIIHFWLYHURDGSODQHKDVEHHQ E
E
V
LQGLFDWHG7KHYLUWXDOURDGSURILOHGHILQHGDVWKHSDWKRIWKHZKHHOFHQWUHWKDW ZRXOGRFFXUDWFRQVWDQWQRUPDOORDGKDVEHHQGUDZQWRJHWKHUZLWKWKHDFWXDO URDGSURILOH7KHGLVWDQFHRIWKHZKHHOFHQWUHORFDWLRQZLWKUHVSHFWWRWKHYLUWXDO URDGSURILOHFRUUHVSRQGVWRWKHLQFUHDVHRIWKHDFWXDOUDGLDOW\UHGHIOHFWLRQ7KH GLVWDQFHRIWKHZKHHOFHQWUHWRWKHLQGLFDWHGHIIHFWLYHURDGSODQH WUDQVODWHGWR D[OHOHYHOZLWK
Z DQG
UHJDUGHG LVWKHLQFUHDVHLQHIIHFWLYHUDGLDOGHIOHFWLRQ
ZKLFKWRJHWKHUZLWKWKHLQLWLDOGHIOHFWLRQEHFRPHVHTXDOWR
]
7KH7ZR3RLQW)ROORZHU7HFKQLTXH ,Q)LJ DQDOWHUQDWLYHWHFKQLTXHLVLQWURGXFHGXVLQJDVLQJOHEDVLFFXUYH ZLWKIXOOKHLJKW
K DQGDWZRSRLQWIROORZHU,IWKHWZRSRLQWVDUHPRYHGDORQJ E
WKHEDVLFFXUYHWKHPLGSRLQWGHVFULEHVDFXUYHWKDWUHSUHVHQWVWKHFKDUDFWHULVWLF
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
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
7KHSDUDPHWHUVWKDWFRQWUROWKH
VL]HDQGSRVLWLRQRIWKHEDVLFFXUYHDUHWKHOHQJWK
O
RIIVHW I7KHKHLJKW
O WKHVKLIW O DQGVRFDOOHG E
V
K LVHTXDOWRWKHVWHSKHLJKW7KHRIIVHWLVDQHZTXDQWLW\ E
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
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
Z
basic function
OV
effective road height
-β
OV
two-point follower
-Z
-Z
β
KE
OE
β
effective road slope
)LJ $OWHUQDWLYHPHWKRGWRGHWHUPLQHWKHHIIHFWLYHURDGKHLJKWDQGVORSHXVLQJWKHEDVLF
O
K
FXUYHZLWKKHLJKW EDQGWKHWZRSRLQWIROORZHUZLWKOHQJWKHTXDOWRWKHVKLIW V
Z
OI
elementary basic curve
effective road height
OV
Z
KE
Z
β
OE
β
effective road slope
)LJ 7KH HIIHFWLYH URDG UHVSRQVH FXUYHV UHVXOWLQJ IURP WKH HOHPHQWDU\ EDVLFFXUYH DVVRFLDWHGZLWKWKHVWHSFKDQJHLQURDGOHYHOE\XVLQJWKHWZRSRLQWIROORZHUFRQFHSW
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
rigid wheel
path center rigid wheel path lowest point rigid wheel
Kstep OE )LJ 7KHULJLGZKHHOUROOLQJRYHUDQXSZDUGVWHSFKDQJHLQURDGOHYHO
constant vertical force:
150
9=
)
2000 4000 6000 N
path lowest point of rigid wheel
OE
[mm] rigid wheel
0 140
OI
elementary basic function
length
shift
OV
[mm]
KE Kstep
OV
2D 40 50
offset
OI
[mm]
0 -10 OE
0
K step
[mm]
30
)LJ 3DUDPHWHU YDOXHV IRU WKH HOHPHQWDU\ EDVLF FXUYH HVWDEOLVKHG DW FRQVWDQWYHUWLFDO ORDGVE\FRPSXWDWLRQVXVLQJDIOH[LEOHULQJPRGHOZLWKWUHDGHOHPHQWV=HJHODDU
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
,Q )LJ =HJHODDU¶V FDOFXODWHG UHVXOWV KDYH EHHQ SUHVHQWHG IRU D VHULHV RI
)97KHGLDJUDPVVKRZWKDWWKHEDVLFFXUYHOHQJWKOEDQGWKHRIIVHW OI GR FKDQJH ZLWK VWHS KHLJKW KVWHS EXW DUH DSSUR[LPDWHO\ LQGHSHQGHQW RI WKH YHUWLFDOORDG)97KHFXUYHOHQJWKPD\EHHVWLPDWHGIURPWKHFLUFOHFXUYHOHQJWK
YHUWLFDOORDGV
)LJ
OE ' U R & UR& K
VWHS
O
D
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
7KH7DQGHP(JJ7HFKQLTXH ,QVWHDG RI XVLQJ WKH EDVLF SURILOH DQG UXQQLQJ RYHU WKDW ZLWK WKH WZRSRLQW IROORZHUZHPD\JHWDVWHSFORVHUWRWKHDFWXDOW\UHVKDSHPRYLQJRYHUWKHDFWXDO URDGVXUIDFHSURILOH6FKPHLW]GLVFRYHUHGWKDWWKHSULQFLSOHRIWKHFLUFOHPRYLQJ RYHUWKHVXUIDFH )LJ PD\EHDGRSWHGEXWWKDQZLWKDQHOOLSVHLQVWHDGRIWKH
DH
EH [H
]H
path of egg centre
basic curve
)LJ µ(JJ¶PRYLQJRYHUVWHSURDGSURILOHSURGXFLQJEDVLFFXUYH
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
OV
β
-Z basic curve
)LJ 7KHWZRHJJWDQGHPFRQILJXUDWLRQRI6FKPHLW]JHQHUDWLQJWKHHIIHFWLYHURDGKHLJKW DQGVORSH
FLUFOH 7KH HOOLSVH WKDW WDNHV WKH VKDSH RI D VWDQGLQJ HJJ KDV D KHLJKW DSSUR[LPDWHO\HTXDOWRWKDWRIWKHW\UHEXWDFXUYDWXUHUDGLXVDWWKHORZHVWSRLQW FRQVLGHUDEO\VPDOOHUWKDQWKDWRIWKHIUHHW\UH,QWKDWZD\WKHHJJWRXFKHVWKH VWHSODWHUWKDQWKHFLUFOH%\FKRRVLQJDQRSWLPDOVKDSHRIWKHHOOLSVHWKHUROHRI WKH RIIVHW
OI WKDW FKDQJHV ZLWK VWHS KHLJKW RI WKH EDVLF FXUYH LQGLFDWHG LQ
)LJ FDQEHWDNHQFDUHRIDXWRPDWLFDOO\,Q)LJ WKHHJJPRYLQJRYHUD VWHSKDVEHHQLOOXVWUDWHG7KHGLPHQVLRQVRIWKHHJJDUHGHILQHGE\WKHHOOLSVH SDUDPHWHUV,QWHUPVRIWKHFRRUGLQDWHV
[H DH
FH
%
\H EH
FH
'
[HDQG]HWKHHOOLSVHHTXDWLRQUHDGV
7KH HIIHFWLYH URDG KHLJKW DQG VORSH FDQ EH DVVHVVHG E\ XVLQJ WZR HJJV IROORZLQJ HDFK RWKHU DW D GLVWDQFH HTXDO WR WKH VKLIW OHQJWK
OV 7KH FKDQJH LQ
KHLJKW RI WKH PLGSRLQW RI WKH FRQQHFWLQJ OLQH DQG WKH LQFOLQDWLRQ RI WKLV OLQH UHSUHVHQWWKHHIIHFWLYHURDGKHLJKWDQGHIIHFWLYHVORSH)LJXUH LOOXVWUDWHVWKLV µWDQGHP HJJ¶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
OV GRHV
GHSHQGRQWKHYHUWLFDOORDG ,QDPRUHHIILFLHQWZD\RIRSHUDWLRQWKHEDVLFFXUYHRUSURILOHLVDVVHVVHG ILUVWWKDWLVEHIRUHWKHDFWXDOZKHHOUROOVRYHUWKHURDGVHFWLRQFRQVLGHUHG7KLV
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
belt
path of egg centre
2-point follower basic profile
actual road profile
)LJ 7KHHJJJHQHUDWLQJWKHEDVLFURDGSURILOHDQGWKUHHSRLQWIROORZHUPRYLQJRYHULW
LVDFKLHYHGE\VHQGLQJRQHHJJRYHUDVKRUWVHFWLRQRIWKHURDGDKHDGDQGKDYLQJ WKDWGHWHUPLQHWKHEDVLFFXUYH7KHWZRSRLQWIROORZHULVVXEVHTXHQWO\PRYHG FRQFXUUHQWO\ ZLWK WKH DFWXDO ZKHHO IRUZDUG PRWLRQ RYHU WKH EDVLF SURILOH WKHUHE\JHQHUDWLQJWKHHIIHFWLYHURDGLQSXWVZKLFKDUHIHGLQWRWKHW\UHPRGHO )LJXUH LOOXVWUDWHV WKH SURFHGXUH :KHQ WUDYHUVLQJ D VLQJOH VWHS LW LV RI FRXUVHPRUHHIILFLHQWWRXVHDQDQDO\WLFH[SUHVVLRQIRUWKHEDVLFFXUYHEDVHGRQ (T
7KH(IIHFWLYH5ROOLQJ5DGLXVZKHQ5ROOLQJRYHUD&OHDW 7KHWKLUGHIIHFWLYHLQSXWLVFRQVWLWXWHGE\WKHHIIHFWLYHUROOLQJUDGLXVWKDWFKDQJHV ZKHQ D URDG XQHYHQQHVV LV WUDYHUVHG ,Q )LJ WKHVH YDULDWLRQV KDYH EHHQ VKRZQLQWKHORZHUGLDJUDPV7KHFXUYHVKDYHEHHQGHULYHGIURPPHDVXUHPHQWV E\XVLQJWKH(TV ,QWKHHIIRUWWRPRGHOWKHDVSHFWRIUROOLQJRYHUDQREVWDFOHLWLVLPSRUWDQWWR UHDOLVHWKDWZHKDYHWKUHHHOHPHQWVWKDWFRQWULEXWHWRWKHYDULDWLRQRIWKHUROOLQJ UDGLXV
LQFUHPHQWLQQRUPDOORDG
WKHORFDOVORSH
WKHORFDOFXUYDWXUH
)LJXUH LOOXVWUDWHVWKHPDWWHU7KHILUVWLWHPKDVEHHQGHDOWZLWKEHIRUHFI 6HF)LJ(T DQG(T $FFRUGLQJWRWKHODWWHUHTXDWLRQ WKHHIIHFWLYHUDGLXVLVDIXQFWLRQRIYHUWLFDOORDGDQGVSHHGRIUROOLQJ,Q)LJ PRGHO FRQVLGHUDWLRQV DQG WKH IURP H[SHULPHQWV UHVXOWLQJ JUDSK KDYH EHHQ UHSHDWHG)RUVPDOOYDULDWLRQVLQUDGLDOGHIOHFWLRQZHPD\HPSOR\WKHHTXDWLRQ
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
9[
Ω
-Z
)1
V
9[
Ω
9[
β
ORDGFKDQJH
VORSH
FXUYDWXUH
)LJ 7KUHHFRQWULEXWLRQVWRWKHFKDQJHLQHIIHFWLYHUROOLQJUDGLXV
a
UH
'&
a
]
7KHVHFRQGFRQWULEXWLRQDFFRXQWVIRUWKHIDFWWKDWDWDVORSHWKHUXQQLQJVSHHG SDUDOOHOWRWKHURDGVXUIDFHLVVPDOOHUWKDQWKHIRUZDUGKRUL]RQWDOYHORFLW\:H KDYHIRUWKHFKDQJHLQ
U
H
9
[
UHVORSH ' UHR & FRV a
7KHWKLUGFRQWULEXWLRQFRPHVIURPWKHURDGVXUIDFHFXUYDWXUH7KHDQDO\VLVWKDW DWWHPSWVWRPRGHOWKHUHODWLRQEHWZHHQFXUYDWXUHDQGHIIHFWLYHUROOLQJUDGLXVLV PRUHGLIILFXOWDQGUHTXLUHVVSHFLDODWWHQWLRQ)LJXUH XQUDYHOVWKHSURFHVVRI UROOLQJRYHUDFXUYHGREVWDFOHDQGLQGLFDWHVWKHFRQQHFWLRQZLWKUROOLQJRYHUD GUXPVXUIDFHZLWKHTXDOFXUYDWXUH,QFRQWUDVWWRWKHGUXPWKHREVWDFOHGRHVQRW URWDWH&RQVHTXHQWO\WRFRPSDUHWKHSURFHVVRIUROOLQJRYHUDFXUYHGREVWDFOH ZLWKWKDWRIUROOLQJRYHUDURWDWLQJGUXPVXUIDFHZHPXVWDGGWKHHIIHFWRIWKH
9[
Ω
UR UE
UH
) 1=
UE
UO
0
UH
UE
6
)1
UH )1
6above road surface
310
6below road surface
[mm]
UH UO
290
UR
0
UO
UH
4000 [N]
)1
)LJ 7KHHIIHFWLYHUROOLQJUDGLXVYDU\LQJZLWKDFKDQJHLQQRUPDOORDG
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
V
9[
Ω
9[
+
=
Ω ΩGU
.
slope rate -β
drum
)LJ 8QUDYHOOLQJWKHSURFHVVRIUROOLQJRYHUDFXUYHGREVWDFOH
W\UHVXSSRUWHGE\DFRXQWHUURWDWLQJVXUIDFHWKDWGRHVQRWPRYHIRUZDUGV7KH OHIWKDQGGLDJUDPRI)LJ GHSLFWVDSRVVLEOHWHVWFRQILJXUDWLRQZLWKDSODQN WKDW FDQ EH WLOWHG DERXW D WUDQVYHUVH OLQH LQ WKH FRQWDFW VXUIDFH :KHQ EHLQJ
6
WLOWHGSRLQW WKDWLVDWWDFKHGWRWKHZKHHOULPPXVWPRYHDORQJZLWKWKHSODQN LQORQJLWXGLQDOGLUHFWLRQVLQFHWKHZKHHOVOLSLV]HUR
9
V[
DVEUDNHRUGULYH
WRUTXHLVQRWDSSOLHG&RQVHTXHQWO\WKHZKHHOURWDWHVVOLJKWO\DQGWKHIROORZLQJ HTXDWLRQDSSOLHV
a
9V[ ' & UHR & UO G & UHR ' GW
7KH DSSDUHQW YDULDWLRQ LQ HIIHFWLYH UROOLQJ UDGLXV GHULYHV IURP WKH HTXDWLRQ
UH ' a
9[
&
9[
.&
9[
a
R
R
VRWKDWZHREWDLQIRUWKHFRQWULEXWLRQIURPWKHVORSHUDWH
UH a
VORSH UDWH
'
UHR &UO R
G G
W
7KH UHODWLRQVKLS KDV EHHQ FRQILUPHG WR KROG WKURXJK HODERUDWH H[SHULPHQWVFRQGXFWHGE\=HJHODDU RQWKHWLOWLQJSODQNRIWKH'HOIWIODW SODQNPDFKLQH 7KH RWKHU FRQWULEXWLRQ WKDW FRPHV IURP WKH GUXP DQDORJXH LV IRXQG E\ FRQVLGHULQJWKHVLPSOHPRGHOVKRZQLQWKHULJKWKDQGGLDJUDPRI)LJ 7KH
G DUHDVVXPHGWRVWDQGSHUSHQGLFXODUO\RQWKHGUXP
WUHDGHOHPHQWVZLWKOHQJWK
W
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
VXUIDFH7KHGUXPKDVDFXUYDWXUHZLWKUDGLXV 5GU7KHEHOW ZLWK UDGLXV UELV FRQVLGHUHGLQH[WHQVLEOH$VDFRQVHTXHQFHZHILQGWKHIROORZLQJUHODWLRQEHWZHHQ WKHZKHHODQGGUXPYHORFLWLHV
UE '
GU 5GU
%
G
W
'
%
9
[
G
W
5
GU
DQG IRU WKH HIIHFWLYH UROOLQJ UDGLXV IRU WKH W\UH UROOLQJ IUHHO\ RYHU WKH GUXP VXUIDFH
U
HGUXP
9
'
[
U
'
%
E
G
W 5GU
)RUWKHPRGHOWKHHIIHFWLYHUROOLQJUDGLXVRIWKHW\UHUROOLQJRYHUDIODWVXUIDFH LVHTXDOWRWKHUDGLXVRIWKHEHOW U
HR
'
U
E
1RZWKHWUHDGGHSWKLV G
W
' R& U
U
HR
+HQFH WKH H[SUHVVLRQ IRU WKH HIIHFWLYH UROOLQJ UDGLXV RQ WKH GUXP FDQ EH UHZULWWHQDV
U
HGUXP
'
U
HR
% R& U
HR 5GU
U
.
U
HR
&
U
R
&
U
HR
5
GU
7KHYDULDWLRQRIWKHUDGLXVEHFRPHV a
U
HGUXP
' & R& U
HR
U
U
HR
5
GU
7KHGUXPUDGLXVLVHTXDOWRWKHUDGLXVRIFXUYDWXUHRIWKH HIIHFWLYH URDGVXUIDFH
WHVW
tyre
Ω
~
Ω UO
tilting plank
6
. β
G
W
9[
UHR
9V[
UE
5GU drum
)LJ 7ZRFRPSRQHQWVRIUROOLQJRYHUDFXUYHGREVWDFOH
ΩGU
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
SURILOH7KLVFXUYDWXUHFRUUHVSRQGVWRFKDQJHLQVORSH ZLWKWUDYHOOHGGLVWDQFH
V1RWHWKDWIRUWKHFRQYH[GUXPVXUIDFHWKH UDWHRIFKDQJHLVQHJDWLYH6RZH KDYH
'&G
5
GV
GU
7KHVORSHUDWHRIFKDQJHPD\EHZULWWHQDV
G GV
'
G
'
[ GW
9
'
G
H GW
U
7KHGUXPFRQWULEXWLRQLVQRZH[SUHVVHGDV a
U
HGUXP
U
'
R
&
U
HR G
GW
$GGLQJ XS WKH WZR FRQWULEXWLRQV DQG \LHOGV WKH YDULDWLRQ LQ HIIHFWLYHUROOLQJUDGLXVGXHWRREVWDFOHFXUYDWXUH a
U
HFXUYDWXUH
'
a
U
R&
U
'
H VORSH UDWH U
O G
R ZKHUH
]
GW
%
a
U
H GUXP
'
G
] UHR
GV
LVWKHUDGLDOFRPSUHVVLRQRIWKHW\UH%\DGGLQJXSDOOWKHFRQWULEXWLRQV
ZHILQDOO\REWDLQIRUWKHYDULDWLRQRIWKHHIIHFWLYHUROOLQJUDGLXVZLWKUHVSHFWWR WKHLQLWLDOFRQGLWLRQZKHUH a
U
H
'&
a
]
%
HR
U
G
GV
& FRV %
DQG
] UHR
G
]
]R
GV
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
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
6LPXODWLRQVDQG([SHULPHQWDO(YLGHQFH
=HJHODDUDQG6FKPHLW]KDYHSHUIRUPHGQXPHURXVH[SHULPHQWVRQWKHGUXPWHVW VWDQG )LJ DQGWKHIODWSODQNPDFKLQH )LJ DQGXVHGWKH6:,)7ZLWK REVWDFOH PRGHO WR FDUU\ RXW WKH VLPXODWLRQV )LJXUH VKRZV WKH REVWDFOH SDUDPHWHUYDOXHVIRUWKHTXDUWHUVLQHEDVLFFXUYHRI=HJHODDU7DEOH JLYHVWKH SDUDPHWHUVXVHGE\6FKPHLW]ZKLFKDUHEDVHGRQWKHHOOLSVHFRQFHSW WDQGHPHJJ WHFKQLTXH 7DEOH 3DUDPHWHUYDOXHVXVHGIRUHQYHORSPHQWFDOFXODWLRQV7\UH 5EDU
U R
P
UHR
P
VORSHHIIHFWLYHUROOLQJUDGLXVFKDUDFWHULVWLF
YHUWLFDOW\UHVWLIIQHVVDW)1
&)]
1PP
UROOLQJUHVLVWDQFHFI(TV
TV\
KDOIFRQWDFWOHQJWKFI(T
D
FI7DEOH
KDOIHOOLSVHOHQJWKXQORDGHGUDGLXVDHUR
SDH
KDOIHOOLSVHKHLJKWXQORDGHGUDGLXVEHUR
SEH
HOOLSVHH[SRQHQWFH
SFH
VKLIWOHQJWKFRQWDFWOHQJWKOVD
SVK
XQORDGHGUDGLXV
HIIHFWLYHUROOLQJUDGLXVDW)1
1
1
surface canting measurung hub mechanism
wheel carrier
cleat
road surface (sledge)
air spring system
)LJ 7KH'HOIWIODWSODQNPDFKLQHZLWKVWURNHRIPDQGURDGFDQWLQJV\VWHPFI)LJ
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
15
)9 = 2000N
Z
[mm] 0 0.4
)+
0
[kN] -0.4
egg test
0.35
UH
0.30
[m] 0.25
-0.2
-0.1
0
V
0.1
0.2
[m]
15
)9 = 4000N
Z
[mm] 0 0.5
)+
0
[kN] -0.5
egg test
0.35
UH
0.30
[m] 0.25
-0.2
-0.1
0
V
0.1
0.2
[m]
15
)9 = 6000N
Z
[mm] 0 0.5
)+
0
[kN] -0.5
egg test
0.35
UH
0.30
[m] 0.25
-0.2
-0.1
0
V
0.1
0.2
[m]
)LJ 5ROOLQJ RYHU D WUDSH]LXP FOHDW OHQJWK PP KHLJKW PP 0HDVXUHGDQG FDOFXODWHGYDULDWLRQRIYHUWLFDOD[OHGLVSODFHPHQWKRUL]RQWDOORQJLWXGLQDOIRUFHDQG HIIHFWLYH UROOLQJ UDGLXV DW WKUHH GLIIHUHQW FRQVWDQW YHUWLFDO ORDGV 0HDVXUHPHQWV FDUULHG RXW RQ WKH 'HOIW IODW SODQN PDFKLQH YHU\ ORZ VSHHG DQG FDOFXODWLRQV FRQGXFWHG ZLWK WKH XVH RI WKH WDQGHP HJJ WHFKQLTXH )LJ DQG (T 7DQGHPHJJSDUDPHWHUVDFFRUGLQJWR7DEOH
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
7KHWDEOHVKRZVWKDWWKHKHLJKWDQGOHQJWKRIWKHHOOLSVHDUHVOLJKWO\ODUJHUWKDQ WKHGLPHQVLRQVRIWKHIUHHW\UH,WLVWKHH[SRQHQW WKDW JLYHV ULVHWRWKHODUJHU FXUYDWXUHRIWKHHOOLSVHQHDUWKHJURXQG )LJXUH SUHVHQWVWKHPHDVXUHGDQGFDOFXODWHGYDULDWLRQVRIWKHHIIHFWLYH
ZWKHIRUHDQGDIWIRUFH)+DQGWKHHIIHFWLYHUROOLQJUDGLXVUH
URDGSODQHKHLJKW
ZLWKWKHYHUWLFDOORDGNHSWFRQVWDQWZKLOHVORZO\UROOLQJRYHUWKHWUDSH]LXPFOHDW 7KHHIIHFWLYHURDGSODQHVORSHWDQ IROORZVIURPWKHUDWLRRIWKHKRUL]RQWDOIRUFH YDULDWLRQDQGWKHYHUWLFDOORDG7KHFDOFXODWLRQVDUHEDVHGRQWKHHJJWDQGHP FRQFHSWRI)LJ 7KHWDQGHPLVPRYHGRYHUWKHRULJLQDOURDGSURILOHDQGWKH HIIHFWLYHURDGSODQHKHLJKWDQGVORSHDUHREWDLQHG)URPWKHGHULYDWLYHRI
WKH
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
)9R = 2000 N 9[ = 39 km/h
3 fixed axle height
∆)9
2
4
10
699
1
[kN]
0
10
0 -1
-2
10
10
4
50
)+
10
6++
2 0
0
[kN]
10 -2
egg test
Ω
-2
10
-2
10
6ΩΩ
2
-6
10
0
[rad/s] -2 -0.05
-10
0
W
0.05
[s]
0.1
0.15
10
0
Q
50
[Hz]
100
)LJ 7LPHWUDFHVDQGSRZHUVSHFWUDRIYHUWLFDO IRUFHKRUL]RQWDOORQJLWXGLQDOIRUFHDQI DQJXODUYHORFLW\IRUZKHHOUXQQLQJDWJLYHQVSHHGRI NPKDQGLQLWLDOYHUWLFDOORDG RI1RYHUWUDSH]LXPFOHDWDWFRQVWDQWD[OHKHLJKWXVLQJ6:,)7W\UHSDUDPHWHUV DQGREVWDFOHSDUDPHWHUVRI7DEOH 6FKPHLW]
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
)9R = 4000 N
2 fixed axle height
∆)9
9[ = 25 km/h
1
5
10
699 0
10
[kN] 0 -1
[kN]
10
2
6++
0
10
0
-2 -4
egg test
5
Ω
5
50
4
)+
-5
10
10
-5
10
0
10
6ΩΩ -5
0
10
[rad/s] -5 -0.05
-10
0
W
0.05
[s]
0.1
0.15
10
0
Q
50
[Hz]
100
)LJ 6DPHDV)LJ EXWDWGLIIHUHQWVSHHGDQGLQLWLDOYHUWLFDOORDG
2
∆)9
)9R = 4000 N
fixed axle height
9[ = 39 km/h
1
4
10
699
0
[kN]
0
10
-1 -2 4
)+ [kN]
5
50
10
2
6++
0
10
0
-2 -4
egg test
4
Ω
-2
10
10
-5
10
0
10
6ΩΩ -4
0
10
[rad/s] -4 -0.05
-8
0
W
0.05
[s]
0.1
0.15
)LJ 6DPHDV)LJ EXWDWGLIIHUHQWVSHHG
10
0
Q
50
[Hz]
100
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
)9R = 4000 N 9[ = 59 km/h
fixed axle height
∆)9
2
4
10
699 0
10
[kN] 0 -2
[kN]
10
2
6++
0
10
0
-2 -4
egg test
3
Ω
4
50
4
)+
-4
10
10
-4
10
-2
10
6ΩΩ -6
0
10
[rad/s] -3 -0.05
-10
0
W
0.05
[s]
0.1
0.15
10
0
Q
50
Q
50
[Hz]
100
)LJ 6DPHDV)LJ EXWDWGLIIHUHQWVSHHG
)9R = 6000 N 9[ = 39 km/h
fixed axle height
∆)9
2
4
10
699 0
0
10
[kN] -2
-4
10
10
6
50
)+ [kN]
6++
4
2
10
0
0
10 -4
egg test
5
Ω
10
-2
10
0
10
6ΩΩ -4
0
10
[rad/s] -5 -0.05
-8
0
W
0.05
[s]
0.1
0.15
10
0
)LJ 6DPHDV)LJ EXWDWGLIIHUHQWLQLWLDOYHUWLFDOORDG
[Hz]
100
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
15
10
15
290
105
105mm
[kN]
221
120
)9R= 4000N
4 fixed axle height
∆)9
10
237
50
5
10
2
9[ = 39km/h 6 99
0
10
0
-2 -4
)+
-5
10
egg test
6
10
4
6++
0
10
2
[kN] -4 -2
10
0
10
10
6ΩΩ
Ω
-4
0
10
[rad/s] -10
-8
0
W
0.1
[s]
10
0.2
Q
0
50
[Hz]
100
)LJ 5XQQLQJRYHUDVHULHVRIFOHDWVPRXQWHGRQWKHGUXPVXUIDFHDWFRQVWDQWD[OHKHLJKW
)9R = 4000 N, 9[= 35 km/h free rolling
15mm
500mm
medium brake torque
large brake torque
3
∆)+
2 1
[kN] 0 -1 -2 test model
-3 -4
0.0
0.1 time [s]
0.2
0.0
0.1 time [s]
0.2
0.0
0.1 time [s]
0.2
)LJ 7KHKRUL]RQWDOIRUFHYDULDWLRQZKHQWUDYHUVLQJDSRWKROHZKLOHWKHZKHHOLVEHLQJ
0%
EUDNHG DW WKUHH GLIIHUHQW OHYHOV RI EUDNH WRUTXH
FD 1P
UHVSHFWLYHO\ =HJHODDU ([SHULPHQWV RQ P GUXP FDOFXODWLRQV XVLQJ REVWDFOHSDUDPHWHUVDFFRUGLQJWR)LJ
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
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
(IIHFWLYH5RDG,QSXWVDW:KHHODQG5RDG&DPEHU
,Q WKLV VXEVHFWLRQ DQ DWWHPSW LV PDGH WR GHDO ZLWK QRQV\PPHWULF URDG XQHYHQQHVVHV DQG FDPEHUHG ZKHHOV ,I RQH ZRXOG EH DEOH WR FRQGXFW DQ H[SHULPHQWZLWKDW\UHVORZO\UROOLQJRYHUDQXQHYHQURDGVXUIDFHWKDWORFDOO\ VKRZV D IRUZDUG DQG WUDQVYHUVH VORSH HJ DQ REOLTXH FOHDW ZLWK WKH VXUIDFH EHLQJ PDGH YHU\ VOLSSHU\ VR WKDW IULFWLRQDO IRUFHV DUH DYRLGHG DV PXFK DV SRVVLEOHWKHHIIHFWLYHURDGSODQHWUDQVYHUVHVORSHPD\EHDVVHVVHGWRJHWKHUZLWK WKHIRUZDUGVORSHDQGWKHHIIHFWLYHURDGSODQHKHLJKW$JDLQWKHSUHVXPSWLRQLV
vertical plane through wheel axis
-α
γ
γJ )
+\
)
+[
)
+[
UR
)9 U\R
\FR
-α
&R
&
ρ190
effective road plane
ρ] )1
)LJ 6LWXDWLRQZLWKZKHHOFDPEHU DQGURDGFDPEHU DWDIRUZDUGVORSH
β
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
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µD[OHV¶RIWKHIRXUSRLQWIROORZHU7KHXOWLPDWHSURFHGXUHLV\HW WREHGHYHORSHG ,IWKHZKHHOVKRZVDQLQFOLQDWLRQDQJOHZLWKUHVSHFWWRWKHYHUWLFDO WKHJOREDO FDPEHUDQJOH
J
LWLVSUREDEO\EHVWWRGHILQHWKHDVVHVVHGURDGLQSXWVWRRFFXU
DWWKHKRUL]RQWDOFRRUGLQDWHVRIWKHORZHVWSRLQWRIWKHXQGHIRUPHGW\UHLQWKH ZKHHO FHQWUH SODQH &R )LJXUH GHSLFWV WKH VLWXDWLRQ $VVXPH WKDW WKH HIIHFWLYH URDG SODQH KHLJKW DQG WKH HIIHFWLYH VORSHV DUH DYDLODEOH DQG ZHUH GHWHUPLQHG ZLWK DQ XSULJKW ZKHHO
J
WKDW LV YHUWLFDO HJJV )LJXUH
VKRZVWKHGHWDLOHGFRQILJXUDWLRQLQWKHYHUWLFDOSODQHWKURXJKWKHZKHHOVSLQD[LV 7KHSUREOHPLVQRZWRILQGWKH XQFRUUHFWHG QRUPDOW\UHFRPSUHVVLRQ
]
RIWKH
W\UH WKDW UXQV DW D JOREDO FDPEHU DQJOH DW D JLYHQ KHLJKW ZLWK UHVSHFW WR D KRUL]RQWDOUHIHUHQFHSODQHFI)LJ ,QFRQWUDVWWRWKHVLWXDWLRQZLWKRXWURDGDQGZKHHOFDPEHUDVGHDOWZLWKLQ WKH SUHFHGLQJ VHFWLRQV LW LV QRZ QHFHVVDU\ WR NQRZ WKH DFWXDO ORFDWLRQ RI WKH LQFOLQHGHIIHFWLYHURDGSODQHZLWKWKHKHLJKWGHILQHGEHORZSRLQW&R,WWXUQVRXW WKDW D IRU
FRUUHFWHGHIIHFWLYHURDGSODQHKHLJKWZ KDVWREHLQWURGXFHGDV
LQGLFDWHG LQ )LJ 7KH FRUUHFWLRQ LV QHHGHG EHFDXVH RI WKH ILQLWH ODWHUDO GLPHQVLRQVRIWKHW\UH LWLVQRWDWKLQGLVF :HKDYHWRDFFRXQWIRUWKHDFWXDO XQGHIRUPHGFRQWRXURIWKHW\UHFURVVVHFWLRQJLYHQE\WKHIUHHUDGLXVU\RDVD IXQFWLRQRIWKHODWHUDOGLVWDQFHWRWKHZKHHOFHQWUHSODQH\FR 7KH FRQWRXUKDV EHHQXVHGEHIRUHWRFRPSXWHWKHRYHUWXUQLQJFRXSOH0[DWFDPEHUFI6HF
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
WHVW
γJ
constant )9R
0
]DR ( Z) UOR
) +[R
UOR
corrected eff. height Zα
UR
) )
9R
+\R U\R
\FR
α
ρ9R() 9R) ρ19R &R ) 19R =ρ9R() 9R)/cosα
γ
α =
undeflected contour
vertical plane through wheel axis
)
)LJ 6XJJHVWHGWHVWFRQILJXUDWLRQSHUIRUPHGDWFRQVWDQWYHUWLFDOZKHHOORDG
9R
WRDVVHVV
HIIHFWLYHURDGSODQHKHLJKWDQGVORSHV RQIULFWLRQOHVVVXUIDFH 7KHIRUWUDQVYHUVH
Z
VORSHFRUUHFWHGHIIHFWLYHKHLJKW
VLPXODWLRQ ]D
γJ
UOR
Zα
ρ9
ρ190
γ
α
\FR
U
\R UR
&R
ρ]0 = ρ190 /cosβ
vertical plane through wheel axis )LJ 7KHXVHRIWKH FRUUHFWHG HIIHFWLYHURDGLQSXWVZKLFKDUHFDOFXODWHGDVIXQFWLRQRI WKHKRUL]RQWDOSRVLWLRQRISRLQW DQJOH
PD\RFFXU J
&
R
LQDVLPXODWLRQZKHUHDJOREDOZKHHOFDPEHU
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
(T 7RVLPSOLI\WKHPDWWHULWLVDVVXPHGWKDWWKHW\UHVWLIIQHVVQRUPDO WR WKH W\UH WUHDG VXUIDFH LV LQGHSHQGHQW RI WKH ORFDWLRQ DW WKH FURVV VHFWLRQ FRQWRXU 7KH YHUWLFDO W\UH WKDW LV ORDGHG DJDLQVW D KRUL]RQWDO VXUIDFH ZLWK D QRUPDO IRUFH
)
9R
VKRZV D UDGLDO FRPSUHVVLRQ
9R
)LJXUH LQGLFDWHV WKDW
GHIOHFWLRQ7RDFKLHYHWKHVDPHYHUWLFDOIRUFHEXWQRZORDGHGDJDLQVWDVXUIDFH ZLWK WUDQVYHUVH VORSH DQJOH
DQG SRVVLEO\ ZLWK D IRUZDUG VORSH WKH QRUPDO
GHIOHFWLRQ
SURMHFWHG DJDLQVW WKH YHUWLFDO SODQH WKURXJK WKH ZKHHO VSLQ D[LV ]R
SURGXFLQJ
19R
VKRXOGEHHTXDOWR
19R ' ^U \R & UR & 9R &]DR %Z ` FRV
& \ FR VLQ
7RJHWWKHVDPHYHUWLFDOIRUFHWKHIROORZLQJFRQGLWLRQKROGV
9R )9R
19R '
FRV
:KLOHE\GHILQLWLRQWKHXQFRUUHFWHGHIIHFWLYHKHLJKWZRXOGUHDG
Z ' ]DR
WKH FRUUHFWHG HIIHFWLYH URDG SODQH KHLJKW GHILQHG DW KRUL]RQWDO FRRUGLQDWHV RI
& Z ' Z&
SRLQW
)LJ QRZEHFRPHVE\XVLQJ
R
9R WDQ
& \FR WDQ & U R&U\R
:LWKWKHFRUUHFWHGKHLJKWLQWURGXFHGZHFDQKDQGOHWKHFDOFXODWLRQRIWKHQRUPDO W\UHGHIOHFWLRQDWZKHHO JOREDO FDPEHUFI)LJ 7KHYHUWLFDOGLVWDQFH
9
LVIRXQGWREH
9 ' U R FRV J & UOR & ] D % Z
U
ZKHUH
OR
)
LVWKHORDGHGUDGLXVDWQRPLQDOYHUWLFDOORDG
YR
7KHSURMHFWHGQRUPDO
GHIOHFWLRQWKHQUHDGV
19 '
9 FRV
& U R&U\R FRV % \ FR VLQ
$W D IRUZDUG VORSH GHILQHG DFFRUGLQJ WR )LJ WKH QRUPDO GHIOHFWLRQ QRW FRUUHFWHGIRUW\UHJURZWKHWF EHFRPHV
] '
19
FRV
)LQDOO\WKHDFWXDOQRUPDOW\UHFRPSUHVVLRQLVGHWHUPLQHGZLWKW\UHJURZWKDQG WKH LQWHUDFWLRQ ZLWK ODWHUDO DQG ORQJLWXGLQDO GHIOHFWLRQ WDNHQ LQWR DFFRXQW DQDORJRXVWR(T
]'
] %
U FRV & T )F[
[ UR &T)F\ \ UR
6+257:$9(/(1*7+,17(50(',$7()5(48(1&<7<5(02'(/
,Q D G\QDPLF VLWXDWLRQ ZH GHDO ZLWK WKH EHOW UDWKHU WKDQ ZLWK WKH ZKHHO ULP PRWLRQ$WJLYHQHIIHFWLYHURDGSODQHWKHQRUPDOGHIOHFWLRQPD\EHDVVHVVHGE\ XVLQJLQ(TV WKHEHOWFHQWUHKHLJKWYDULDWLRQ DQJOH
]
E
DQGWKHEHOWFDPEHU
LQVWHDG RI WKRVH RI WKH ZKHHO ULP $V D UHVXOW WKH QRUPDO UHVLGXDO E
GHIOHFWLRQ
]U
LV REWDLQHG 7KH QRUPDO IRUFH
)
1
PD\ ILQDOO\ EH FDOFXODWHG E\
XVLQJ(T 7KLVQRUPDOORDGLQLWVWXUQJRYHUQVWKHOHQJWK SRLQWIROORZHU
O
V
RIWKH
&KDSWHU 02725&<&/('<1$0,&6
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¶VVWDELOLW\XVLQJ DSURSHUW\UHPRGHO/DWHUWKHWRUVLRQDOFRPSOLDQFHRIWKHIUDPHZDVLQWURGXFHG 6KDUS DDQG6SLHULQJ ZKLFKDSSHDUHGWRKDYHDPDUNHGHIIHFWRQ
02725&<&/('<1$0,&6
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
02725&<&/('<1$0,&6
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
0RGHO'HVFULSWLRQ
,Q)LJWKHPRWRUF\FOHKDVEHHQGHSLFWHGZKLOHLWPRYHVDWDUROODQJOH
RI
WKHPDLQIUDPHDQGZLWKDVWHHUDQJOH RIWKHKDQGOHEDUDERXWDVWHHULQJD[LV
g
WKDWLQWKHQHXWUDOXSULJKWSRVLWLRQVKRZVDVWHHULQJKHDG UDNH DQJOH ZLWK UHVSHFWWRDYHUWLFDOOLQHDQGDFDVWHUOHQJWKWF7KHUHIHUHQFHSRLQW$WKDWOLHVRQ WKHOLQHRILQWHUVHFWLRQRIWKHSODQHRIV\PPHWU\RIWKHYHKLFOHDQGWKHURDGSODQH DQG LV ORFDWHG LQ WKH XSULJKW SRVLWLRQ EHORZ WKH FHQWUH RI JUDYLW\ RI WKH
DI
δ
PU
[U P
PP KU
ϕ
]P
U
KP E EF
PU
\
\
) \
&
[V VV PV HV ]V
VF ε
D DF
ϕU δ
PU
ϕ = γ
PJ
β
KI
U
KV WF
O
PPU
) ]
H I [I PI
K
$
(offset)
PD
DV
γ
PP
\
δ
$
]
[
X Y
\
U ]
)LJ0RWRUF\FOHPRGHOFRQILJXUDWLRQ
02725&<&/('<1$0,&6
PDLQIUDPH PRYHV IRUZDUGV ZLWK D YHORFLW\ X DQG LQ ODWHUDO GLUHFWLRQ ZLWK D YHORFLW\Y7KHOLQHRILQWHUVHFWLRQPRYHVRYHUWKHURDGVXUIDFHDQGVKRZVD\DZ WKHWLPHUDWHRIZKLFKLVGHQRWHGE\U7KHPDLQIUDPHUROODQJOH
DQJOH
LV
PHDVXUHGDVWKHDQJOHEHWZHHQWKHSODQHRIV\PPHWU\DQGWKHQRUPDOWRWKHURDG VXUIDFH $V GHSLFWHG LQ WKH ILJXUH DQ DGGLWLRQDO GHJUHH RI IUHHGRP PD\ EH LQWURGXFHGDVVRFLDWHGZLWKWKHWRUVLRQDOIOH[LELOLW\RIWKHIURQW VWHHUDEOH IUDPH SRVVLEO\LQFOXGLQJDSRUWLRQRIWKHPDLQIUDPHZLWKUHVSHFWWRWKHFHQWUHSDUWRI WKHPDLQIUDPH7RPRGHOWKLVDQD[LVRIURWDWLRQLVLQWURGXFHGSHUSHQGLFXODUWR WKHVWHHULQJD[LVDERXWZKLFK DSDUWRI WKHIURQWIUDPHFDQURWDWHZLWKWZLVW 7KH ULGHU KDV D OHDQ GHJUHH RI IUHHGRP UHODWLYH DQJOH
DQJOH
U
DERXW D
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
*HRPHWU\DQG,QHUWLD
7KHJHRPHWULFDOGLPHQVLRQVRIWKHPRWRUF\FOHDQGWKHORFDWLRQRIWKHFHQWUHV RIJUDYLW\RIWKHIRXUFRQQHFWHGERGLHV PDLQIUDPHLQFOXGLQJORZHUSDUWRIWKH ULGHU DQG UHDU ZKHHO XSSHU WRUVR RI WKH ULGHU IURQW XSSHU IUDPH DQG IURQW VXEIUDPHLQFOXGLQJIURQWZKHHO DUHGHILQHGE\TXDQWLWLHVJLYHQLQWKHILJXUH7KH IROORZLQJUHODWLRQVH[LVWEHWZHHQJHRPHWULFDOSDUDPHWHUV
DI DV VV K
VN KN
g VLQg H
g D ^K W `FRVg V ^K H W VLQg `FRVg V FRVgW VLQg V W WDQg W VLQg
DF^KIVLQ HIWF `FRV F
V
F
V
F
F
V
V
F
F
F
F
F
7KHODVWWKUHHGLPHQVLRQVKDYHQRWEHHQLQGLFDWHGLQWKHILJXUH 7KHPDVVHVRIWKHPDLQIUDPHWKHIURQWXSSHUIUDPHWKHIURQWVXEIUDPHDQG WKHXSSHUWRUVRDUHGHQRWHGDVPPPIPVDQGPUUHVSHFWLYHO\7KHPDJQLWXGH RIWRWDOPDVVPWKHSRVVLEO\VKLIWHGFJRIPPU
PPPUWKHZKHHOEDVHODQG
WKHORFDWLRQRIWKH FHQWUHRIWKHWRWDOPDVVFHQWUHZLWKUHVSHFWWRWKHUHDUDQG IURQWZKHHOD[OHV GLVWDQFHVDDQGE DQGDERYHWKHJURXQG KHLJKWK EHFRPH
02725&<&/('<1$0,&6
ϕU
U
PJ
U
V
PU
\
U
K
P
(offset)
PU
PJ
rider upper torso
PD\ K
ϕ
P
PJ
K
ϕ
ϕ
P
PJ
K
main frame UF
)\
0[ &
)]
\
)]
]
)LJ5HDUYLHZRIPRWRUF\FOHZLWKW\UHFURVVVHFWLRQDQGSRVVLEOHFJRIIVHWLQGLFDWHG
P
PPPIPVPU
PPU O
PPPU
DFEF
E
^PPUEFPI DIEF PV DVEF `P
D
OE
K
KPPPKIPIKVPVKUPU P
7KHLQHUWLDOTXDQWLWLHV SURYLGHGZLWKVXEVFULSW WKDWZRXOGDSSO\LQFDVHRID ULJLG ULGHU OHDQ DQJOH UHPDLQV ]HUR PD\ EH UHWULHYHG E\ XVLQJ WKH IROORZLQJ HTXDWLRQV ZKHUH ,U[ LV PRPHQW RI LQHUWLD RI WKH ULGHU¶V XSSHU WRUVR DERXW LWV ORQJLWXGLQDOD[LV ,U]LVDVVXPHGWREHLQFRUSRUDWHGLQWKHPDLQIUDPHLQHUWLD
PP
PPU
PPPU
KP
KPPPKUPU PP
,P[
,P[KP PP,U[KU PUKP PP
,P[]
,P[],U[]
3URGXFWVRILQHUWLDH[FHSW,P[]RIWKHPDLQIUDPHZLOOEHQHJOHFWHG 7KHFURVVVHFWLRQVRIWKHW\UHVDUHDVVXPHGWRKDYHDILQLWHFURZQUDGLXVUF DQGUF IURQWDQGUHDU 7KHVHUDGLLDUHUHVSRQVLEOHIRUWKHFUHDWLRQRIWKHPDMRU SDUWRIWKHW\UHRYHUWXUQLQJFRXSOHV0[0RUHRYHUWKHKHLJKWVRIWKHFJ¶VDUH DIIHFWHG E\ WKHVH FURZQ UDGLL ZKHQ UXQQLQJ DW ODUJH UROO DQJOHV )LJXUH
02725&<&/('<1$0,&6
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
UF
EO UF DO UF
ZHILQG
K
KUF FRV FRV
$W ODUJH UROO DQJOHV DOVR WKH FDVWHU OHQJWK WF VKRXOG EH DGDSWHG :H ILQG DSSUR[LPDWHO\ZLWK
WF
DVVXPHGVPDOO
g FRV
WFUFVLQ
FRV
,Q WKH OLQHDU DQDO\VLV UHVWULFWHG WR VPDOO DQJOHV WKHVH H[WHQVLRQV DUH RI QR LPSRUWDQFH
7KH6WHHU&DPEHUDQG6OLS$QJOHV
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
0 7KHPDLQIUDPHURWDWHVDERXWWKH[D[LVJLYLQJULVHWRWKHUROO
7KHURWDWHGV\VWHPRID[HV [ \ ] LVDWWDFKHGWRWKHPDLQIUDPH,Q
WKH PDLQIUDPH FHQWUH SODQH WKH VWHHULQJ D[LV LV SRVLWLRQHG DW DQ DQJOH RI
g
LQFOLQDWLRQ WKHUDNHDQJOH ZLWKUHVSHFWWRWKH] D[LV7KHWULDG [g\g]g LV DOVRDWWDFKHGWRWKHPDLQIUDPHEXWZLWKD]gD[LVDORQJWKHLQFOLQHGVWHHULQJD[LV 7KHV\VWHP [ \ ] LVDWWDFKHGWRWKHXSSHUSDUWRIWKHIURQWIUDPHWKDWLV
02725&<&/('<1$0,&6
ε
. ϕ
δ
side view
ε
steer axis
γ
wheel axis
ε
δ
[
V
WF
U
δ
[
\
ε
]
&
[ϕ
δ ψ
[o
$
\
o
]
δ
\
[
ϕ
δ V
ϕ
\
ϕ
]ϕ
[
ε ] δ
\
\
ψ
$
]
o
]
o
\
[
.
δ
\
o
top view
&
δ
)LJ 9LHZ RI IURQW ZKHHO DVVHPEO\ ZLWK YDULRXV FRRUGLQDWH V\VWHP WULDGV WR DVVHVVWKH DQGFDPEHUDQJOH RIWKHIURQWZKHHOXVLQJWKHXQLWYHFWRU V
JURXQGVWHHUDQJOH
DORQJWKHZKHHOVSLQD[LV
URWDWHGZLWKVWHHUDQJOH ZLWKUHVSHFWWRWKH [g\g]g IUDPH)LQDOO\ZHPD\ LQWURGXFHWKHWZLVWDQJOH QRWFRQVLGHUHGLQ)LJFI)LJ JLYLQJULVH WRWKHV\VWHPRID[HV [ \ ] ZLWKWKH\ D[LVUXQQLQJDORQJWKHZKHHOVSLQ D[LV:HQRZLQWURGXFHDXQLWYHFWRUVGLUHFWHGDFFRUGLQJWRWKHZKHHOVSLQD[LV WKDWLVDORQJWKH\ D[LV7KHFRPSRQHQWVRIWKLVYHFWRUDORQJWKH D[HV RI WKH PRYLQJ KRUL]RQWDO V\VWHP [ \ ] ZLOO EH QRZ EH GHWHUPLQHG E\ VXFFHVVLYH URWDWLRQ WUDQVIRUPDWLRQV 7KH UHVXOW RI HDFK VXFFHVVLYH VWHS LV LQGLFDWHG E\ D VXEVFULSWWKDWGHQRWHVWKHIUDPHZLWKUHVSHFWWRZKLFKWKHXQLWYHFWRULVUHJDUGHG :HKDYH
V
'
V
'
FRV
&VLQ
VLQ
FRV
V
V
g
'
FRV
&VLQ
VLQ
FRV
V
FRV V
n
'
g
g
VLQ
&VLQ g
FRV
V
g
g
V
'
FRV
VLQ
n
&VLQn
n
FRV
n
V
n
02725&<&/('<1$0,&6
EF α
DF
α
δ X
U Y
.
.
δ
β
)LJ7RSYLHZRIWKHVLQJOHWUDFNYHKLFOHVKRZLQJWKHIURQWDQGUHDUVOLSDQJOHV
,QWKHVXEVHTXHQWDQDO\VLVZHZLOODSSUR[LPDWHWKHVLWXDWLRQE\DVVXPLQJVPDOO VWHHUDQGWZLVWDQJOHV7KLVLVFHUWDLQO\DGPLVVDEOH)RUWKHQRQOLQHDUVWHDG\ VWDWHFRUQHULQJSUREOHPWKHUROODQJOHVKRXOGEHDOORZHGWRDWWDLQPDJQLWXGHV
E
g
ODUJHUWKDQ 2IFRXUVHWKHUDNHDQJOH ZKLFKLVDV\VWHPSDUDPHWHUZLOOEH UHJDUGHGWREHODUJH:LWK DQG DVVXPHGVPDOOWKHXQLWYHFWRUUHGXFHVWR
& FRVg % VLQ g V ' FRVn & VLQn VLQ g % VLQn % FRVn VLQ g %
g
FRV
g
FRV
ZKLFKLQFDVHRIDFRPSOHWHO\OLQHDUDQDO\VLVZLWKDOVRWKHUROODQJOHDVVXPHG VPDOOUHGXFHVWR
& FRVg % VLQ g V ' FRVg & n VLQ g % n % VLQ g % FRVg 7KHJURXQGVWHHUDQJOH
NDQGWKHFDPEHUDQJOH
FDQQRZHDVLO\EHGHWHUPLQHG
IURPWKHFRPSRQHQWVRIWKHXQLWYHFWRU:HILQGIRUWKHQRQOLQHDUH[SUHVVLRQV
N
WDQ
VLQ
'&
'
V
V
[
V
\
'
]
g&
'
FRV
n & VLQ n
FRV
VLQ
n % FRV n
VLQ
VLQ
VLQ
g
g%
g%
g
FRV
g
FRV
DQGIRUWKHOLQHDULVHGDSSUR[LPDWLRQV
N
' '
g&
FRV
n%
VLQ
VLQ
g%
g
g
FRV
)RUWKHQRQVWHHUHGUHDUZKHHOZHVLPSO\KDYHDFDPEHUDQJOH
'
n
7KH VOLS DQJOHV DUH DVVXPHG WR UHPDLQ VPDOO DQG UHDG DW VWHDG\ VWDWH E\
02725&<&/('<1$0,&6
FRQVLGHULQJ)LJ
'
&
N
'&
Y
X
D
F U
&
Y
X
%
E
F U
)RU WKH G\QDPLF QRQVWHDG\VWDWH VLWXDWLRQ WKH OLQHDULVHG PRGHO ZLOO EH HPSOR\HG7KHVOLSDQJOHVWKHQEHFRPHLQFOXGLQJWKHWLPHUDWHRIFKDQJHVRI DQG
'
'&
FRV
g&
Y
X
&
VLQ
g&
X
Y
%
D U
F
&
W
F
0
&
V
F
0
E U
F
:LWKWKHFDPEHUDQGVOLSDQJOHVQRZGHULYHGZHFDQIRUPXODWHWKHUHVXOWLQJVLGH IRUFHDQGPRPHQWV)LUVWKRZHYHUWKHQRUPDOORDGVDUHWREHHVWDEOLVKHG7KHVH DUHDIIHFWHGE\WKHIRUHDQGDIWORDGWUDQVIHUFDXVHGE\DHURG\QDPLFGUDJDQG EUDNLQJRUGULYLQJIRUFHV
$LU'UDJ'ULYLQJRU%UDNLQJDQG)RUHDQG$IW/RDG7UDQVIHU
$SDUWIURPWKHIRUFHVDQGPRPHQWVDFWLQJIURPURDGWRW\UHVZHZLOOFRQVLGHU WKHDHURG\QDPLFGUDJIRUFH)GWKDWLVDVVXPHGWRDFWLQORQJLWXGLQDOEDFNZDUG GLUHFWLRQLQWKHSUHVVXUHFHQWUHDGLVWDQFHKGDERYHWKHURDGVXUIDFH LQXSULJKW SRVLWLRQ :HZLOOKHUHGHILQHWKHGUDJIRUFHWRGHSHQGTXDGUDWLFDOO\RQWKHVSHHG XDVIROORZV )
G
'
&
G$ X
'XHWRWKHDFWLRQRIWKHGUDJIRUFH)GDQGWKHORQJLWXGLQDOW\UHIRUFHV)[LORDG WUDQVIHUDULVHVIURPWKHIURQWWRWKHUHDUZKHHO7KHLQFUHDVHRIWKHUHDUQRUPDO ORDGZKLFK E\QHJOHFWLQJWKHRYHUDODHURG\QDPLFOLIW LVHTXDOWRWKHGHFUHDVH RIWKHIURQWQRUPDOORDG 7KHVXPRIWKHORQJLWXGLQDOW\UHIRUFHVLVGHQRWHGDV)[WRW7KHUHPDLQLQJ IRUFHIRUWKHDFFHOHUDWLRQRIWKHYHKLFOHLQORQJLWXGLQDOGLUHFWLRQEHFRPHV )
D[
'
)
[WRW
&
)
G
ZKLFKUHVXOWVLQWKHIRUZDUGDFFHOHUDWLRQ
D
[
'
P
)
D[
02725&<&/('<1$0,&6
DQGIURPWKLVWKHDFFHOHUDWLRQIRUFHVDFWLQJRQWKHIRXULQGLYLGXDOPDVVHV)D[P D[PP)D[U
D[PU)D[I
D[PIDQG)D[V
D[PV
:LWKWKHPRPHQWDUPVKn FI)LJ DQGKGnWKHDPRXQWRIORDGWUDQVIHU EHFRPHV
)
]
'
%
K
Gn )G
O
n ) D[ FRVn
K
ZKLFKIRUVPDOOUROODQJOHVUHGXFHVWR
)
]
'
'
)
]
K )
G
O
%
G
K)
D[
7KHUHVXOWLQJYHUWLFDOZKHHOORDGVQRZEHFRPH )
]
'
)
]R
&
)
]
)
]
'
)
]R
%
)
]
DQGDWVPDOOUROODQJOHV
)
]
'
)
]
'
)
]R
&
)
]
)
]
'
)
]
'
)
]R
%
)
]
ZLWKWKHLQLWLDOZKHHOORDGV
)
]R
'
E
PJ
)
]R
O
'
D
PJ
O
7KHLPSRVHGEUDNLQJIRUFHLVDVVXPHGWREHGLVWULEXWHGRYHUWKHIURQWDQGUHDU ZKHHOVLQSURSRUWLRQWRWKHZKHHOORDGVDVZRXOGRFFXULQVWUDLJKWDKHDGPRWLRQ WKDWLV DFFRUGLQJWRWKHORDGV :HKDYHDWEUDNLQJ )[WRW
[
)
'
)
]
PJ
)
[WRW
)
[
'
)
]
PJ
)
[WRW
DQGDWGULYLQJ
)
[
'
)
[
'
)
[WRW
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
02725&<&/('<1$0,&6
7\UH)RUFHDQG0RPHQW5HVSRQVH
/LQHDU0RGHO :HZLOOWDNHLQWRDFFRXQWWKHWUDQVLHQWUHVSRQVHRIWKHVLGHIRUFH)\DQGDOLJQLQJ WRUTXH0]WRFKDQJHVLQVOLSDQJOHDQGFDPEHUDQJOH+RZHYHUWKHQRQODJJLQJ SDUW RI WKH UHVSRQVH ZLOO EH GLVUHJDUGHG 2Q WKH RWKHU KDQG WKH RYHUWXUQLQJ FRXSOH0[LVDVVXPHGWRUHVSRQGLQVWDQWDQHRXVO\WRFKDQJHVLQFDPEHU)RUWKH WUDQVLHQWUHVSRQVHVWKHUHOD[DWLRQOHQJWK ZLOOEHXVHGDVSDUDPHWHULQWKHILUVW RUGHU GLIIHUHQWLDO HTXDWLRQV 7KHVH HTXDWLRQV GHVFULEH WKH UHVSRQVHV RI WKH WUDQVLHQW VOLS RU GHIOHFWLRQ DQJOHV
N DQG N ZKLFK LQ VWHDG\VWDWH FRQGLWLRQV
EHFRPHHTXDOWRWKHLQSXWVOLSDQGFDPEHUDQJOHV L
DQG :HREWDLQIRUW\UHL
RU
%
0 L LN
X
%
0 L LN
X
LN LN
'
L
'
L
ZKLFKJLYHVULVHWRWKHVLGHIRUFHDWVPDOOVOLSDQGFDPEHUDQJOHV )
\L
'
) L LN
&
%
) L LN
&
)RUWKHDOLJQLQJWRUTXHVWLIIQHVVDJDLQVWZKHHOFDPEHUZHLQWURGXFHWKHHIIHFWRI WKHORQJLWXGLQDOIRUFH)[LE\FRQVLGHULQJDILQLWHFURZQUDGLXVUFLDVVXPLQJWKDW WKH ODWHUDO VKLIW RI WKH OLQH RI DFWLRQ RI WKH ORQJLWXGLQDO IRUFH FKDQJHV LQVWDQWDQHRXVO\ZLWKWKHFDPEHUDQJOH:HILQGIRUWKHDOLJQLQJWRUTXH 0
]L
'&
0 L NL
&
%
0 L NL
&
&
U
FL )[L L
1RWHWKDWZHKDYHGLVUHJDUGHGKHUHWKHRWKHULQIOXHQFHVRIWKHORQJLWXGLQDOIRUFH )[LRQWKHVLGHIRUFHDQGWKHDOLJQLQJWRUTXHDVH[SUHVVHGHJE\(TV )LQDOO\ZHKDYHWKHRYHUWXUQLQJFRXSOHLQGLFDWHGLQ)LJDVVXPHGKHUH WRGHSHQGRQO\RQWKHFDPEHUDQJOH,QWKHOLQHDULVHGYHUVLRQZHKDYH 0
[L
'&
&
0[ L L
2EYLRXVO\ZHKDYHQHJOHFWHGKHUHWKHVPDOOHIIHFWRIWKHODWHUDOGLVWRUWLRQGXH WRWKHVLGHIRUFH7KHFRHIILFLHQWVDUHDVVXPHGWRGHSHQGRQWKHQRUPDOORDGDV IROORZV RPLWWLQJVXEVFULSWL &
)
'
&
) R
%
G
LQWKHOLQHDUPRGHOG
) R' ) ' 0 '
& & &
G
G
H
)
G
ZLWK
)
]
&
)
]R
]
)
)
]R %
]
02725&<&/('<1$0,&6
'
&
0
DQG
H
'
&
0[
H
)
]
) ZLWK H
]
'
U
F
:HLQWURGXFHWKHSQHXPDWLFWUDLOVW W
R
! RIWKHVLGHIRUFHGXHWRVLGHVOLSDQG
RIWKHFDPEHUIRUFH R
W
&
'
R
0
W
R
&
) R
'&
&
0
&
)
$OVRWKHUHOD[LRQOHQJWKVGHSHQGRQWKHQRUPDOORDG7KLVDSSHDUVWRRFFXU LQ D ZD\ VLPLODU WR WKDW RI WKH FKDQJH LQ FRUQHULQJ VWLIIQHVV ([SHULPHQWDO HYLGHQFH VKRZV WKDW WKH UHOD[DWLRQ OHQJWK IRU VLGH VOLS LV FORVH WR WKH RQH IRU FDPEHU 7KH QRQODJJLQJ SDUW WKDW DOWKRXJK VPDOO H[LVWV LQ WKH UHVSRQVH WR FDPEHUFKDQJHVLVGLVUHJDUGHGKHUH:HGHILQH
'
'
I
)
]R
%
I )
]
&
)
]R
1RQ/LQHDU0RGHO )RUWKHQRQOLQHDUIRUFHDQGPRPHQWGHVFULSWLRQZHZLOOPDNHXVHRIWKH0DJLF )RUPXODLQDVLPSOLILHGYHUVLRQ7KHYDOXHVRIWKHSDUDPHWHUVLQYROYHGKDYHEHHQ OLVWHGLQ7DEOHDWWKHHQGRIWKHSUHVHQW6HFWLRQ7KHVLPLODULW\PHWKRG ZLOOEHHPSOR\HGWRLQFRUSRUDWHWKHHIIHFWRIWKHLPSRVHGIRUHDQGDIWIRUFH)[ :HZLOODVVXPHKHUHWKDWWKHFRUQHULQJVWLIIQHVV& ) LVQRWDIIHFWHGE\)[DQG WKDW WKH YHUWLFDO VKLIW LV VPDOO ZLWK UHVSHFW WR ' :H KDYH IRU WKH VLGH IRUFH DJDLQRPLWWLQJVXEVFULSWL
&
.
'
'
'
%
'
6
+I
6
9
'
6
+
'
)
G
%
)
]
'
&
)
G
[
. &'
'
)HT
&
'
'
'
G
&
&
)
G
6
'
)
+I
]
&
' '
9 &)
6
' '
)
%
6
+I
&
6
+I
02725&<&/('<1$0,&6
)
'
\
' VLQ >& DUFWDQ ^ %
)HT
%
6
+ ` @
%
6
9
DQGIRUWKHDOLJQLQJWRUTXHXVLQJWKHSQHXPDWLFWUDLODQGWKHVLGHIRUFHVROHO\GXH WRVLGHVOLS
'
HT
'
)
%
&
%
&
W
0
]UR
0
]U
0
]
\
'
W W
'
U
'
U
' VLQ >& DUFWDQ %
HT @
H
H
H
'
'
' '
H
W
%
H
H
RFRV>&W DUFWDQ %W HT @
'
'
%
DUFWDQ H
&
0
0
'&
W
%
H
H
]UR FRV>&U DUFWDQ %U HT @ )
\
%
0
]U
&
U
F )[ WDQ
LQZKLFKWKHWHUPZLWKWKHSURGXFW)[)\KDVEHHQGLVUHJDUGHG)LQDOO\ZHGHILQH IRUWKHRYHUWXUQLQJFRXSOHXVLQJIRUPXOD ZKLOHQHJOHFWLQJWKHYHUWLFDO DQGODWHUDOW\UHGHIOHFWLRQ
0
[
'&
U
F )]
WDQ
6LQFHWKHQRQOLQHDUDQDO\VLVZLOOEHOLPLWHGWRVWHDG\VWDWHFRQGLWLRQVWKHLQSXW VOLS DQG FDPEHU DQJOHV DQJOHV
DQG PD\ EH XVHG GLUHFWO\ LQVWHDG RI WKH WUDQVLHQW
NDQG N
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
ZLWK
WKHVLPLODULW\PHWKRGHPSOR\HGWRLQFOXGHWKHHIIHFWVRIWKHJLYHQORQJLWXGLQDO
02725&<&/('<1$0,&6
4
)\
3
80
front tyre )] = )]R = 1732N )[ = 0
60
2
40
1
20
0
0
[kN]
[Nm]
-20
-1
γ = -10, 0, 10, 20, 30, 40
-2 -3
o
-8
4
)\
0]
3
o
-4
0
o
4
o
-40 o
α
-60
8
rear tyre )] = )]R = 2094N )[ = 0
80 60
2
40
1
20
0
0
-1
-20
[kN]
0] [Nm]
γ = -10, 0, 10, 20, 30, 40
-2 -3
-40
o
o
-8
o
-4
0
o
4
α
o
-60
8
)LJ 6LGHIRUFHDQGDOLJQLQJWRUTXHFKDUDFWHULVWLFVIRUWKHIUHHO\UROOLQJIURQWDQGUHDU PRWRUF\FOHW\UHPRGHODWDVHULHVRIFDPEHUDQJOHV
IRUFH)[,ILQVWHDGRIWKHIRUFH)[WKHORQJLWXGLQDOVOLS
LVLPSRVHGRUUHVXOWV
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
02725&<&/('<1$0,&6
4 3
)\
front tyre )] = )]R = 1732N )[ = -1132N
80 60
2
40
1
20
0
0
-1
-20
[kN]
[Nm]
γ = -10, 0, 10, 20, 30, 40
-2 -3
o
-8 4 3
)\
0]
o
-4
o
0
4
-40
o
α
-60
o
8
rear tyre )] = )]R = 2094N )[ = -1368N
80 60
2
40
1
20
0
0
[kN]
0] [Nm]
γ = -10, 0, 10, 20, 30, 40
-1
-20
o
-40
-2 -3
o
-8
o
-4
o
0
4
α
-60
o
8
)LJ 6LGH IRUFH DQG DOLJQLQJ WRUTXH FKDUDFWHULVWLFV IRU WKH EUDNHG IURQWDQGUHDU PRWRUF\FOHW\UHDWDVHULHVRIFDPEHUDQJOHV
7DEOH3DUDPHWHUVRIK\SRWKHWLFDOIURQW DQGUHDU PRWRUF\FOHW\UHPRGHO
G
G
G
I
G
G
G
I
H
H
H
U
F
I
H
H
H
U
F
I
G
G
G
G
G
G
G
G
G
H
H
H
H
H
H
H
02725&<&/('<1$0,&6
/LQHDU(TXDWLRQVRI0RWLRQ
)RUWKHG\QDPLFDQDO\VLVRIWKHYHKLFOHPRWLRQZHUHVWULFW RXUVHOYHV WRVPDOO GHYLDWLRQVIURPWKHUHFWLOLQHDUSDWK7KHGLIIHUHQWLDOHTXDWLRQVZLOOEHNHSWOLQHDU ZLWKWKHIRUZDUGVSHHGXFRQVLGHUHGDVDFRQVWDQWSDUDPHWHU:HZLOOGHULYHWKH HTXDWLRQVXVLQJWKHPRGLILHGHTXDWLRQVRI/DJUDQJH GHULYHGLQ&KDSWHU $IWHUDGGLQJWKHGLVVLSDWLRQIXQFWLRQ'WRLQFOXGHGDPSLQJLQWKHV\VWHPWKHVH HTXDWLRQV UHDG IRU WKH YDULDEOHV Y DQG U UHPDLQLQJJHQHUDOLVHGFRRUGLQDWHVTM KHUH M
G
M % M ' Y M M M & M % M ' U M M M M &M %M %M ' M JM M M M M M JM 7
GW G
Y
G GW
GW DQG VXEVHTXHQWO\ IRU WKH
RU
4
X
7
GW
7
U
R
G
7
Y
7
X
4
U
X
7
7
8
'
T
T
T
T
Y
4
M
7KHJHQHUDOLVHGIRUFHVDUHIRXQGIURPWKHYLUWXDOZRUN XVLQJ
:
'
3
LQVWHDGRI
M
'
4
M
T
M
ZLWKTMUHIHUULQJWRWKHTXDVLFRRUGLQDWH\DQGWRWKHJHQHUDOLVHGFRRUGLQDWHV
DQG 7RHVWDEOLVKWKHHTXDWLRQVZHZLOODVVHVVVXFFHVVLYHO\WKHNLQHWLF
U
DQGSRWHQWLDOHQHUJLHVWKHGLVVLSDWLRQIXQFWLRQDQGWKHYLUWXDOZRUN7KHWHUPV DSSHDULQJLQWKHUHVXOWLQJH[SUHVVLRQVZLOOEHUHVWULFWHGWRWHUPVRIWKHVHFRQG RUGHURIPDJQLWXGH7KHYHORFLW\XDQGWKHFRQQHFWHGZKHHOVSHHGVRIUHYROXWLRQ L
XUL WKHYHKLFOHZHLJKWPJDQGWKHDHURG\QDPLFGUDJ)GDQGORQJLWXGLQDO
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
RIWKHPDLQIUDPHZKLFKGRHV QRWUHSUHVHQWDGHJUHHRI
IUHHGRPLVRIWKHVHFRQGRUGHURIPDJQLWXGHDQGVKRXOGEHDFFRXQWHGIRULQWKH H[SUHVVLRQVIRUWKHHQHUJLHV([SUHVVHGLQWHUPVRIWKHJHQHUDOLVHGFRRUGLQDWHV ZHILQGWKHVHFRQGGHJUHHFRQVWUDLQWHTXDWLRQFI(TV
02725&<&/('<1$0,&6
' & KN
O
g%
VLQ
& VN %
%
g
FRV
g%
VLQ
g%
VLQ
g
FRV
%
g
FRV
7KH.LQHWLF(QHUJ\ 7UDQVODWLRQDODQGDQJXODUYHORFLWLHVRIWKHVL[ERGLHVDUHWREHH[SUHVVHGLQWHUPV RI Y DQG WKH JHQHUDOLVHG FRRUGLQDWHV DQG WHQWDWLYHO\
DQG WKHLU WLPH
GHULYDWLYHV:HILQGIRUWKHORQJLWXGLQDOODWHUDODQGYHUWLFDOYHORFLWLHVRIWKHFJ RIWKHPDLQIUDPHSOXVUHDUZKHHO H[FHSWSRODUPRPHQWRILQHUWLD,Z\ ZLWKPDVV PP ERG\
' X & KPUn & K P 0 & \PU 0 Y ' Y%K n P P Z ' &E 0 P F X
P
DQGIRUWKHDQJXODUYHORFLWLHV
' n0 ' 0 P\ 'U P] P[
7KHYHORFLWLHVRIWKHULGHUXSSHUWRUVRZLWKPDVVPU ERG\
' X & KUUn & V UUnU & KU 0 & \UU 0 0 % 0\ % V n Y ' Y%K n U U U U U Z ' &E 0 U F X
U
DQGIRUWKHDQJXODUYHORFLWLHV
' n0 % n0 U ' 0 U\ 'U U] U[
7KHYHORFLWLHVRIWKHIURQWIUDPHZLWKPDVVPI ERG\ZLWK]D[LVSDUDOOHOWR VWHHUD[LV
' X & K I Un & HI U & KI 0 0 % H 0 %D U Y ' Y%K n I I I I Z ' & E % D 0 I F I X
I
DQG
I[
'
0 FRVg & U VLQg n
02725&<&/('<1$0,&6
' ' I]
0
I\
0 VLQg % U FRVg % n
0
)RU WKH IURQW VXEIUDPH SOXV IURQW ZKHHO H[FHSW SRODU PRPHQW RI LQHUWLD ,Z\ ZLWKPDVVPV ERG\ZLWK]D[LVSDUDOOHOWRVWHHUD[LV
' X & K V Un & HV U % VV U & KV 0 0 % H 0 & V 0 %D U Y ' Y%K n V V V V V 0 Z ' & E % D V F V X
V
DQG
' ' V\ ' V]
0 FRVg & U VLQg % 0 n
V[
0
0 VLQg % U FRVg % 0 n
)RU WKH IURQW ZKHHO DQJXODU YHORFLWLHV ERGLHV DQG RQO\ SRVVHVVLQJ SRODU PRPHQWVRILQHUWLD,Z\SRVVLEO\H[WHQGHGZLWKHIIHFWLYHPRPHQWRILQHUWLDQJ,H\ RIRWKHUURWDWLQJSDUWVUHGXFHGWRWKHZKHHOURWDWLRQDOVSHHG
'&
Z\
%
0
% U
DQGIRUWKHUHDUZKHHO
'&
Z\
%nU
ZKHUH
'
X
U
'
X
U
ZLWKUDQGUGHQRWLQJWKHIURQWDQGUHDUZKHHO HIIHFWLYHUROOLQJ UDGLL'XHWR V\PPHWU\ RI WKH XQGLVWXUEHG V\VWHP WKH
¶V GR QRW QHHG WR EH UHJDUGHG DV
YDULDEOHV LQ RXU OLQHDULVHG V\VWHP HTXDWLRQV DQG QRQKRORQRPLF FRQVWUDLQW HTXDWLRQVGRQRWRFFXU7KHNLQHWLFHQHUJ\EHFRPHVLQJHQHUDOVXPPHGRYHUWKH VL[ERGLHV
7
' %
%YN %ZN %
P X N N
N'
^,
N'
[N
%
[N ,\N
%
\N ,]N
&
]N ,[\N
[N
%,\]N
\N
7KHSURGXFWVRILQHUWLDZLOOEHQHJOHFWHGH[FHSW,][
\N
%,][N
]N
]N
`
[N
,P[]RIWKHPDLQIUDPH7KH
YHORFLWLHVXNHWFDSSHDULQJLQ FRUUHVSRQGWRWKHTXDQWLWLHVJLYHQE\WKH H[SUHVVLRQV 7KHWLPHGHULYDWLYHRIWKHSLWFKDQJOH H[SUHVVLRQ
LVREWDLQHGIURP
02725&<&/('<1$0,&6
7KH3RWHQWLDO(QHUJ\DQGWKH'LVVLSDWLRQ)XQFWLRQ 7KH SRWHQWLDO HQHUJ\ LV FRPSRVHG RI FRQWULEXWLRQV IURP WRUVLRQDO VSULQJ GHIOHFWLRQV WZLVWDQJOH
DQGOHDQDQJOH
n
U
ZLWKVWLIIQHVVHVF DQGFnU UHVSHFW
LYHO\ DQG WKH KHLJKWV RI WKH FHQWUHV RI JUDYLW\ RI WKH YDULRXV ERGLHV 7KHVH KHLJKWVEHFRPHH[SUHVVHGLQWHUPVRIWKHJHQHUDOLVHGFRRUGLQDWHVDQG )RUERG\ZLWKFJODWHUDORIIVHW\P
&]P '
K
P
&
n
% EF & \Pn
)RUERG\ZLWKFJODWHUDORIIVHW\U
&]U '
K
U
&
K
&VU n &
U
n%nU
V
U
% EF & \Un
)RUERG\
&]I '
K
I
&
K
I
n
& HI
n&
H
g % EF%DI
VLQ
I
)RUERG\
&]V'KV&
K
n
V
&HV n%VV n&
H
V
g%VV
g%
VLQ
VLQ
n
)XUWKHUPRUHZHKDYHWKHWRUVLRQDOGHIOHFWLRQV
U
V
g% EF%DV
FRV
V
DQG 7KHFRPSOHWHSRWHQWLDO
HQHUJ\LVQRZZULWWHQDVIROORZV
8
' & P PJ ]P & P UJ ]U & P I J ]I & PVJ ]V %
nUnU
F
%
F
$JDLQ(T VKRXOGEHHPSOR\HGWRH[SUHVV DSSHDULQJLQWKHIRUPXODH LQWHUPVRIWKHJHQHUDOLVHGFRRUGLQDWHV ,I ZH FRQVLGHU YLVFRXV GDPSLQJ WR EH SUHVHQW LQ WKH VWHHU EHDULQJV DQG SRVVLEO\ DOVR DURXQG WKH OHDQ DQG WZLVW D[HV ZH REWDLQ IRU WKH GLVVLSDWLRQ IXQFWLRQZLWKN NnUDQGN GHQRWLQJWKHUHVSHFWLYHGDPSLQJFRHIILFLHQWV
'
'
N
0
%
0 nUnU
N
%
N
0
7KH9LUWXDO:RUN 7KURXJK WKH YLUWXDO ZRUN WKH JHQHUDOLVHG IRUFHV HDFK DVVRFLDWHG ZLWK D JHQHUDOLVHG FRRUGLQDWH FDQ EH DVVHVVHG 7KH IRUFHV ZKLFK DFW IURP WKH HQYLURQPHQWXSRQWKHYHKLFOHDUHWKHKRUL]RQWDOJURXQGIRUFHVDQGPRPHQWVWKH DHURG\QDPLF IRUFHV DQG WKH JUDYLWDWLRQDO IRUFH FRPSRQHQW LQ ORQJLWXGLQDO GLUHFWLRQ LQ FDVH RI D IRUZDUG VORSH RU WKH G\QDPLF ORQJLWXGLQDO G¶$OHPEHUW IRUFHVDFWLQJLQWKHPDVVFHQWUHVZKLFKDUHLQHTXLOLEULXPZLWKWKHORQJLWXGLQDO IRUFHVJHQHUDWHGE\WKHW\UHV7KHRSWLRQRIFRQVLGHULQJDIRUZDUGVORSHWKDWDW JLYHQDHURG\QDPLFGUDJDQGGULYLQJRUEUDNLQJIRUFHVLVMXVWDEOHWRPDLQWDLQD
02725&<&/('<1$0,&6
FRQVWDQWIRUZDUGYHORFLW\ZRXOGPDNH WKHDQDO\VLVFRUUHFWDVLQWKDWFDVHWKH FRHIILFLHQWV RI WKH OLQHDU HTXDWLRQV DUH WLPH LQGHSHQGHQW FI (T ZLWK ZKHUHWKLVLVQRWWKHFDVH 7KHORQJLWXGLQDODFFHOHUDWLRQD[DSSHDULQJLQ WKHHQVXLQJHTXDWLRQVPD\EHFRQVLGHUHGHTXDOWRWKHORQJLWXGLQDOFRPSRQHQWRI WKHDFFHOHUDWLRQGXHWRJUDYLW\WKDWZRXOGDULVHLIWKHYHKLFOHFRQWLQXRXVO\UXQV RYHUDURDGZLWKHTXLYDOHQWIRUZDUGVORSH :LWKWKHYDULRXVIRUFHVDQGPRPHQWVWKDWDFWXSRQWKHYHKLFOHLQDQGDURXQG WKHUHVSHFWLYHSRLQWVRIDSSOLFDWLRQWKHYLUWXDOZRUNEHFRPHV
:
' )[ [ % )\ \ % 0] % % 0[ % )[ [ % )\ \ % 0] % 0[ n % & )G [G% 0 % 0nU nU % & D[ PP [ P % PU [U % PU [ U % PV [V
%
7KHYLUWXDOGLVSODFHPHQWVH[SUHVVHGLQWHUPVRIWKHJHQHUDOLVHGFRRUGLQDWHVWXUQ RXWWRUHDG
' FRVg & VLQg \ % D F % WF %VF [ ' \ ' \%D F
' FRVg & VLQg ' n % VLQg % FRVg \ ' \&E F [ ' &K n & KG G G [ ' & K n % \ & KP P P P [ ' & K n % V n % \ & KU U U U U U [ ' & K n % H & KI I I I [ ' & K n % H & VV & KV V V V [
%K
ZKHUH
FDQEHH[SUHVVHGLQWKHJHQHUDOLVHGFRRUGLQDWHVE\WDNLQJWKHYDULDWLRQ
RI 1RZ ZH PD\ FRPSDUH DIWHU KDYLQJ VXEVWLWXWHG KHUHLQ WKH H[SUHVVLRQV ZLWK WKH IRUPXODWLRQ RI WKH YLUWXDO ZRUN DFFRUGLQJ WR (T ZKLFKEHFRPHV
:
' '
3
M
'
4
4
Y
T
M \
M
'
% 4U
% 4n
n % 4nU nU % 4
%4
$VDUHVXOWWKHJHQHUDOLVHGIRUFHV4MDUHREWDLQHGZKLFKDUHWREHLQVHUWHGDWWKH
02725&<&/('<1$0,&6
ULJKWKDQGVLGHVRIWKH/DJUDQJHDQHTXDWLRQV
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
HWFGHSHQGRQWKHFXUUHQWYHUWLFDOZKHHOORDG)]L
)]L
(TV 7KHWRWDORUGHURIWKHV\VWHPWXUQVRXWWREH :LWKDJLYHQVSHHGRIWUDYHOXDQGWKHWRWDOORQJLWXGLQDOW\UHIRUFH)[WRWWKH DLUGUDJ)GWKHYHUWLFDOW\UHORDGV)])]DQGWKHLQGLYLGXDOORQJLWXGLQDOW\UH IRUFHV)[FDQEHDVVHVVHGXVLQJWKHHTXDWLRQV 7KHLQLWLDOZKHHOORDGV DWVWDQGVWLOO )]R)]RDUHGLUHFWO\DVVRFLDWHGZLWK WKHYHKLFOHPDVVGLVWULEXWLRQ(T ,QWKHHTXDWLRQVWKHLPSRVHGKDQGOHEDUWRUTXH0 DQGOHDQWRUTXH0nUDSSHDU LQWKHULJKWKDQGPHPEHUV7KHW\UHVLGHIRUFHVDQGPRPHQWVDUHWHQWDWLYHO\SXW RQ WKH ULJKWKDQG VLGH DV ZHOO ,Q WKH HTXDWLRQ IRU WKH VWHHU DQJOH WKH FRQWURO WHUPVKDYHEHHQDGGHG
nn
7KHHTXDWLRQVIRUVXFFHVVLYHO\WKHYDULDEOHVYU
U
DQG
EHFRPHDVIROORZV
Y
0 XU PIDIPVDV U0 PPPIPVPU Y PPKPPIKIPVKVPUKU
g
g
)[ FRV VLQ
P n
U
VU
P n U
I
P
HIPVHV
)\)\
V
VV
U
PIDIPVDV Y 0 XU ^PIDI PVDV ,P] ,I[,V[ VLQ
g
g
g ` U0
,I],V] FRV
g `n
^PIKIDIPVKVDV,P[] ,I],V],I[,V[ VLQ FRV
0 n 0 ^P H D P H D , , FRVg` , U XVLQg P V D , VLQg , U XFRVg 0 ) K n ) ^ W D FRVg V D VLQg ` D ^PK n P V n P H P H P V ` U
^,Z\U ,Z\QJ,H\ U`X
G
[
I
I
I
V
V
V
V
V[
G
V
V
I]
[
F
U
U
V]
Z\
F
I
I
Z\
F
V
V
F
V
V
DF)\EF)\0]0]PPUD[\PU
02725&<&/('<1$0,&6
n
PPKPPIKIPVKVPUKU Y 0 XU ^,Z\U ,Z\QJ,H\ U`XU
g
g ` U0 FRV g , , VLQ g ` n P V J n K n U U
^PIKIDIPVKVDV,P[] ,I],V],I[,V[ VLQ FRV
^PIKI PVKV PPKP PUKU ,P[,U[ ,I[,V[
n ,
PPKPPIKIPVKVPUKU J
g
^PIHIKIPVHVKV ,I],V] VLQ `
g
PVVVKV,V[FRV
g 0
U XVLQ Z\
U
U
g 0 W )
U X FRV Z\
,
I]
PUVU U[
,
V]
U
F
PIHIJPVHVJ ]
VF)]PVVVJ
0[0[PPUJ\PU
n
U
PUVU Y 0 XU ,U[PUVUKU ,U[PUVU
n
U
N
U
P n
U
0 F n U
U
n J n U
VUJ
PUVU
0U
PIHIPVHV Y 0 XU ,Z\U VLQ
g XU^P H D , g`n
I
^PIHIKIPVHVKV ,I],V] VLQ WF)]PIHIJPVHVJ
,
PVHVVV
Z\
U X
0 ^ V
F
F
g
)]PIHIJPVHVJ VLQ
g
)]PVVVJ VLQ )[K ` G
0 J
g
0 gn
U XFRV Z\
0 J n J n
JYYJUUJG
g ` U0
PVHVDV ,I],V] FRV
I
n
N 0 W
PIHI PVHV ,I],V]
I
WF)\0]FRV
g
0[VLQ
0
g U0 ^P V , U FRVg `X U , U X VLQg n 0 V ) P V J n g n 0 , U X V ) P V J VLQg P H V 0 P V , N ^F V ) P V J FRVg ` V ) 0 VLQg 0 FRVg
PVVV Y 0 PVVVDV,V[VLQ
PVVVKV,V[FRV V
V
V
Z\
V
Z\
V
Z\
F
]
V
F
]
F
V
]
V
V
V
V
V[
F
\
]
V
V
[
WUDQVLHQWVOLSDQGFDPEHUDQJOHV
N 0N 0N 0N
0 X
X
X
X
N' N' N' N'
X
X
X X
W\UH IRUFHV DQG PRPHQWV FRHIILFLHQWV GHSHQG RQ ZKHHO ORDG
02725&<&/('<1$0,&6
)\
&)
)\
&)
0 &0 ] 0]
&)
&0[
0[
&0[
N
N
&0 ¶
0[
UF)[
&0
N N
&)
&0 ¶
N N N N
UF)[
VOLSDQGFDPEHUDQJOHV
' Y D U W 0 V 0 X FRVg ' n VLQg FRVg ' Y E U X 'n
F
F
F
g
VLQ
F
WKHULGHUIHHGEDFNFRQWUROJDLQVRFFXUULQJLQ(T GHSHQGRQWKHVSHHG RIWUDYHOXDQGDUHDVVHVVHGKHUHE\WULDODQGHUURU
JY Jn
JYRXJU JnRJG
JURXJGn JG
XJ R
JGnR XXF X J
R
7KH VWHHU WRUTXH 0 LV FRQVLGHUHG WR DFW LQ DGGLWLRQ WR WKH VWDELOLVDWLRQ VWHHU WRUTXH WKDW UHVXOWV IURP IHHGEDFN FRQWURO DQG LV UHSUHVHQWHG E\ WKH WHUPV FRQWDLQLQJWKHFRQWUROJDLQVLQ(T )RUWKHPDVVFHQWUHODWHUDORIIVHWZHKDYHLQWURGXFHGIRUDEEUHYLDWLRQ
PPU\PU PP\PPU\UZLWKPPU PPPU
,IWKHVWHHUDQGUROODQJOHVUHPDLQVXIILFLHQWO\VPDOOVRWKDWLWPD\EHDVVXPHG WKDWJHRPHWULFOLQHDULW\VWLOOSUHYDLOVZHPD\LQYHVWLJDWHWKHLQIOXHQFHRIODUJHU ZKHHOVOLSE\XVLQJIRUWKHW\UHVLGHIRUFHVDQGPRPHQWVWKHQRQOLQHDUIXQFWLRQV RI
L L
L
ZKHHO ORDG )]L DQG WKH EUDNLQJGULYLQJ IRUFH )[L DFFRUGLQJ WR WKH
(TV )RU WKH EDVHOLQH FRQILJXUDWLRQ RI WKH OLQHDU YHKLFOH PRGHO WKH ULGHU LV FRQVLGHUHG ULJLG Fn
U
64 :LWK WKH ULGHU XSSHU WRUVR UHOHDVHG WKH SDUDPHWHU
YDOXHVRI7DEOHKROG7KHW\UHVSDUDPHWHUYDOXHVKDYHEHHQFKRVHQDVOLVWHG LQ7DEOH7KHPRGHOFRQILJXUDWLRQPD\EHFRQVLGHUHGDVW\SLFDOIRUDKHDY\ PRWRUF\FOH:KHQDULJLGO\FRQQHFWHGULGHUXSSHUWRUVRLVFRQVLGHUHG EDVHOLQH FRQILJXUDWLRQZLWK
n
U
WKHUHOHYDQWSDUDPHWHUVUHVXOWIURP(TV
)RUWKHOLQHDUV\VWHPGHILQHGWKHHLJHQYDOXHVDQGWKHVWHDG\VWDWHVWHHUDQJOH DQGVWHHU WRUTXH SHUXQLWSDWKFXUYDWXUHKDYHEHHQHVWDEOLVKHGIRUDVHULHVRI YDOXHVRIWKHVSHHGRIWUDYHOX
9 ,QDGGLWLRQDQXPEHURISDUDPHWHUVUHODWHG
02725&<&/('<1$0,&6
WR ERWK WKH YHKLFOH FRQVWUXFWLRQ DQG WKH W\UHV KDYH EHHQ FKDQJHG LQ YDOXH WR H[DPLQHWKHLULQIOXHQFH 7DEOH3DUDPHWHUVRIYHKLFOHLQEDVHOLQHFRQILJXUDWLRQEXWZLWKULGHUXSSHUWRUVRUHOHDVHG
Fn 64 5LGHUFRQWUROJDLQVJZLWKFURVVRYHUYHORFLW\XF
LQEDVHOLQHFRQILJXUDWLRQ
PP
NJ
,P[
NJP
,I[
NJP
,Z\
NJP
DF
U
PI
NJ
,P]
NJP
,I]
NJP
,Z\
NJP
P
EF
VF
P
KP VU
N J X
PV
NJ
,P[]
NJP
,V[
NJP
,H\
NJP
P
HI
WF
P
P
KI
P
P
KU
P
1PVUDG PV
F
PV
F Q J Jn
PU
NJ
,U[
NJP
,V]
NJP
P
HV
P
U
P
U
P
KV
P
KG
P
N
1PVUDG
N1PUDG
Nn
J
g
YR
R
J J
U
G
R
1PVUDG UDG
UR
Fn & Jn J
U
1PUDG
G$
NJP
G
R
R
6WDELOLW\$QDO\VLV
)UHH8QFRQWUROOHG0RWLRQ ,QDVHULHVRIGLDJUDPVWKHYDULDWLRQRIWKHHLJHQYDOXHVDVDIXQFWLRQRIVSHHGKDV EHHQSUHVHQWHG7KHGLDJUDPRI)LJUHSUHVHQWVWKHVLWXDWLRQFRUUHVSRQGLQJ WRWKHEDVHOLQHFRQILJXUDWLRQGHILQHGE\7DEOHVEXWZLWKWKHULGHUUHJDUGHG DVDULJLGHOHPHQWRIWKHPDLQIUDPHERG\7KHXSSHUGLDJUDPJLYHVWKHUHDOSDUW WK
RIWKHHLJHQYDOXHVRIWKLV RUGHUV\VWHPYHUVXVVSHHGLQGLFDWLQJWKHGHJUHHRI LQVWDELOLW\ LI SRVLWLYH RU GDPSLQJ VWDEOH LI QHJDWLYH 7KH VHFRQG GLDJUDP SUHVHQWVWKHYDULDWLRQRIWKHLPDJLQDU\SDUWRIWKHHLJHQYDOXHVWKDWUHSUHVHQWWKH IUHTXHQFLHV RI WKH GLIIHUHQW PRGHV RI YLEUDWLRQ LQ UDGV 7KH WKLUG GLDJUDP GHSLFWVWKHYDULDWLRQRIWKHHLJHQYDOXHVLQWKHFRPSOH[SODQH$ORQJWKHFXUYHV WKHVSHHGFKDQJHVDVLQGLFDWHG
02725&<&/('<1$0,&6
5 real part eigenvalues [1/s]
weave wobble 0
weave -5
-10
imaginary part eigenvalues [rad/s]
capsize
0
40
[km/h]
9
120
160
60 wobble 40 weave 20
capsize
0 0
imaginary part eigenvalues [rad/s]
80
40
80
9[km/h]
120
160
wobble
9
50
9 9
weave
9 0 -10
capsize -20
-10 real part eigenvalues [1/s]
0
5
)LJ %DVHOLQHFRQILJXUDWLRQ ULJLGULGHUZLWKDLUGUDJ]HURIRUZDUGDFFHOHUDWLRQQR
9
FRQWURO 5HDODQGLPDJLQDU\SDUWVRIHLJHQYDOXHVYHUVXVVSHHGRIWUDYHO
02725&<&/('<1$0,&6
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
7KH PRGHO VKRZV WKDW DEVHQFH RI DLU GUDJ FDVH VOLJKWO\ VWDELOLVHV WKH ZHDYHPRGHDQGGHVWDELOLVHVWKHZREEOHPRGH
7RUVLRQDOULJLGLW\RIWKHPDLQIUDPH FDVHODUJHF DSSHDUVWREHRIFUXFLDO LPSRUWDQFHHVSHFLDOO\IRUWKHPDQQHULQZKLFKWKHZREEOHPDQLIHVWVLWVHOI +LJKVWLIIQHVVRUGLVUHJDUGLQJWKHWRUVLRQDOFRPSOLDQFHJLYHVULVHWRD WRR KLJKFULWLFDOVSHHG$PRUHIOH[LEOHIUDPH EDVHOLQHFRQILJXUDWLRQ FDXVHVWKH ZREEOHWRRFFXULQDOLPLWHGUDQJHRIVSHHGDURXQGNPKZKLFKWXUQVRXW WREHH[SHULHQFHGDOVRLQUHDOLW\
$OPRVW YDQLVKLQJ UHOD[DWLRQ OHQJWKV UDSLG W\UH UHVSRQVH GUDVWLFDOO\ VXSSUHVVHVWKHZREEOHLQVWDELOLW\
02725&<&/('<1$0,&6
real part eigenvalues [1/s]
5
weave
4 6
7
9 5
0
0: base line 1: air drag =0 2: Fβ x10
9
5
160
0: base line 1: air drag =0 2: Fβ x10 3: σ s /10 & s x10 )α 4: & Μα s /10 σ s /10 5: & Μα s /10 6: & Μγ s /10 s /10 7: ,Z\ 8: ε /10 9: ε /10, WF /10
2 8 0
-5
3
6 -10
0
40
4 80
120
9[km/h]
capsize
5 real part eigenvalues [1/s]
160
3: σ s /10 & s x10 )α 4: & Μα s /10 σ s /10 5: & Μα s /10 6: & Μγ s /10 s /10 7: ,Z\ 8: ε /10 9: ε /10, WF /10
1
1
0 6 9 5
-5
0,-1, 2,3,4,8 7
7
-10
0
40
80
120
9[km/h]
wobble
5 real part eigenvalues [1/s]
0: base line 1: air drag =0 2: Fβ x10
7
4 8 0
6 5
2
9 -5
1 3
-10
0
40
0 80
9[km/h]
120
3: σ s /10 & s x10 )α 4: & Μα s /10 σ s /10 5: & Μα s /10 6: & Μγ s /10 s /10 7: ,Z\ 8: ε /10 9: ε /10, WF /10
160
)LJ(IIHFWRIGUDVWLFFKDQJHVLQPRGHOSDUDPHWHUYDOXHVRQVWDELOLW\FKDUDFWHULVWLFV
02725&<&/('<1$0,&6
0DNLQJWKHW\UHDOPRVWULJLG ZLWKLQDGGLWLRQWRYHU\VPDOO
YHU\ODUJH
FRUQHULQJVWLIIQHVVDQGYHU\VPDOOSQHXPDWLFWUDLOFDVH ZRXOGGUDVWLFDOO\ VXSSUHVVWKHKLJKVSHHGZHDYHEXWJLYHVULVHWRYLROHQWZREEOHRYHUDODUJH UDQJHRIVSHHG$PXFKODUJHUFDPEHUIRUFHVWLIIQHVVFDQQRWEHWHFKQLFDOO\ UHDOLVHGHYHQZLWKDYHU\VWLIIVROLGZKHHO,WZRXOGLQFUHDVHWKHORZVSHHG XQVWDEOHZHDYHUDQJHDQGVWDELOLVHWKHKLJKVSHHGZHDYH ZREEOHLVDOPRVW QRWDIIHFWHG QRWVKRZQ
0DNLQJRQO\WKHDOLJQLQJWRUTXHVWLIIQHVVHVYHU\VPDOOGHVWDELOLVHVWKHKLJK VSHHGZHDYHPRGHDQGVWDELOLVHVWKHZREEOHPRGH7KHFDSVL]HURRWLVPDGH PRUHQHJDWLYH
(YHQDFRQVLGHUDEOHGHFUHDVHRIWKHFDPEHUDOLJQLQJPRPHQWVWLIIQHVVKDV YHU\OLWWOHHIIHFWDWOHDVWLQWKHFDVHRI]HURRUYHU\VPDOOORQJLWXGLQDOW\UH IRUFHV7KHVDPHKROGVIRUWKHRYHUWXUQLQJFRXSOHVWLIIQHVVWKDWLV DVPDOO FURZQ UDGLXV QRW VKRZQ 7KH VPDOOHU FDPEHU DOLJQLQJ PRPHQW VWLIIQHVV GRHVKDYHDQLQIOXHQFHRQWKHFDSVL]HVWDELOLW\$VLQGLFDWHGLQWKHGLDJUDP FDSVL]HLQVWDELOLW\QRZVKRZVXSDQGRFFXUVEH\RQGDVSHHGRIDURXQG NPK
7KH IXQGDPHQWDO UROH RI WKH J\URVFRSLF FRXSOLQJ WHUPV EHFRPHV HYLGHQW ZKHQWKHSRODUPRPHQWVRIWKHZKHHODUHFRQVLGHUDEO\UHGXFHG FDVH )LUVW WKH ORZ VSHHG XQVWDEOH ZHDYH UDQJH LV VWUHWFKHG WR DOPRVW NPK DQG VHFRQG WKH ZREEOH EHFRPHV YLROHQWO\ XQVWDEOH XS WR YHU\ KLJK OHYHOV RI VSHHG,WPD\EHFRQFOXGHGWKDWWKHJ\URVFRSLFHIIHFWRIWKHURWDWLQJZKHHOV LVODUJHO\UHVSRQVLEOHIRUWKHIDFWWKDWWKHPRWRUF\FOHLVFDSDEOHRIPRYLQJLQ DVWDEOHPDQQHU
5HGXFWLRQRIWKHUDNHDQJOHWRDQDOPRVWYHUWLFDOVWHHUD[LVRULHQWDWLRQZLWK WKHFDVWHUOHQJWKNHSWWKHVDPHZRXOGVWURQJO\GHVWDELOLVHWKHZREEOHPRGH
$WWKHVDPHWLPHUHGXFLQJWKHFDVWHUOHQJWKDVZHOO FDVH ZRXOGVWURQJO\ GHVWDELOLVH WKH ZHDYH PRGH ZKLFK XQYHLOV WKH IXQGDPHQWDO HIIHFW RI LQWURGXFLQJWKHVWHHUD[LVEDFNZDUGVLQFOLQDWLRQ
(IIHFWVRIPRUHUHDOLVWLFOHYHOVRIYDULDWLRQRIWKHV\VWHPSDUDPHWHUVKDYHEHHQ SUHVHQWHGLQWKHVWDELOLW\GLDJUDPVRI)LJV &RPSDULVRQZLWKWKHFXUYHVWKDW UHSUHVHQW WKH EDVHOLQH FRQILJXUDWLRQ UHYHDOV WKDW DJDLQ FXUYH QXPEHUV FRUUHVSRQGWROLVWQXPEHUVEHORZ
6
$VPDOOHUIURQWUHOD[DWLRQOHQJWKVWDELOLVHVWKHZREEOHPRGHDQGKDV SUDFWLFDOO\QRHIIHFWRQWKHZHDYHDQGFDSVL]HPRGHV
6
$ VPDOOHU UHDU UHOD[DWLRQ OHQJWK VWDELOLVHV WKH ZREEOH PRGH LQ D OHVVHU GHJUHHDQGVWDELOLVHVWKHZHDYHPRGH1RHIIHFWRQWKHFDSVL]HPRGH
6
0RYLQJ WKH FHQWUH RI JUDYLW\ RI WKH PDLQIUDPH SOXV ULGHU FP IRUZDUGV WXUQVRXWWRVWURQJO\VWDELOLVHWKHZHDYHPRGH+RZHYHULWGHVWDELOLVHVWKH
02725&<&/('<1$0,&6
weave
real part eigenvalues [1/s]
5
8 0 5 0 -5
0
3
40
80
2
6
9[km/h]
120
160
capsize
5 real part eigenvalues [1/s]
14
7 6 -10
0-6,8 -5
7
0
40
80
9[km/h]
160
6
0 5
2
7
8
0
-5
4
0
40
5: σα,γ 2 /1.2 6: DF -0.1m, E +0.1m F 7: KP -0.1m, K -0.1m U Nms 8: Nδ =10 rad
0: base line 1: &)α1 x1.2 2: &)α2 x1.2 3: &0α1 x1.2 4: σα,γ 1 /1.2
1
3
-10
120
wobble
5
5: σα,γ 2 /1.2 6: DF -0.1m, E +0.1m F 7: KP -0.1m, K -0.1m U Nms 8: Nδ =10 rad
0: base line 1: &)α1 x1.2 2: &)α2 x1.2 3: &0α1 x1.2 4: σα,γ 1 /1.2
0
-10
real part eigenvalues [1/s]
7
8
0: base line 1: &)α1 x1.2 2: &)α2 x1.2 3: &0α1 x1.2 4: σα,γ 1 /1.2
80
9[km/h]
6
120
160
5: σα,γ 2 /1.2 6: DF -0.1m, E +0.1m F 7: KP -0.1m, K -0.1m U Nms 8: Nδ =10 rad
)LJ (IIHFWRIUHDOLVWLFFKDQJHVLQSDUDPHWHUYDOXHVRQVWDELOLW\FKDUDFWHULVWLFV
02725&<&/('<1$0,&6
ZREEOH&DSVL]HLVQRWDIIHFWHG
6
/RZHULQJERWKFHQWUHVRIJUDYLW\RYHUFPVWDELOLVHVERWKWKHZHDYHDQG ZREEOHPRGHVDOLWWOH7KHFDSVL]HPRGHEHFRPHVPRUHVWDEOH
6
$GGLQJVWHHUGDPSLQJGRHVRIFRXUVHVWDELOLVHWKHZREEOHPRGH+RZHYHU LWDGYHUVHO\DIIHFWVWKHZHDYHPRGHVWDELOLW\
)RUWKHIUHHFRQWUROVLWXDWLRQZHILQDOO\FRQVLGHUWKHLQIOXHQFHRIGULYLQJRU EUDNLQJDQGWKHDGGLWLRQRIDGHJUHHRIIUHHGRP WKHULGHUOHDQDQJOHPHDVXUHG ZLWK UHVSHFW WR WKH PDLQIUDPH UROO DQJOH )LJXUH VKRZV WKH VWDELOLW\ GLDJUDPRIWKHEDVHOLQHFRQILJXUDWLRQDW]HURQHWDFFHOHUDWLRQIRUFH )D[ UHDU ZKHHO GULYLQJ IRUFH MXVW ZLWKVWDQGV WKH DLU GUDJ VR WKDW VSHHG LV
baseline
)D[ = 0
5
) D[
real part eigenvalues [1/s]
wobble
) D[
= -1500N = 1500N
0
-5 capsize weave -10
)LJ
0
80
40
9[km/h] 120
160
(IIHFWRIWKHDSSOLFDWLRQRIEUDNLQJ )D[)[ DQGGULYLQJIRUFHV )D[)[! RQWKHVWDELOLW\GLDJUDP EDVHOLQHFRQILJXUDWLRQ
baseline + rider lean
5 real part eigenvalues [1/s]
)D[ = 0
wobble 0 lean mode capsize
-5
-10
0
weave
40
80
9
[km/h]
120
160
)LJ7KHVWDELOLW\GLDJUDPZLWKWKHULGHUOHDQGHJUHHRIIUHHGRPLQWURGXFHG
02725&<&/('<1$0,&6
FRQVWDQW DWEUDNLQJ )D[
1DQGDWGULYLQJ )D[
1
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
6WHS5HVSRQVHVRI&RQWUROOHG0RWLRQ :HZLOOVWXG\WKHUHVSRQVHVRIWKHYHKLFOHPRWLRQWRDXQLWVWHSRIWKHLPSRVHG VWHHUWRUTXH0*DQGRIWKHOHDQWRUTXH
0 7KLVDVDILUVWDWWHPSWWRLQYHVWLJDWH U
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
)
RI]HUR
D[
U
7KHILJXUHGHSLFWVWKHYDULDWLRQVRIWKH\DZUDWH WKHVWHHUDQJOH
02725&<&/('<1$0,&6
100
δ
.
δ
J
U
0
ϕ
.
ϕ
Y
-100 0
20
40
9 [km/h]
60
80
100
)LJ+\SRWKHWLFDOULGHUVWHHUWRUTXHIHHGEDFNVWDELOLVLQJFRQWUROJDLQVYHUVXVVSHHG
rider controlled
5
=0 = -1500N ) D[ = 1500N ) D[
real part eigenvalues [1/s]
) D[
lean mode 0
weave -5 capsize wobble -10
0
40
80
9[km/h]
120
160
)LJ7KHVWDELOLW\GLDJUDPZLWKIHHGEDFNVWHHUWRUTXHFRQWURORI)LJDFWLYDWHG
nDQGWKHXSSHUWRUVRUROODQJOHnn 7ZRYDOXHVRI
WKHPDLQIUDPHUROODQJOH
U
VSHHGKDYHEHHQFRQVLGHUHG DQG NPK7KHUHVXOWVDUHPRVWLQWHUHVWLQJ
:HREVHUYHWKDWDVDUHVXOWRIDSRVLWLYHFKDQJHLQVWHHUWRUTXH WRWKHULJKW ILUVW D SRVLWLYHVWHHUDQJOHDULVHVZLWKDOVRSRVLWLYH\DZUDWH WRWKHULJKW $IWHUD VKRUWWLPHDFKDQJHLQVLJQRFFXUVDQGDQHJDWLYH\DZUDWHLVEHLQJGHYHORSHG ZLWKDQHJDWLYHVWHHUDQJOH,WLVVHHQWKDWGXULQJWKLVVKRUWWUDQVLWLRQWLPHWKH UROODQJOHEHJLQVWREXLOGXSLQWKHFRUUHFWGLUHFWLRQ WRWKHOHIW WKDWEHORQJVWR WKHILQDOVWHDG\VWDWHFXUYLQJVLWXDWLRQ7KLVYDULDWLRQLQWKHPRWLRQYDULDEOHVLV VLPLODUWRZKDWZRXOGEHREVHUYHGLQUHDOLW\7KHXQFRQWUROOHGYHKLFOHLVVWDEOH DWWKHVSHHG
9
NPK )LJ ,IDWWKLVVSHHGWKHVWDELOLVDWLRQFRQWUROOHU
ZRXOG QRW EH DFWLYDWHG WKH PRWRUF\FOH H[KLELWV D VLPLODU UHVSRQVH EXW ZLWK DQJOHVDQG\DZUDWHEHFRPLQJODUJHUDQGZLWKOHVVGDPSHGZREEOHDQGZHDYH YLEUDWLRQV
02725&<&/('<1$0,&6
response to step steer torque
)D[ = 0
δ
UO
response to step lean torque
9 = 35 km/h
U
0.01
10
0δ =1Nm
X 0
10 δ
ϕ
ϕ
X
U
=1Nm
9 = 35 km/h )D[ = 0
δ
ϕ
ϕ
0 [rad]
-0.02 [rad] 0
1
2
tim e 3
response to step steer torque
4
[s]
5
-0.04
0δ =1Nm
10 δ
ϕ
ϕ
-0.02
U
10
δ
0
1
X
0
U
4
5
=1Nm
9 = 70 km/h
ϕ
ϕ
[s]
5
)D[ = 0
U δ
0 [rad]
2 time 3
4
[s]
ϕ ϕU
UO
10 δ
ϕ ϕU
[rad]
2 time 3
1
0.01
UO X 0
0
response to step lean torque
9 = 70 km/h )D[ = 0
0.01
10
UO
10 δ
ϕ ϕU
U
ϕ ϕU
0.01
10
0
-0.04
0
2 time 3
1
[s]
4
5
)LJ 7KHVWHSUHVSRQVHVRIPRWLRQYDULDEOHVWRXQLWVWHHUDQGULGHUOHDQWRUTXHDW]HUR IRUZDUGDFFHOHUDWLRQ)D[
response to step steer torque
10
0δ =1Nm
U
0.01
response to step lean torque
9 = 35 km/h )D[ = -1500N
δ
UO
ZLWKULGHUVWDELOLVLQJFRQWURO
0.01
10
ϕ ϕU ϕ
X 0
10 δ
ϕ
1
2 time 3
response to step steer torque
4
[s]
5
)D[= -1500N
10
0
ϕ
ϕ
-0.02
U
δ
[s]
4
5
[s]
0
U
4
5
=1Nm
9 = 70 km/h )D[ = -1500N
ϕ ϕU
X
ϕ
U
0 [rad]
2 time 3
2 time 3
UO
-0.04
)LJ
1
ϕ
ϕϕU 1
0
10 δ
[rad] 0
δ
ϕ
0.01
=1Nm
9 = 35 km/h )D[ = -1500N
response to step lean torque
9 = 70 km/h
UO X
-0.04
0δ =1Nm
0.01
10 δ
U
0 [rad]
0
10
X
ϕ
[rad]
U
ϕ ϕU
UO
10 δ
-0.02
0
0
1
δ 2 time 3
[s]
4
5
7KHVWHSUHVSRQVHVRIPRWLRQYDULDEOHVWRXQLWVWHHUDQGULGHUOHDQWRUTXHZKLOH EUDNLQJ)D[
1 ZLWKULGHUVWDELOLVLQJFRQWURO
02725&<&/('<1$0,&6
7KHOHDQWRUTXHUHVSRQVHVKRZVLQLWLDOO\DQHJDWLYHUROODQJOHZKLFKLVWKH UHVXOWRIFRXQWHUDFWLQJWKHLPSRVHG LQWHUQDO OHDQWRUTXH$IWHUWKHLQLWLDOSKDVH WKHDQJOHEHFRPHVSRVLWLYH$OVRWKHVWHHUDQJOHDQGZLWKWKDWWKH\DZUDWHWXUQ RXWWREHFRPHQHJDWLYHLQWKHLQLWLDOSKDVHZKLFKFDXVHVWKHFRQWDFWSRLQWVWR PRYHLQWKHULJKWGLUHFWLRQ WRWKHOHIW 7KHVWHDG\VWDWHVLWXDWLRQLVUHDFKHGDIWHU ORZIUHTXHQF\GDPSHGOHDQPRGHRVFLOODWLRQV,IWKHFRQWUROOHULVQRWDFWLYDWHG D YHU\ VLPLODU UHVSRQVH ZLWK QRZ ORZ IUHTXHQF\ GDPSHG ZHDYH RVFLOODWLRQV )LJ RFFXUV1RZLWEHFRPHVFOHDUWKDWLWLVPDLQO\WKHJ\URVFRSLFFRXSOH SOXVFDPEHUDOLJQLQJPRPHQW WKDWVWHHUVWKHIURQWZKHHOLQLWLDOO\LQWKHULJKW GLUHFWLRQ$QDO\VLVVKRZVWKDWWKHKDUGO\YLVLEOHVPDOOSRVLWLYHVWHHUDQJOHSHDN ULJKW DIWHU WKH OHDQ WRUTXH VWHS FKDQJH LV GXH WR WKH PHFKDQLFDO WUDLO DQG WKH DOLJQLQJVWLIIQHVVRIWKHIURQWZKHHO WFDQG&0 $WWKHKLJKHUVSHHGRI9
NPKWKHUHVSRQVHVVKRZWKHVDPHWHQGHQFLHVEXW
ZLWKPXFKVPDOOHUVWHHUDQJOHVDQG\DZUDWHV7KHZHDYHLVPRUHGDPSHGDQG WKHZREEOHOHVV:LWKRXWFRQWUROOHUWKHZREEOHEHFRPHVXQVWDEOHDQGDYLROHQW VWHHULQJRVFLOODWLRQDWFD +]LVGHYHORSHG :LWKWKHOHDQGHJUHHRIIUHHGRPGLVUHJDUGHGRQO\WKHUHVSRQVHWRVWHHUWRUTXH FDQEHFRQVLGHUHG7KHUHVXOWLQJPRWLRQUHVSRQVHVWXUQRXWWRJHWFORVHWRWKRVH
nn GHOHWHG
GHSLFWHGLQWKHOHIWKDQGGLDJUDPVRI)LJZLWKWKHFXUYHVIRU
U
7KH ORZ IUHTXHQF\ RVFLOODWLRQ WKDW LV SUHGRPLQDQW DW WKH ORZHU VSHHG LV QRZ DWWULEXWHGWRWKHZHDYHPRGH7KHXQFRQWUROOHGV\VWHP )LJ LVMXVWVWDEOH DWNPKDQGEHKDYHVVLPLODUWRWKHFRQWUROOHGRQHDOEHLWWKDWVRPHZKDWODUJHU ZHDYHDQGZREEOHYLEUDWLRQVDQGDODUJHUVWHDG\VWDWHUHVSRQVHDULVH$WNPK WKHXQVWDEOHZREEOHYLEUDWLRQGHYHORSVDJDLQ )LJXUHSUHVHQWVWKHUHVXOWVIRUWKHV\VWHPZKLOHWKHEUDNHVDUHDSSOLHG )D[
1 7KHFRXUVHRIWKHYDULRXVVLJQDOVVKRZVLPLODUWHQGHQFLHVDV
RFFXUVLQWKHXQEUDNHGVLWXDWLRQ7KHREYLRXVGLIIHUHQFHVDUHWKHORZO\GDPSHG ZREEOHYLEUDWLRQWKDWLVVHHQWRRFFXUDWNPK FI)LJ DQGWKHPXFK VPDOOHUVWHDG\VWDWHUHVSRQVHV FI)LJRIQH[WVHFWLRQ 7KHVHGLIIHUHQFHV DUHSULPDULO\GXHWRWKHDFFRPSDQLHGORDGWUDQVIHU
$QDO\VLVRI6WHDG\6WDWH&RUQHULQJ
:KHQDVWHDG\VWDWHFRUQHULQJFRQGLWLRQ LVUHDFKHGHTXLOLEULXPRIIRUFHVDQG PRPHQWV H[LVWV :H FRQVLGHU WKH HTXLOLEULXP LQ ODWHUDO GLUHFWLRQ DERXW WKH YHUWLFDOD[LVDQGDERXWWKHOLQHRILQWHUVHFWLRQRIPDLQIUDPH FHQWUHSODQHDQG URDGSODQH WKLVLQDGGLWLRQWRWKHVWDWLFHTXLOLEULXPRIIRUFHVLQYHUWLFDOGLUHFWLRQ ,QWKHDSSUR[LPDWHDQDO\VLVWKHVPDOOFJIRUZDUGRIIVHWVRIWKHVWHHUDEOHIURQW DQGVXEIUDPHVHIDQGHVZLOOEHQHJOHFWHG,QDGGLWLRQWKHWZLVWDQJOH DQGWKH
02725&<&/('<1$0,&6
ULGHU OHDQ DQJOH
n ZLOO EH GLVUHJDUGHG ,QVWHDG ZH ZLOO VWXG\ WKH HIIHFW RI D U
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
n GHJUHH RI IUHHGRP GHOHWHG $IWHU WKDW WKH WKHRU\ LV U
H[WHQGHGWRODUJHUUROODQJOHVDQGWKHQRQOLQHDUKDQGOLQJGLDJUDPLVHVWDEOLVKHG IRUWKHVWHDGLO\FRUQHULQJPRWRUF\FOH
/LQHDU6WHDG\6WDWH7KHRU\
7KH WKHRUHWLFDO H[SUHVVLRQV IRU WKH FRHIILFLHQWV FRQFHUQLQJ WKH VWHDG\VWDWH WXUQLQJEHKDYLRXUDQGWKHFJODWHUDORIIVHWHIIHFWVZLOOEHGHYHORSHGLQVHYHUDO VXFFHVVLYH VWDJHV VWDUWLQJ ZLWK WKH VLPSOHVW FDVH LQ ZKLFK W\UH PRPHQWV J\URVFRSLF HIIHFWV DQG DLU GUDJ DUH GLVUHJDUGHG DQG HQGLQJ ZLWK WKH XOWLPDWH FRQILJXUDWLRQLQZKLFKDOOWKHVHLWHPVDUHLQFOXGHGDQGGULYLQJRUEUDNLQJPD\ EHFRQVLGHUHG)LUVWWKHUHTXLUHGUROODQGVWHHUDQJOHVZLOOEHDVVHVVHGDQGZLWK WKHVHUHVXOWVWKHH[SUHVVLRQIRUWKHVWHHUWRUTXHGHWHUPLQHG
:KHHO0RPHQWVDQG$LU'UDJ1HJOHFWHG $V DQ LQWURGXFWLRQ ZH ZLOO ILUVW GLVFXVV WKH VLPSOH FDVH ZKHUH WKH DLU GUDJ ORQJLWXGLQDO W\UH IRUFHV DQG DOO WKH W\UH PRPHQWV DERXW WKH YHUWLFDO D[LV DUH GLVUHJDUGHG7KDWPHDQVWKDWWKHDOLJQLQJPRPHQWWKHRYHUWXUQLQJFRXSOHDQG WKH VSLQ FDPEHU PRPHQW DUH VHW HTXDO WR ]HUR 0RUHRYHU ZH QHJOHFW WKH J\URVFRSLFFRXSOHV )RUWKHODWHUDOZKHHOIRUFHVZHILQGDVLQ&KDSWHU6HFWLRQ(T IRU WKH DXWRPRELOH ZLWK SQHXPDWLF WUDLOV QHJOHFWHG HTXLOLEULXP LQ ODWHUDO GLUHFWLRQDQGDERXWWKHYHUWLFDOD[LV ) )
\
]R
'
) )
\
]R
7KHUROODQJOH
'
D
\
J
'
XU
J
n QHHGHGWRPDLQWDLQHTXLOLEULXPDERXWWKHORQJLWXGLQDO[D[LV \
ZKHQDODWHUDORIIVHW\PURIWKHFHQWUHRIJUDYLW\RIWKHFRPELQHGPDLQIUDPHDQG ULGHUH[LVWV FI)LJV HTXDOV
02725&<&/('<1$0,&6
P
n\ ' &
PU \ PU
PK
7KHDGGLWLRQDOUROODQJOHRIWKHPRWRUF\FOHWKDWDULVHVZKLOHFRUQHULQJEHFRPHV D
n & n\ '
'
\
J
XU
J
7KHIURQWDQGUHDUFDPEHUDQJOHVUHDGLQWHUPVRIWKHUROODQJOHDQGWKHJURXQG VWHHUDQJOHFI(TV
' '
n%
N WDQ g
n
7KHVOLSDQJOHVUHVXOWIURP(T SULPHVRPLWWHGZLWK :H ILQG &
)
&
)
' '
& ] R&
% n &
) n
)]R
&
)
&
)
&
)
g & )] Rn\
WDQ
]Rn\
)
7KHVHHTXDWLRQVVKRZWKHLPSRUWDQFHRIWKHGLIIHUHQFHEHWZHHQFDPEHUVWLIIQHVV 1UDG DQGLQLWLDOW\UHORDG 1 LQYLHZRIWKHVLJQRIWKHVOLSDQJOHV2EYLRXVO\ LQJHQHUDOVLGHVOLSLVQHHGHGWRDFFRPSOLVKODWHUDO\DZHTXLOLEULXP,WPD\EH UHDOLVHGWKDWDWKLJKHUVSHHGVWKHVWHHUDQJOHLVPXFKVPDOOHUWKDQWKHUROODQJOH
n_
_ _ _
:LWKWKHZKHHOEDVHO
DFEFDQGWKH(TV ZHILQGWKHUHODWLRQVKLS
¶
EHWZHHQ WKH JURXQG VWHHU DQJOH GLPHQVLRQDO SDWK FXUYDWXUH O5
g
FRV RQ WKH RQH VLGH DQG WKH QRQ
OUX DQG WKH GLIIHUHQFH RI VOLS DQJOHV RQ WKH
RWKHU
N
'
%
O 5
&
:LWKWKHXVHRI WKHJURXQGVWHHUDQJOHPD\QRZEHZULWWHQLQWHUPV RIWKHQRQGLPHQVLRQDOSDWKFXUYDWXUHO5WKHODWHUDODFFHOHUDWLRQD\DQGWKH FJRIIVHW\PU
N
'
O 5
%
D
P
%
\
J
\
PU \ PU
PK
ZKHUHZHKDYHLQWURGXFHGWKHVWHHUDQJOHFRHIILFLHQW ZKLFKLVDELWODUJHUWKDQ XQLW\
'
RR
' %
&
)
)
&
WDQ g
WKHXQGHUVWHHUFRHIILFLHQW
RIWKHVLPSOLILHGV\VWHP
02725&<&/('<1$0,&6
'
)
'
RR
]R
&
)
&
) &
)
\'
&
\RR '
)
)
&
&
&
)
)
&
)
)
&
DQGWKHFJRIIVHWFRHIILFLHQW
&
&
]R
\
&
&
)
)
&
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
&
' &
O 5
%
D
\
J
%
P
\
PU \ PU
PK
ZKHUHWKHILUVWWHUPLQWKHULJKWKDQGPHPEHUYDQLVKHVLIWKHUDNHDQJOHHTXDOV ]HURPDNLQJ
:KHQWKHPRWRUF\FOHPRYHVVWUDLJKWDKHDG O5
D\
(TV
VKRZ WKDW D VWHHU DQJOH DQG GLIIHUHQFH LQ VOLS DQJOHV ZRXOG DULVH VROHO\ DV D UHVXOWRIWKHFJRIIVHWDQGGHSHQGRQWKHGLIIHUHQFHRIWKHUDWLRVRIFDPEHUDQG VLGHVOLSVWLIIQHVVHVIURQWDQGUHDU
ϕ
ϕ
ζδ (= ζδ FRVε)
ξ
O
5
η
P \ PUPU
ξ
P\
\
\
PK
η
ζδ
0
D\
PUPU
PK
)LJ
J
0
7KH JURXQG VWHHU DQJOH
¶ DQG WKH UROO DQJOH
=
9
2
5J
GXH WR SDWK FXUYDWXUH 5DQG
FJODWHUDORIIVHW\PUDQGWKHYDULDWLRQZLWKODWHUDODFFHOHUDWLRQD\
02725&<&/('<1$0,&6
7KH JHQHUDO FRXUVH RI VWHHU DQJOH DQG UROO DQJOH YHUVXV VSHHG RI WUDYHO DW JLYHQSDWKFXUYDWXUHDQGFJODWHUDORIIVHWKDVEHHQLOOXVWUDWHGLQ)LJ7KH FRHIILFLHQWV DQG
\
ZLOOEHGHILQHGEHORZ
7KHDFWXDOVWHHUDQJOH RIWKHKDQGOHEDUDERXWWKHLQFOLQHGVWHHUD[LVZLWK
g
UDNHDQJOH LVREWDLQHGIURPWKHJURXQGVWHHUDQJOH
N E\GLYLGLQJWKLVDQJOHE\
g
FRV DVKDVEHHQIRUPXODWHGE\(T ZLWKWZLVWDQJOH VHWHTXDOWR]HUR ,IIRUHDQGDIWORDGWUDQVIHUGXHWRDHURG\QDPLFGUDJLVFRQVLGHUHGWKHOLQHV RI )LJ DUH QRW TXLWH VWUDLJKW DQ\PRUH 7KH XQGHUVWHHU FRHIILFLHQW FRUUHVSRQGVWRWKHVORSHRIWKHOLQHLQWKHGLDJUDPLIDLUGUDJLVQHJOHFWHG7KLV PD\EHFRPSDUHGZLWKWKHWKHRU\IRUWKHDXWRPRELOHZKHUH LVWDNHQHTXDOWR XQLW\ DQG DLU GUDJ LV GLVUHJDUGHG )LJ ,I DW KLJK VSHHG DLU GUDJ LV QRW QHJOLJLEOH LVIRXQGDVWKHVORSHRIWKHFKDUDFWHULVWLFWKDWDULVHVLI
NO5LV
SORWWHGDJDLQVWD\ JZLWKWKHVSHHG9KHOGFRQVWDQWZKLOHWKHFXUYDWXUHLQFUHDVHV
ZLWKJURZLQJODWHUDODFFHOHUDWLRQ,IZHZRXOGSORW
NYHUVXVD JDJDLQLQWKLV
\
OLQHDUWKHRU\ZLWKFRQVWDQWVSHHGDVWUDLJKWOLQHDULVHVZKLFKQRZKDVDVORSH HTXDO WR OJ9
)RU WKH VDNH RI FRQYHQLHQFH WKH DERYH GHILQLWLRQ IRU WKH
GHJUHHRIXQGHUVWHHUKDVEHHQDGRSWHGDOWKRXJKDPRUHSURSHUGHILQLWLRQPLJKW EH
N M M D
J 5
\
g
FRV
'LVFXVVLRQ RI QXPHULFDO UHVXOWV 7DEOH IROORZV DIWHU PRUH DGHTXDWH DSSUR[LPDWLRQVKDYHEHHQGHYHORSHG)RUWKHVLPSOHWKHRU\GHYHORSHGVRIDUWKH FRHIILFLHQWV DQG
\
DSSHDULQJLQWKHGLDJUDPDUHHTXDOWRXQLW\%HORZZHZLOO
VHH WKH HIIHFW RI W\UH ZLGWK DQG J\URVFRSLF FRXSOHV RQ WKHVH UROO DQJOH FRHIILFLHQWV
7\UH2YHUWXUQLQJDQG*\URVFRSLF&RXSOHV,QFOXGHG &ORVHUFRQVLGHUDWLRQRIWKHUROOHTXLOLEULXPDURXQGWKHOLQHRILQWHUVHFWLRQ UHYHDOVWKDWLQVWHDGRI ZHDFWXDOO\KDYHZLWKDOVRWKHFDVWHUOHQJWKWF WDNHQLQWRDFFRXQW FI(T ZLWK
n& PKD \& ,Z\
PJK
%
U
%
Z\ U XU
,
%
)
] WF
0
%
[
0
&
[
P
PU J \PU
'
7KHWKLUGWHUPUHSUHVHQWVWKHVXPRIJ\URVFRSLFFRXSOHV:LWK(TV ZHREWDLQIRUWKHUROODQJOH
n'
%
PK
%
Z\ U
,
% 0[ VLQg& & 0[ & 0[
Z\ U D\
,
PJK
&
&
)
&
]W F
&
P
PU J \ PU
,QVSHFWLRQ RI WKH YDOXHV RI SDUDPHWHUV DV OLVWHG LQ 7DEOHV OHDGV WR WKH FRQFOXVLRQ WKDW WKH VWHHU DQJOH
nDQG\
EHWZHHQUROODQJOH
KDV D QHJOLJLEOH HIIHFW RQ WKH UHODWLRQVKLS
nDQGODWHUDODFFHOHUDWLRQD
DQGDOVREHWZHHQ PU
\
XU
02725&<&/('<1$0,&6
LI a D\J :LWK H[SUHVVHGLQUDGLDQVLWLVH[SHFWHGWKDWDWQRWWRRORZ VSHHG WKLV FRQGLWLRQ ZLOO VRRQ EH VDWLVILHG LQ WKH PRUH LQWHUHVWLQJ UDQJH RI RSHUDWLRQ :HZLOOKHQFHIRUWKXVHWKHDSSUR[LPDWHH[SUHVVLRQIRUWKHUROODQJOHZLWKWKH WHUPLQ QHJOHFWHG
D
n'
P
&
\
J
\
\
PU
PU
PK
ZKHUHWKHWLOWFRHIILFLHQWV DQG
'
PK J
PJK
'
\
%
,
Z\
&
% &
U
,
&
Z\
\
UHDG
U
&
0[
0[
P JK PJK
&
&
0[
&
&
0[
7KHTXDQWLWLHVDSSHDUWREHDOLWWOHODUJHUWKDQXQLW\GXHWRWKHJ\URVFRSLFDFWLRQ DQGWKHZLGWKRIWKHW\UHV ILQLWHFURZQUDGLL 7KHWLOWFRHIILFLHQWVEHFRPHLQWKH EDVHOLQHFRQILJXUDWLRQ 7DEOHV
'
\
' nN
$QHIIHFWLYHWLOWDQJOH
WKHDQJOHRIWKHGDVKHGOLQHWKURXJKW\UHFRQWDFWSRLQW
DQG PDVV FHQWUH LQGLFDWHG LQ )LJ ZLWK UHVSHFW WR WKH YHUWLFDO PD\ EH GHILQHG:HKDYH
n
nN '
\
:LWKWKHDGGLWLRQDOPRPHQWVDERXWWKH[D[LVQRZLQWURGXFHGZHILQGIRUWKH XQGHUVWHHUFRHIILFLHQW
'
R
'
) &
)
)
&
]R
&
]R
)
&
&
)
&
)
&
&
)
&
)
DQGWKHFJRIIVHWFRHIILFLHQW
\
'
\R
'
&
)
\
&
)
&
&
)
&
)
ZKLOHWKHVWHHUDQJOHFRHIILFLHQW UHPDLQVXQFKDQJHG
'
R
' %
&
)
&
)
WDQ g
7KHLQ7DEOHOLVWHGQXPHULFDOYDOXHVZLOOEHGLVFXVVHGODWHURQ
02725&<&/('<1$0,&6
7\UH
)
\
)
\
'
&
)
'
&
)
%
%
&
)
&
)
'
'
EPD
O DPD
O
\
&
\
%
K
K
G
G
)
)
n&0
]
n%0
]
G
G
& %
0
0
]
]
LQZKLFKWKHODWHUDODFFHOHUDWLRQD\FDQZLWK EHH[SUHVVHGLQWHUPVRI DQGWKHFJRIIVHW\PU7KHDOLJQLQJWRUTXHVDUH
WKHUROODQJOH
0
]L
'&
&
)
L
W
L
L
%
&
0 L
L
ZLWKFDPEHUDOLJQLQJWRUTXHVWLIIQHVVWKDWZLWKRXWWKHDFWLRQRIWKHORQJLWXGLQDO W\UHIRUFHLVGHILQHGE\ 7KHFDPEHUDQJOHVFDQEHH[SUHVVHGLQWHUPVRI WKHUROODQJOHDQGWKHVOLSDQJOHVE\XVLQJWKHUHODWLRQV :H KDYH
' '
n%
N WDQ g
' n%
O 5
% &
WDQ g
n
)URP WKH WKXV FUHDWHG WZR HTXDWLRQV WKH WZR XQNQRZQ VOLS DQJOHV
L
FDQ EH
nDQG
VROYHGDQGH[SUHVVHGLQWHUPVRIWKHNQRZQSDWKFXUYDWXUHO5UROODQJOH FJRIIVHW\PU%\VXEVWLWXWLQJWKHH[SUHVVLRQ
N
'
%
O 5
&
LQWKHUHODWLRQVKLS
N
'
O 5
D
%
\
J
P
%
\
PU
\
PU
PK
DQGXVLQJDJDLQ WKHVWHHUDQJOHXQGHUVWHHUDQGFJRIIVHWFRHIILFLHQWV DQG
\
FDQILQDOO\EHDVVHVVHG7KHUHVXOWLQJH[SUHVVLRQVUHDG &
' %
)
&
)
%
&
)
%
W
&
)
%
&
)
O
(
&
0
WDQ g
ZLWKWKHHIIHFWLYHZKHHOEDVH
O
(
' & % O
W
W
02725&<&/('<1$0,&6
DQG )
'
]R
&
&
)
&
)
&
(
]R
)
&
)
%
&
0
&
%
&
%
)
)
%
&
&
&
O
)
&
0
)
G
K
G
DQG
\
'
&
)
\
&
)
&
)
(
O
)
%
&
&
&
)
%
' % ' &
O
)
(
&
)
O
%
&
(
%
&
)
&
)
&
)
&
&
0
)
W
%
&
%
0
)
G
K
G
LQZKLFKWKHFRHIILFLHQWV DQG
%
KDYHEHHQLQWURGXFHG
&
)
W
ZKLFK DUH FORVH WR XQLW\ DURXQG DQG UHVSHFWLYHO\ LQ WKH EDVHOLQH FRQILJXUDWLRQFKDQJLQJDOLWWOHGHSHQGLQJRQORDGWUDQVIHU ,Q 7DEOH DSSHDULQJ IXUWKHU RQ WKH YDOXHV WKDW
DQG
\
WDNH LQ WKH
EDVHOLQHFRQILJXUDWLRQ 7DEOHV IRUWKHFDVHVZLWKRXWDQGZLWKDLUGUDJ LQ WKH ODWWHU FDVH DW VSHHGV DQG NPK KDYH EHHQ OLVWHG WRJHWKHU ZLWK WKH FXUUHQWZKHHOORDGVDQGDLUGUDJIRUFH
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
nDQGSRVVLEO\DFJODWHUDORIIVHW\
YHUWLFDO]D[LVLIZHKDYHDUROODQJOH
PU
7KH
IURQWDQGUHDUODWHUDOW\UHIRUFHVQRZEHFRPHLQVWHDGRIWKHULJKWPRVWPHPEHUV RI(TV
02725&<&/('<1$0,&6
)
'
'
\
)
\
EPD
O
DPD
O
&
O)
\
%
K
\
&
[
)
G
K
)
G
n%0
G
n%0
G
%
]
%
]
0
%
]
0
PD
[
%
]
PD
n%P
K
[
n%P
K
PU
D \ [
PU
PU
D
[
\
PU
ZLWKWKHW\UHDOLJQLQJWRUTXHVDFFRUGLQJWR EXWZLWKWKHWRWDOFDPEHU DOLJQLQJWRUTXHVWLIIQHVVQRZH[SUHVVHGDV DFFRUGLQJWR &
0 L
'
&
&
0 L
U
FL
)
[L
7KHVDPHSURFHGXUHLVIROORZHGDVEHIRUHDQGZHILQGWKHFRHIILFLHQWV DQG
DXJPHQWHGIRUWKHSUHVHQFHRIYHKLFOHDFFHOHUDWLRQD[DQGORQJLWXGLQDO W\UH \ IRUFHV)[7KHWHUPVWKDWDUHWREHDGGHGWRWKHH[SUHVVLRQV IRU]HURDFFHOHUDWLRQ)D[
)[
)
'
&
[
)
)[
O
\)[
'
'&
(
&
)
O
%
\
(
FI UHDGUHVSHFWLYHO\
&
)
&
)
%
PKD
&
&
0[
)
[
%
D &
0[
[
J
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
\
02725&<&/('<1$0,&6
7DEOH+DQGOLQJFRHIILFLHQWVZLWKW\UHDOLJQLQJPRPHQWVQHJOHFWHG )]R
9
1)]R
g
1FRV
(TV
)G
)]
)]
NPK 1
1
1
\
R
RR
R
\RR
\R
7DEOH+DQGOLQJFRHIILFLHQWVZLWKW\UHDOLJQLQJPRPHQWVDQGGULYLQJDQGEUDNLQJLQFOXGHG )]R
9
)G
1)]R
g
1FRV
(TV
)D[
)[ )[ )]
)]
NPK 1
1
1
1
1
1
\
\
7KHVWHHUDQJOHFRHIILFLHQW LVPXFKOHVVDIIHFWHGE\WKHW\UHPRPHQWVZKLOHWKH UROO DQJOH FRHIILFLHQWV DQG
\
DUH LQIOXHQFHG FRQVLGHUDEO\ E\ WKH J\URVFRSLF
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
02725&<&/('<1$0,&6
EHFRPHV XQVWDEOH EHFDXVH LQ WKDW VLWXDWLRQ WKH ODVW WHUP RI WKH FKDUDFWHULVWLF HTXDWLRQ RI WKH V\VWHP EHFRPHV QHJDWLYH :H WKHQ KDYH GLYHUJHQW LQVWDELOLW\ FRUUHVSRQGLQJWRWKHFDSVL]HPRGHWKDWEHFRPHVXQVWDEOHEH\RQGDFULWLFDOVSHHG DVLOOXVWUDWHGLQWKHGLDJUDPRI)LJFDVHZLWKYHU\VPDOOFDPEHUDOLJQLQJ VWLIIQHVV 7KH VLJQ RI WKH FJ RIIVHW FRHIILFLHQW DSSHDUV WR FKDQJH IURP SRVLWLYH WR QHJDWLYHLQWKHFDVHRIDFFHOHUDWLQJDWORZVSHHG:KHQWKHULGHUKDQJVWRULJKW
n ZKLOHPRYLQJVWUDLJKWDKHDGVWHHULQJWRWKHULJKW
KDQGVLGH \PU!PDNLQJ
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
WZRFRHIILFLHQWV:KHQGULYLQJZLWK)D[
FKDQJHVIURP
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¶V SDWK $VDQH[DPSOHZHPLJKWIXUWKHUDQDO\VHWKHFDVHUHSUHVHQWHGE\WKHODVWURZ RI 7DEOH ,I WKH PRWRUF\FOH ZRXOG EH QHJRWLDWLQJ D FLUFXODU SDWK ZLWK D UDGLXV RI WLPHV WKH ZKHHO EDVH O5
5
PV WKHODWHUDODFFHOHUDWLRQEHFRPHVD\
FRQVHTXHQFHWKHUROODQJOHEHFRPHV
n
D\ J
7KHJURXQGVWHHUDQJOHWDNHVWKHYDOXH r UDG
9 5
P DW 9
PV
î
O5
E
D\J
DQGWKHKDQGOHEDUVWHHUDQJOH
NPK
J$VD
E
UDG
g
rFRV
E
,WLV
REYLRXVWKDWZLWKWKLVKLJKODWHUDODFFHOHUDWLRQWKHDVVXPSWLRQRIOLQHDULW\LVQRW YDOLG$VPDOOHUSDWKFXUYDWXUHZRXOGKDYHEHHQDEHWWHUFKRLFH,IWKHFHQWUHRI JUDYLW\RIWKHPDLQIUDPHSOXVULGHULVORFDWHGDGLVWDQFH\PU WKH YHKLFOH FHQWUH SODQH D UROO DQJOH î
UDG
E
n
PPU\PU PK \
FPWRWKHULJKWRI îî
UHVXOWVDWVWUDLJKWOLQHUXQQLQJ7KHJURXQG
VWHHU DQJOH LV SUHGLFWHG WR DPRXQW WR î
î UDG
E
r
\
PPU\PU PK
îî
DQGWKHKDQGOHEDUVWHHUDQJOH
E
02725&<&/('<1$0,&6
)D[ = 0
δ
0
0
δ
approx. exact
α2
α2
α1
α1
9
10000
5000 2
2
)D[ = -1500N
ϕ 10
ϕ 10
1
0
2
[km /h ]
9
10000
5000 2
2
2
[km /h ]
)LJ 6WHHUUROODQGVOLSDQJOHVSHUXQLWQRQGLPHQVLRQDOSDWKFXUYDWXUHO5YHUVXVVSHHG
VTXDUHG9
D\
5
OHIW QRQDFFHOHUDWLQJULJKW EUDNLQJ DFFRUGLQJWRWKHH[DFWDQG
DSSUR[LPDWHWKHRULHV
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
n1RWHWKDWWKHGLIIHUHQFHLQVOLSDQJOHV
DPRXQWVWRFDRIWKHUROODQJOH
SUDFWLFDOO\HTXDOVO5PLQXVWKHJURXQGVWHHUDQJOH r EHLQJYHU\FORVH
WRXQLW\ :HPD\FDOFXODWHWKHVOLSDQJOHVIROORZLQJWKHDQDO\WLFDODSSURDFK WKURXJKDQH[SOLFLWH[SUHVVLRQE\XVLQJWKHHTXDWLRQV IRUWKHODWHUDO W\UHIRUFHFRPSRQHQWVFDXVHGE\VLGHVOLSJLYHQODWHURQ 7KHULJKWKDQGGLDJUDPRI)LJUHIHUVWRWKHVLWXDWLRQWKDWDULVHVZKHQ WKH EUDNHVDUHDSSOLHG ILIWKDQGODVWURZRI7DEOH $JDLQDQH[FHOOHQW FRUUHVSRQGHQFHEHWZHHQWKHDSSUR[LPDWHDQGH[DFWVROXWLRQVRFFXUV2EYLRXVO\ WKHYHKLFOHSHUIRUPVQRZLQDOHVVRYHUVWHHUHGPDQQHUDVZDVDOUHDG\FRQFOXGHG IURP WKH OHVV QHJDWLYH YDOXH RI
LQ 7DEOH 7KH IURQW VOLS DQJOH LV OHVV
QHJDWLYH WR PDNH WKH VLGH IRUFH PRUH SRVLWLYH WR FRXQWHUDFW WKH VLGHZD\V
02725&<&/('<1$0,&6
)D[ = 0 α1
0.2
)D[ = -1500N α1 α2
0.2
α2
δ
0
δ
0 approx. exact
ϕ 10
ϕ 10 -0.2
0
9
50 [km/h ]
100
-0.2
0
50 [km/h ]
9
100
)LJ 6WHHUUROODQGVOLSDQJOHVDQGWKHVWHHUWRUTXHSHUPODWHUDOFJRIIVHW\PUWKDWRFFXU DWVWUDLJKWOLQHPRWLRQ DFFRUGLQJWRWKHH[DFWDQGDSSUR[LPDWHWKHRULHV
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
LQFUHDVHVRQO\DOLWWOH
WREDODQFHERWKWKHLQFUHDVHGFDPEHUIRUFHDQGWKHVLGHZD\VFRPSRQHQWRIWKH IURQWZKHHOEUDNLQJIRUFHWKHVWHHUDQJOHQHHGVWREHFRPHODUJHUWRNHHS HTXDOWR
,QERWKFDVHV
LVHTXDOWRFD RI
r
nDQGZLOOKDYHDUHODWLYHO\
ODUJHHIIHFWRQWKHDFWXDOYDOXHRIVWHHUDQJOHRIWKHKDQGOHEDU FI(T
02725&<&/('<1$0,&6
7KH6WHHU7RUTXH %\FRQVLGHULQJ(T ZLWK QHJOHFWHGWKHIROORZLQJVWHDG\VWDWHH[SUHVVLRQ IRUWKHVWHHULQJFRXSOHLVREWDLQHG 0
'
%
W )
F \
&F ] % % Z\ \ VLQ g % & [ VLQ g % & ] FRV g % % I I % V V \ & W )
,
U
0
0
D
P H
P H
D
J
/LNHZHGLGZLWKWKHVWHHUDQJOHZHZLVKWRGHYHORSDQH[SUHVVLRQIRUWKHVWHHU WRUTXHLQWHUPVRISDWKFXUYDWXUHO5ODWHUDODFFHOHUDWLRQD\DQGFJRIIVHW\PU )RUWKLVILUVWWKHUHODWLRQVKLSZLWKD\
n
DQG rZLOOEHHVWDEOLVKHG:LWKWKH
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
)
\
%
)
\
'
PD
\ & O )[
&
&
)
&
) n
&
7KHIROORZLQJUHODWLRQIRUWKHIURQWZKHHOVLGHVOLSIRUFHLVILQDOO\REWDLQHG
\ O&W
)
%
' %
W
EPD
W
\& O%W )[
&
PD\
&
)
&
&
G G%PKD[ n&P PU \PUD [%
K )
) n &&0
&
&
&
0 n&O &)
7KHH[SUHVVLRQIRU)\
FDQEHXVHGWRGHWHUPLQHWKHVOLSDQJOH
RUGLUHFWO\WKHDOLJQLQJWRUTXHGXHWRVLGHVOLS0] SQHXPDWLFWUDLOW
6LPLODUO\ZHREWDLQ0]
E\PXOWLSO\LQJ)\
)\
&)
ZLWKWKH
7KHUHPDLQLQJSDUWRIWKHDOLJQLQJ
WRUTXHGXHWRZKHHOFDPEHU0] LVREYLRXVO\IRXQGE\PXOWLSO\LQJWKHFDPEHU DOLJQLQJ WRUTXH VWLIIQHV &0 L ZLWK WKH FDPEHU DQJOH H[SUHVVLRQIRUWKHVWHHUWRUTXHWXUQVRXWWRUHDG
L
7KH UHVXOWLQJ
02725&<&/('<1$0,&6
0
W
'
& &
F%W FRVg E%W PD\& O%W &) ( O
&
0
W
%
0 n & O )[
&
F )
U
& ] &
) & &0[
&
Z\ VLQg
,
D
&
W
K )
&
PKD
) n %
&
& G G% [ n & VLQg % % 0 FRVg
P
PU \ PU D[
%
\ % PIHI%P VHV D\& J
7KH IURQW ZKHHO FDPEHU DQJOH
PD\ EH ZULWWHQ LQ WHUPV RI
nDQG
(T 2QWKHLUWXUQ
n DQG
r ZLWK
rFDQEHH[SUHVVHGLQO5D\DQG\PUZLWKWKH
XVHRI(TV DQG $WWKLVVWDJHZHPD\LPSURYHWKHUHVXOWIRU 0 E\ DSSUR[LPDWHO\ DFFRXQWLQJ IRU WKH WHUP ZLWK
LQ ZKLFK ZH
QHJOHFWHG'XHWRDVWHHUDQJOHWKHFRQWDFWSRLQWVKLIWVVLGHZD\VDQGDUROODQJOH LVQHHGHGWRNHHSWKHPRWRUF\FOHLQEDODQFH7KHDSSUR[LPDWHFRUUHFWLRQIRU
n
EHFRPHV EW
'&
F
FRVg
OK
ZLWKWKHVWHHUDQJOHHTXDWLRQ
N
'
D
%
O 5
\
J
P
%
\
PU \ PU
PK
DQGWKHUROODQJOHHTXDWLRQ WKHFRUUHFWHGUROODQJOHFDQEHDVVHVVHG
n
D
'
\
J
&
P
\
PU \PU PK
&
EW OK
F
FRVg
DQGDOVRWKHIURQWZKHHOFDPEHUDQJOH
'
n%
N WDQ g
(VSHFLDOO\DWORZVSHHGVDQGZKHQWKHIURQWEUDNHLVDSSOLHGWKHLPSURYHPHQW FDQEHFRPHFRQVLGHUDEOH2EYLRXVO\WKLVLVGXHWRWKHFDPEHUDOLJQLQJWRUTXH LQZKLFKWKHORQJLWXGLQDOIRUFHPD\SOD\DSUHGRPLQDQWUROHFI(T ,W LV SRVVLEOH QRZ WR ZULWH (T LQ WKH IROORZLQJ QRQGLPHQVLRQDO IRUP 0
)
]R O
'
5
O 5
%
D
D
\
J
%
P
\
PU \PU
PJK
7KH H[SUHVVLRQV IRU WKH VWHHU WRUTXH FRHIILFLHQWV DUH WRR HODERUDWH WR EH UHSURGXFHG KHUH 1XPHULFDOO\ KRZHYHU WKHLU YDOXHV FDQ EH HDVLO\ DVVHVVHG GLUHFWO\IURPWKHRULJLQDO(T WRJHWKHUZLWK(TV ,Q)LJ WKHJHQHUDOFRXUVHRIWKHQRQGLPHQVLRQDOVWHHUWRUTXH
02725&<&/('<1$0,&6
0
δ
) O
]1o
µ
P \ PUPU
\
PK µ5
0
O
D\
5
J
µD
=
9
2
5J
)LJ 7KHQRQGLPHQVLRQDOVWHHUWRUTXHGXHWRSDWKFXUYDWXUH5DQGFJODWHUDORIIVHW
PUDQGWKHYDULDWLRQZLWKODWHUDODFFHOHUDWLRQD RUVSHHG9
\
\
KDV EHHQ GHSLFWHG 7KH LQLWLDO YDOXH DW VSHHG QHDUO\ ]HUR LV JRYHUQHG E\ WKH FRHIILFLHQWV5DQG\ZKLOHWKHVORSHHTXDOVD
1XPHULFDO5HVXOWV )RU WKH EDVHOLQH FRQILJXUDWLRQ DQG WKH YDULRXV FRQGLWLRQV H[DPLQHG LQ 7DEOH WKHYDOXHVIRUWKHWKUHHVWHHUWRUTXHFRHIILFLHQWVKDYHEHHQOLVWHGLQ7DEOH 7DEOH6WHHUWRUTXHFRHIILFLHQWV (T
)D[
)[
)[
)]
)]
5
D
\
NPK 1
1
1
1
1
1
9
)G
02725&<&/('<1$0,&6
)D[ = 0
0.4
)D[ = -1500N
0.4
)\1
0.2
0[1
)\1
0.2 ,
0[1 ,
1
Z\
0
-0.4
)]1
0
0
PIV
0δ
-0.2
0δ
PIV
approx.
0 ]1
2500
9 2 [km2/h2]
1
Z\
-0.4
)[1
0δ
) ]1
0
0 ]1
2500
9 2 [km2/h2]
)LJ 7KHQRQGLPHQVLRQDOVWHHUWRUTXH 0 )]RO DQGLWVFRPSRQHQWV FI(T
´
SHUXQLWQRQGLPHQVLRQDOSDWKFXUYDWXUHO5,Q0 ] WKH)[FRQWULEXWLRQLVRPLWWHG
)LJXUHGHSLFWVWKHYDULDWLRQRIWKHVWHHUPRPHQWZLWKWKHVSHHGVTXDUHG FRPSXWHGDFFRUGLQJWRWKHDSSUR[LPDWHDQDO\WLFDOHTXDWLRQ GRWWHGOLQH EXWZLWK
DQGZLWKWKHH[DFWHTXDWLRQV LQFOXGLQJ
n
U
WRJHWKHU
ZLWKFRQWULEXWLQJFRPSRQHQWV7KHVHFRPSRQHQWVFRUUHVSRQGZLWKWKHWHUPVRI H[SUHVVLRQ DQG ZLWK WKH UHOHYDQW WHUPV RI (T ,Q WKH FRQWULEXWLRQIURPWKHIURQWZKHHODOLJQLQJWRUTXH ZHGLVWLQJXLVKWKHSDUW
)D[ = 0
0.3 0
0
)O
δ
0.2
)D[ = -1500N
0.3 δ
0δ
]1o
approx.
)O
]1o
)
1
]
0.1
)
1
1
0.1
1
]
0
0
]
]
)
)
1W F
0
0
P
IV
0
1
[
-0.1
0
1
[
\
9
1
[
50
100
-0.1
0
[km/h ]
9
50
100
[km/h ]
0 )]R O DQGLWVFRPSRQHQWV FI(T \PU,Q0´] WKH)[FRQWULEXWLRQLVUHPRYHG
)LJ 7KHQRQGLPHQVLRQDOVWHHUWRUTXH SHUPFJODWHUDORIIVHW
02725&<&/('<1$0,&6
GLUHFWO\ DWWULEXWHG WR )[ DQG WKH UHPDLQLQJ SDUW LQGLFDWHG ZLWK 0´ 7KH ] FRPSRQHQW LQGLFDWHG LQ WKH GLDJUDP E\ )[ FRQWDLQV WKH IRUPHU SDUW SOXV WKH WHUPZLWK)[DOUHDG\DSSHDULQJLQWKH FRHIILFLHQWLQ(T ,QERWKFDVHV H[DPLQHG ZLWKRXW DQG ZLWK EUDNLQJ WKH DSSUR[LPDWH WRWDO PRPHQW FORVHO\ IROORZVWKHFRXUVHRIWKHH[DFWVROXWLRQ7KHGLIIHUHQFHLVODUJHO\GXHWRWKHIDFW WKDWLQWKHDSSUR[LPDWHWKHRU\WKHWZLVWDQJOHKDVEHHQQHJOHFWHG ,QDOOFDVHVWKHLQLWLDOWRUTXH 9
6 DWFRUQHULQJLVQHJDWLYH
5
ZKLFK
PHDQVWKDWZKHQWXUQLQJWRWKHULJKWDQDQWLFORFNZLVHVWHHUWRUTXHLVUHTXLUHG 7KLVDSSHDUVWREHPDLQO\GXHWRWKHGLUHFWDFWLRQRIWKHIURQWQRUPDOORDGWKH D[LDO FRPSRQHQW RI ZKLFK )] IRUPV ZLWK WKH DUP WF D PRPHQW DERXW WKH VWHHULQJD[LV$OVRWKHW\UHDOLJQLQJWRUTXHZLWKWKHQHJDWLYHVOLSDQJOHRIWKH IURQWZKHHO )LJ KHOSVWRPDNHWKHVWHHUWRUTXHQHJDWLYH,QWKHEUDNLQJ FDVHWKHIURQWZKHHOEUDNHIRUFHFRQVLGHUDEO\LQFUHDVHVWKHQHJDWLYHVWHHUWRUTXH 7KH ZHLJKWV RI WKH VWHHUDEOH IURQW DQG VXEIUDPHV KDYH DQ DOPRVW QHJOLJLEOH HIIHFWZKLOHWKHJ\URVFRSLFDFWLRQDQGWKHIURQWZKHHOVLGHIRUFHKDYHDWHQGHQF\ WRPDNHWKHVWHHUWRUTXHSRVLWLYH )LQDOO\)LJJLYHVWKHPRPHQWUHVSRQVHWRODWHUDORIIVHWRIWKHFHQWUHRI PDVVRIWKHFRPELQHGULGHUDQGPDLQIUDPH7KHDSSUR[LPDWLRQLVOHVVJRRGWKHQ ZH KDYH VHHQ VR IDU 7KH GHYLDWLRQ LV ODUJHO\ GXH WKH RPLVVLRQ RI WKH WZLVW FRPSOLDQFH%HFDXVHRIWKHVPDOOVLGHIRUFHVLQWKLVVWUDLJKWDKHDGPRWLRQZHVHH RQO\DYHU\VPDOOGLUHFWHIIHFWRI)\$WEUDNLQJWKHW\UHWRUTXHVEHFRPHODUJHU LQPDJQLWXGHGXHWRWKHLQFUHDVHGIURQWVOLSDQJOHDQGVWHHUDQJOH FDPEHU FI )LJ $VDQLOOXVWUDWLRQZHFRQVLGHUDJDLQWKHFDVHWKDWWKHPRWRUF\FOHUXQVLQD FXUYHZLWKDUDGLXVRIWLPHVWKHZKHHOEDVHO5 NPK
5
PV WKH ODWHUDO DFFHOHUDWLRQ EHFRPHV D\
PDW9
9 5
PV
J7KHYHKLFOHLVEHLQJEUDNHGDQGWKHODVWURZRI7DEOHDSSOLHV$VD FRQVHTXHQFHWKHQRQGLPHQVLRQDOWRUTXHEHFRPHV0 )]RO î î îî
5O5DD\J
RU LQ GLPHQVLRQDO IRUP 0
1P $W NPK ZH ZRXOG REWDLQ 0
îî
î
DQG0
7KHYDOXHFRUUHVSRQGVWRWKHYDOXHIRXQGLQ)LJDW9
)]R O 1P
NPK
)RU D FJ RIIVHW RI FP RQH ZRXOG ILQG D VWHHU WRUTXH DW VWUDLJKW DKHDG UXQQLQJDWNPK 0 )]RO DQG0
\PPU\PUPK îî î
îî
1P7KHYDOXHî
PD\EHSUHGLFWHGIURP)LJZKHUH\PU
PDWNPK
02725&<&/('<1$0,&6
1RQ/LQHDU$QDO\VLVRI6WHDG\6WDWH&RUQHULQJ
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
$6LPSOH$SSUR[LPDWLRQ 7RLQWURGXFHWKHSUREOHPZHZLOODJDLQILUVWQHJOHFWDOOWKHW\UHPRPHQWVDQG WKH J\URVFRSLF FRXSOHV DV ZHOO DV WKH DLU GUDJ :H WKHQ KDYH WKH HTXLOLEULXP FRQGLWLRQVVWDWHGEHIRUH
) \ )\ D\ ' ' ) ] )] J
' XU J
ZKLOH WKH UROO DQJOH DQG WKH IURQW DQG UHDU FDPEHU DQJOHV EHFRPH ZLWK QHJOHFWHGZLWKUHVSHFWWRWKHODUJH
D\ WDQn ' J
n
DQG
'n 'n
nZHNQRZWKHODWHUDODFFHOHUDWLRQERWKVLGH
&RQVHTXHQWO\IRUDJLYHQUROODQJOH
IRUFHVDQGWKHFDPEHUDQJOHV)URPWKHQRQOLQHDUW\UHVLGHIRUFHFKDUDFWHULVWLF WKDWKROGVIRUWKHFDPEHUDQJOHDWKDQGDQGWKHYHUWLFDOORDG KHUHHTXDOWRWKH VWDWLF ORDG LW PXVW QRZ EH SRVVLEOH WR DVVHVV WKH VOLS DQJOH WKDW SURGXFHV WRJHWKHUZLWKWKHFDPEHUDQJOHWKHGHVLUHGVLGHIRUFH:KHQZHGRWKLVIRUERWK WKH IURQW DQG UHDU ZKHHOV WKH GLIIHUHQFH LQ VOLS DQJOHV FDQ EH IRXQG DQG WKH KDQGOLQJFXUYHFDQEHGUDZQ$WDJLYHQSDWKFXUYDWXUHWKHJURXQGVWHHUDQJOH LVWKHQHDVLO\IRXQGE\XVLQJWKHDSSUR[LPDWHUHODWLRQVKLS
N
' O % 5
&
)LJXUHVKRZVWKHQRUPDOLVHGVLGHIRUFHFKDUDFWHULVWLFVDFFRUGLQJWRWKH
02725&<&/('<1$0,&6
)]1 = 1732 N
1.5 1
)\1 )]1
1
γ1= 55 o
0.5
)\2 )]2
equilibrium curve
0
0
-0.5
)]2
1.5
o
γ2= 55 o
0.5
equilibrium curve
0
0
-0.5
-1
= 2094 N
o
-1
-1.5
-1.5 -0.1
)LJ
-0.05
α1
0
0.05
0.1
-0.05
-0.1
[rad]
α2
0
0.05 [rad]
0.1
7\UHVLGHIRUFHGLDJUDPVDWDVHULHVRIFDPEHUDQJOHV 6LPSOLILHGVLWXDWLRQZKHUH ZKHHO ORDGV UHPDLQ XQFKDQJHG QR DLU GUDJ DQG W\UH PRPHQWV DQG J\URVFRSLF FRXSOHVDUHOHIWRXWRIFRQVLGHUDWLRQ(TXLOLEULXPFXUYHVKDYHEHHQVKRZQZKHUH
.
WKHW\UHVLGHIRUFHDJUHHVZLWKWKHUROODQJOH
1.6
D\ J tanϕ
) )
\L
]L
9= 50
1.4 1.2
40 O
1
α2 α1
handling curve
5
δ
30
α1
α2
0.8
equilibrium curves
0.6 speed lines
0.4
10 m/s
0.2 0 -0.01
)LJ
20
0
α2α1
0.01
0.02
α1 α2
0.03
0.04
0.05
O 5
7KHKDQGOLQJGLDJUDPIRUWKHVLPSOLILHGFDVHZKHUHW\UHPRPHQWVDQGJ\URVFRSLF FRXSOHVDUHGHOHWHGDQGDLUGUDJLVGLVUHJDUGHG7KHQRUPDOLVHGVLGHIRUFHYHUVXV VOLS DQJOHFXUYHV LQWKLVVLPSOLILHGFDVHHTXDOWRWKHHTXLOLEULXPFXUYHV KDYH EHHQGHSLFWHGDVZHOO
EDVHOLQHFRQILJXUDWLRQWRJHWKHUZLWKWKHHTXLOLEULXPFXUYHVZKHUHWKHFDPEHU DQJOHHTXDOVWKHUROODQJOHEHORQJLQJWRWKHVLGHIRUFH,Q)LJWKHUHVXOWLQJ KDQGOLQJ FXUYH ZLWK VSHHG OLQHV IRUPLQJ WKH KDQGOLQJ GLDJUDP KDV EHHQ SUHVHQWHG7KHVSHHGOLQHVLQGLFDWHWKHUHODWLRQVKLSEHWZHHQODWHUDODFFHOHUDWLRQ
02725&<&/('<1$0,&6
DQGSDWKFXUYDWXUHIRUJLYHQYDOXHVRIVSHHG
O ' JO D \ 5 9 J
,Q WKH ILJXUH WKH HTXLOLEULXP FXUYHV RI )LJ KDYH EHHQ UHSURGXFHG %\ KRUL]RQWDO VXEWUDFWLRQ RI WKHVH FXUYHV WKH KDQGOLQJ FXUYH LV REWDLQHG 7KH KRUL]RQWDOGLVWDQFHEHWZHHQVHOHFWHGVSHHGOLQHDQGWKHKDQGOLQJFXUYHHTXDOVWKH JURXQGVWHHUDQJOH7KHVORSHRIWKHKDQGOLQJFXUYHZLWKUHVSHFWWRWKHYHUWLFDO D[LVDW]HURODWHUDODFFHOHUDWLRQFRUUHVSRQGVWRWKHXQGHUVWHHUFRHIILFLHQWLQWKH OLQHDUWKHRU\
M
r M D\ J DWFRQVWDQWO5%\XVLQJ(T ZLWKWZLVW
DQJOHQHJOHFWHGDQGVWHHUDQJOHDVVXPHGVPDOOZLWKUHVSHFWWRWKHUROODQJOHZH ILQGWKHDFWXDOVWHHUDQJOH
'
N FRV g
FRV
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
n
VKLIWRIWKHFHQWUHRIJUDYLW\7KHWZLVWDQJOH LVQHJOHFWHGDQGWKHOHDQDQJOH
U
LVGLVUHJDUGHGVLQFHDWVWHDG\VWDWHWXUQLQJLWVHIIHFWLVVLPLODUWRWKDWRIWKHFJ ODWHUDORIIVHW7KHVWHHUDQJOH LVFRQVLGHUHGVPDOOZLWKUHVSHFWWRWKHUROODQJOH $OVRWKHZKHHOVOLSDQJOHVDUHDVVXPHGWREHUHODWLYHO\VPDOO VPDOOHUWKDQFD GHJUHHV
(TXLOLEULXP&RQGLWLRQVIRUWKH&RPSOH[&RQILJXUDWLRQ 7KHFRQGLWLRQRIHTXLOLEULXPLQORQJLWXGLQDOGLUHFWLRQFDQEHPHWE\PDNLQJ WKHGULYLQJIRUFHHTXDOWRWKHDLUGUDJIRUFH7KLVPHDQVWKDW)[DFF
RU)[
)G :KHQ D GLIIHUHQW GULYLQJ IRUFH RU ZKHQ EUDNLQJ IRUFHV DUH DSSOLHG HTXLOLEULXPFDQRQO\EHDVVXUHG D[
ZKHQWKHYHKLFOHUXQVRQDQXSZDUGRU
GRZQZDUGVORSH,IWKLVLVQRWWKHFDVHZHPD\VSHDNRIDTXDVLHTXLOLEULXP VLWXDWLRQLQWKHORQJLWXGLQDOGLUHFWLRQ :H KDYHWKHHTXDWLRQ ZKHUHWKHVPDOO ORQJLWXGLQDOFRPSRQHQWRIWKHVLGHIRUFH)\ rKDVEHHQQHJOHFWHG
PD [ ' )[ DFF ' )[ % )[ & )G
02725&<&/('<1$0,&6
,QODWHUDOGLUHFWLRQHTXLOLEULXPRFFXUVLI
PD \ ' )\ % )\ % )[
N
DQGLQYHUWLFDOGLUHFWLRQLI
PJ ' )] % )]
$ERXWWKH[D[LVZHKDYHWKHFRQGLWLRQ
& PD \ K FRVn %PJK VLQn %PPU J\ PU FRVn % & ,Z\ %,Z\ U FRVn %)] WFn FRVn %0[ %0[ '
DERXWWKH\D[LV
PD [ Kn FRVn & PPU D [ \PU VLQn % % D)] & E)] % KGn ) G FRVn '
DQGDERXWWKH]D[LV
PD [ Kn VLQn %PPU \ PU D[ FRVn % %D)\ &E)\ % D)[ N % 0] %0] %KGn )G VLQn'
7KHZKHHOQRUPDOORDGV)]DUHIRXQGIURPWKH(TV DQG 7KH ORQJLWXGLQDO W\UH IRUFHV )[ DUH GLVWULEXWHG DFFRUGLQJ WR (T DQG DV EHIRUHWKHDFFHOHUDWLRQIRUFH)[DFFLVXVHGDVSDUDPHWHULQWKHKDQGOLQJGLDJUDPV WREHGHYHORSHG :LWK0\DLQWURGXFHGIRUDEEUHYLDWLRQ
0\D ' PD[ Kn FRVn &PPU \PU D [ VLQn %KGn )G FRVn
ZKHUHKnDQGKGnDUHGHILQHGE\(T ZLWKLIUHOHYDQWKUHSODFHGE\KGZH REWDLQWKHORDGV
)] '
)] '
O O
EPJ & 0\D
DPJ % 0\D
7KH WZR VLGH IRUFHV DUH IRXQG IURP WKH (TV DQG ,I ZH LQWURGXFHIRUDEEUHYLDWLRQ
0]D ' PD[ Kn VLQn%P PU D[ \ PU FRVn %0] %0] %KGn )G VLQn
ZHREWDLQ
)\ '
)\ '
O O
EPD\ & O)[
DPD\ % 0]D
N
& 0]D
02725&<&/('<1$0,&6
ZLWK IURP(T ZLWK
N
'
FRVg
FRVn
)URP (T DQ H[SUHVVLRQ IRU WKH ODWHUDO DFFHOHUDWLRQ FDQ EH REWDLQHG $IWHUXVLQJ(T IRU0[LDQGWKHFRUUHFWHGHIIHFWLYHUROOLQJUDGLLULn
ULn ' UL & UFL % U FL FRVn
DQGQHJOHFWLQJWKHYHU\VPDOOFRQWULEXWLRQRI
['
WRWKHPRPHQWDERXWWKH[D[LV
PKn PKn %
Un
,Z\ %
[
&
Un
,Z\
IRUWKHODWHUDODFFHOHUDWLRQD\
D\ ' J
GHILQHGDV [
ZHILQGE\XVLQJWKHIDFWRU
XU
UF )] % UF )] PJKn FRVn
WDQn
%
P PU \PU PKn
7KH+DQGOLQJ'LDJUDP )RU D JLYHQ WDQ
n DQG \
PU
WKH FRUUHVSRQGLQJ ODWHUDO DFFHOHUDWLRQ D\ FDQ EH
FRPSXWHGE\XVLQJ ,IZHGLVUHJDUGDVDILUVWVWHSWKHLQIOXHQFHRI r DQGWKHDOLJQLQJWRUTXHV0]LQ(TV ZHFDQFDOFXODWHWKHIURQWDQG UHDUVLGHIRUFHV)\EHORQJLQJWRD\:LWKWKHDSSUR[LPDWLRQ WKH FDPEHU DQJOHV DUH WDNHQ HTXDO WR WKH UROO DQJOH RI WKH PDLQ IUDPH ,W LV WKHQ SRVVLEOHWKURXJKLWHUDWLRQWRDVVHVVWKHYDOXHVIRUWKHVOLSDQJOHV WRWKHFDOFXODWHG)\¶VDQGWKHNQRZQFDPEHUDQJOHV
EHORQJLQJ
ZLWKWKHORDGVREWDLQHG
IURP )URPWKHVOLSDQJOHVDQGWKHFDPEHUDQJOHVHVWDEOLVKHGWKH DOLJQLQJWRUTXHV0]FDQEHHVWLPDWHGIRUWKHQH[WFRPSXWDWLRQVWHS IRUDJLYHQ QH[WLQFUHPHQWHGUROODQJOH
n E\H[WUDSRODWLRQ$OVRWKHJURXQGVWHHUDQJOH
r
LVHVWLPDWHGIRUWKHQH[WVWHS7KLVLVGRQHE\XVLQJWKHHTXDWLRQV DIWHU KDYLQJVHOHFWHGDYDOXHIRUWKHIRUZDUGVSHHG9 X1RZDPRUHDFFXUDWHYDOXH IRU WKH VLGH IRUFHV WKDW EHORQJ WR WKH HTXLOLEULXP VWDWH FDQ EH IRXQG IURP
nWKHYDOXHVRI
(TV ZLWK )RUDVHULHVRIVXFFHVVLYHYDOXHVRI DQG
PD\EHFRPSXWHGLQWKLVZD\UHVXOWLQJZKHQSORWWHGDJDLQVWD\JLQ
WKHHTXLOLEULXPFXUYHVIRUWKHIURQWDQGUHDUZKHHOV HTXLYDOHQWWRWKHHIIHFWLYH D[OHFKDUDFWHULVWLFVRI&KDSWHU )URPWKHGLIIHUHQFH LVREWDLQHG,IDLUGUDJ)G
WKHKDQGOLQJFXUYH
&G$X DQGRUDIURQWZKHHOORQJLWXGLQDOIRUFH)[DUH
02725&<&/('<1$0,&6
1.6 1.4
40
O
D\ J
α2-α1
1.2 1
G =0)D[ = 0,
)
9= 50
5
PU= 0
\
30 m/s
δ
20
0δ
α2
0.8
9
0.6
α1
20
0.4
tanϕ
50
10
10
0.2 0
-0.02
δ
0
-0.01
1000
0
0.01
α1 α2
α2α1
0.02
0.03
0.04
tanϕ 20
O 5
0.05
)LJ +DQGOLQJGLDJUDPIRUFRPSOH[PRGHOZLWKDOOW\UHPRPHQWVDQGJ\URVFRSLFFRXSOHV LQFOXGHG$LUGUDJ
9&XUYHVIRU
ZKLFKPDNHVWKHGLDJUDPYDOLGIRUDOOVSHHGV
UROODQJOHIURQWDQGUHDUVOLSDQJOHVDQGVWHHUWRUTXH GLIIHUHQWIRUHDFKVSHHG
FRQVLGHUHG WKH KDQGOLQJ FXUYH FKDQJHV ZLWK VSHHG DQG WKH KDQGOLQJ GLDJUDP KROGVRQO\IRUWKHVHOHFWHGYDOXHRIVSHHGX,IDLUGUDJDQGIURQWZKHHOEUDNLQJ DUHGLVUHJDUGHGDJHQHUDOO\YDOLGVLQJOHKDQGOLQJFXUYHZLWKDVHULHVRIVSHHG OLQHVDULVHVDVGHSLFWHGLQ)LJ7KHVORSHDWD\
M
LV
rM D\ J 5
,QWKHVHGLDJUDPVDOVRWKHVOLSDQJOHVDQGWKHDFWXDOUROO DQJOH KDYHEHHQ SORWWHG LQ DEVFLVVD GLUHFWLRQ $V DQ DGGLWLRQDO LQIRUPDWLRQ WKH GLDJUDP FRQWDLQVFXUYH V IRUWKHVWHHUWRUTXH0 SRVVLEO\FKDQJLQJZLWKVSHHGRIWUDYHO 9
. X +RZWKLVTXDQWLW\LVDVVHVVHGZLOOEHGLVFXVVHGLQWKHVXEVHFWLRQEHORZ
7KH6WHHU7RUTXH 7KHVWHHUWRUTXHWKDWLVDSSOLHGWRWKHKDQGOHEDUE\WKHULGHULQDVWHDG\WXUQFDQ EHIRXQGE\FRQVLGHULQJWKHHTXLOLEULXPDERXWWKHVWHHULQJD[LV7KHUHVXOWLQJ H[SUHVVLRQLVVLPLODUWRWKHRQHJLYHQE\(T EXWQRZH[WHQGHGWRFRYHU ODUJHUROODQJOHV:HILQGZLWKWKHFDVWHUOHQJWKWF DFFRUGLQJWR(T
0 ' WFn )\ FRV & WFn )] VLQ % % U n ,Z\ XU VLQ g FRV n % &0[ VLQ g & 0] FRV g FRV % % P I HI % P V HV XU FRV & J VLQ
ZLWKWKHDOLJQLQJWRUTXHDFFRUGLQJWR(T
0] ' & W )\ % 0]U & UF )[
WDQ
02725&<&/('<1$0,&6
,QWKHFRPSXWDWLRQVZHFDQXVHWKHFRUUHFWH[SUHVVLRQIRUWKHFDPEHUDQJOH DFFRUGLQJWR(T ZLWK
EHFDXVHWKHVWHHUDQJOH
LVQRZDYDLODEOH
7KLV GHSHQGHQF\FDXVHVWKHPRPHQWWRFKDQJHZLWKSDWKFXUYDWXUHDQGWKXV SURGXFHVGLIIHUHQWFXUYHVIRU HDFKVSHHGYDOXHHYHQLIDLUGUDJLVGLVUHJDUGHG )LJ 7KHVORSHRIWKHVWHHUWRUTXHFXUYHZLWKUHVSHFWWRWKHYHUWLFDODWD\
FRUUHVSRQGVWRD)]ROFI7DEOH
5HVXOWVDQG'LVFXVVLRQRIWKH1RQ/LQHDU+DQGOLQJ$QDO\VLV )RUDQXPEHURIVLWXDWLRQVWKHKDQGOLQJGLDJUDPKDVEHHQ HVWDEOLVKHGIRUWKH EDVHOLQHFRQILJXUDWLRQDQGSUHVHQWHGLQWKHGLDJUDPVRIWKH)LJV7KH IROORZLQJFDVHVKDYHEHHQH[DPLQHG
1RDLUGUDJQREUDNLQJRUGULYLQJIRUFHVQRRIIVHWRIFJ )LJ
:LWKDLUGUDJDQGDUHDUZKHHOGULYLQJIRUFHQHFHVVDU\WRZLWKVWDQGWKHDLU GUDJVRWKDWWKHIRUZDUGDFFHOHUDWLRQHTXDOV]HUR )D[ 9
PV
QHXWUDOVLWXDWLRQ
NPK )LJ 1 9
%UDNLQJ WRWDOGHFHOHUDWLRQIRUFH)D[
+DUG EUDNLQJ WRWDO GHFHOHUDWLRQ IRUFH ) D[
NPK )LJ
1 9
NPK
)LJ
1HXWUDOEXWQRZZLWKODWHUDOVKLIW\PU IUDPHDQGULGHU 9
FPRIWKHFRPELQHGPDVVRIPDLQ
NPK )LJ
7KHIROORZLQJLQWHUHVWLQJREVHUYDWLRQVPD\EHPDGH
7KH W\UHZKHHO PRPHQWV KDYH D FRQVLGHUDEOH LQIOXHQFH RQ WKH UHVXOWLQJ KDQGOLQJFKDUDFWHULVWLF)LJ0XFKPRUHRYHUVWHHUDSSHDUVWRRFFXU ZKHQFRPSDUHGZLWK)LJ7KLVZDVDOUHDG\FRQFOXGHGIURPWKHOLQHDU WKHRU\
$LUGUDJGRHVQRWFKDQJHKDQGOLQJEHKDYLRXUYHU\PXFKLIWKHVSHHGLVQRW PXFK KLJKHU WKDQ NPK 7KH VWHHU WRUTXH DW WKH VSHHG RI NPK
DSSHDUVWRUHDFKDSHDNGHFUHDVHVDQGFKDQJHVLQVLJQLQWKHKLJKHUODWHUDO DFFHOHUDWLRQ UDQJH )LJ ZKHUH )\ DQG 0[ LQ EHFRPH GRPLQDQW
6
%UDNLQJFDXVHVWKHJURXQGVWHHUDQJOHWREHFRPHODUJHUZKLOHWKHVWHHUWRUTXH FKDQJHV WRZDUGV WKH QHJDWLYH GLUHFWLRQ HVSHFLDOO\ DW KLJKHU ODWHUDO DFFHOHUDWLRQV )LJQRWHWKHFKDQJHLQVFDOHIRUWKHVWHHUWRUTXH
6
+DUGEUDNLQJ )LJ WHQGVWRFKDQJHWKHVORSHRIWKHKDQGOLQJFXUYHLQWR XQGHUVWHHU DW ODUJH ODWHUDO DFFHOHUDWLRQV $OVR WKH PD[LPXP ODWHUDO DFFHOHUDWLRQWKDWFDQEHUHDFKHGLVUHGXFHG/DUJHUVWHHUDQJOHVDULVHLQWKH KLJKHUD\UDQJHRIRSHUDWLRQ$OVRWKHVWHHUWRUTXHLQFUHDVHVFRQVLGHUDEO\LQ PDJQLWXGH
02725&<&/('<1$0,&6
1.6
\
9= 30 m/s
1.4
D\ J
PU= 0
D[ = 0,
)
1.2
δ
1
α2
0.8 0.6
α2α1
α1
0δ
0.4 0.2
tanϕ
O 5
0
-0.02
-0.01
δ
0 )LJ
1000
0
α1 α2
0.01
α2α1
0.02
0.03
O 5
0.04
0.05
tanϕ 20
7KHKDQGOLQJGLDJUDPZLWKDLUGUDJFRQVLGHUHGDQG]HURIRUZDUGDFFHOHUDWLRQ 9 NPK
D\ J
1.6
)
1.4
9= 30 m/s
D[ = -1500N,
PU= 0
\
1.2
α2α1
1 0.8
α1 α2
δ
0δ
0.6
tanϕ
0.4 0.2
O 5
0
-0.02
δ
0
-0.01
5000
0
α1 α2
0.01
α2α1
)LJ7KHKDQGOLQJGLDJUDPDWEUDNLQJ 9
6
0.02
0.03
O 5
0.04
tanϕ 20
0.05
NPK
7KHODWHUDOFJVKLIWRI FPJLYHVULVHWRODUJHFKDQJHVLQWKHGLDJUDPDV LQGLFDWHG LQ )LJ ,W VKRZV WKDW WKH FKDQJHV DUH ODUJHO\ GXH WR KRUL]RQWDO VKLIWV RI WKH FXUYHV :H PD\ FRPSDUH WKH JUDSK ZLWK WKDW RI )LJ
,QLWLDOO\
DW
]HUR
ODWHUDO
DFFHOHUDWLRQ
FRUUHVSRQGLQJ
WR
)LJV DQHJDWLYHUROODQJOHRFFXUVZKLOHWKHJURXQGVWHHUDQJOHDQG WKHVWHHUWRUTXHDUHSRVLWLYH ZKLFKFRUUHVSRQGV WR
\
PU\PU PKDQG \
P
PPU\PU PK )]ROUHVSHFWLYHO\ $WDFHUWDLQOHYHORIWKHODWHUDODFFHOHUDWLRQ
WKHVWHHUWRUTXHUHDFKHVDPLQLPXP
02725&<&/('<1$0,&6
D\ J
1.6
)
1.4
9= 30 m/s
D[ = -2500N,
PU= 0
\
1.2
α2
α2α1
1 0.8
α1
δ
0.6 0.4
tanϕ
0δ
0.2
O 5
0
-0.02
δ
0
-0.01
5000
0
0.01
α1 α2
α2α1 9
)LJ7KHKDQGOLQJGLDJUDPDWKDUGEUDNLQJ
D\ J
0.02
0.03
tanϕ 20
O 5
0.05
NPK
1.6
)
1.4
9= 30 m/s
D[ = 0,
PU= 5 cm
\
1.2 1
O 5
δ
0.8
α2
0.6
α1
0.2 0
tanϕ
α2α1
0.4
-0.02 0δ
)LJ
0.04
-0.01
1000
0
α1 α2
0.01
0δ
α2α1
0.02
0.03
O 5
0.04
tanϕ 20
0.05
7KH KDQGOLQJ GLDJUDP DW ]HUR IRUZDUG DFFHOHUDWLRQ ZLWK FP ODWHUDO VKLIW RIWKH
9
FHQWUHRIJUDYLW\RIFRPELQHGULGHUDQGPDLQIUDPHPDVV
NPK
7KHSDWKFXUYDWXUHZKHUHWKHVHPLQLPDRIWKHVWHHUPRPHQWRUPD[LPDRI
0
RFFXUFRQVWLWXWHVWKHERXQGDU\RIPRQRWRQRXVLQVWDELOLW\$WODUJHUFXUYDWXUHWKH FDSVL]HPRGHEHFRPHVXQVWDEOH7KLVLVDQDORJRXVWRZKDWZDVIRXQGIRUWKH DXWRPRELOHZKHUHWKHLQSXWLVWKHVWHHUDQJOHDQGLWVPD[LPXPDWDJLYHQVSHHG RIWUDYHOFRUUHVSRQGVWRWKHERXQGDU\RIGLYHUJHQW PRQRWRQRXV LQVWDELOLW\ FI )LJ
02725&<&/('<1$0,&6
0RGHVRI9LEUDWLRQDW/DUJH/DWHUDO$FFHOHUDWLRQV
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
ϕ =0
ϕ = 1rad
9: 25....250km/h Im
Im twist
twist 100
100 rear hop
rear hop front hop front hop/ wobble wobble
rear wobble
weave
rear wobble
pitch
pitch
bounce -50 )LJ
0
Re
weave
bounce -50
0
Re
n
(LJHQYDOXHVRI.RHQHQ¶VPRWRUF\FOHPRGHODWVWUDLJKWDKHDGUXQQLQJ DWDUROODQJOHRIRQHUDGLDQ7KHVSHHGYDULHVIURPWRNPK
DQG
02725&<&/('<1$0,&6
DORQJ D FLUFXODU SDWK DW DQ DYHUDJH UROO DQJOH RI RQH UDGLDQ IURP .RHQHQ¶V ZRUN 7KH ERXQFH SLWFK DQG KRS PRGHV DSSHDU WR EHFRPH DOPRVW VSHHG LQGHSHQGHQWDWYDQLVKLQJUROODQJOH,QDFXUYHDOOWKHPRGHVGRGHSHQGRQWKH VSHHGRIWUDYHO,QWKHVKDUSKLJKODWHUDODFFHOHUDWLRQFXUYLQJPRWLRQFRQVLGHUHG WKHFRUQHULQJZHDYHURRWORFLDSSHDUWREHVKLIWHGFRPSOHWHO\WRWKHULJKWKDQG VLGHRIWKHLPDJLQDU\D[LV7KHZREEOHPRGHLVGHVWDELOLVHGDVZHOO7KHKLJKHU IUHTXHQF\WZLVWPRGH IOH[LEOHIUDPHWRUVLRQPRGH NHHSVDQLQGLYLGXDOFRXUVH
0RWRUF\FOH0DJLF)RUPXOD7\UH0RGHO
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
&
\DQG
&
LVQRWDOORZHGWRH[FHHGWKHYDOXHWRSUHYHQWWKH
VLGHIRUFHIURPEHFRPLQJQHJDWLYHDWODUJHVOLSDQGFDPEHUDQJOHV)LJXUH LOOXVWUDWHVWKHVWUXFWXUHRIWKHIRUPXODJHQHUDWLQJWKH
)
\YV
FKDUDFWHULVWLFIRU
DJLYHQYDOXHRI ,WLVVHHQWKDWZLWKUHVSHFWWRWKHVLWXDWLRQRI)LJWKHEDVLF VLQHFXUYHLVVKLIWHGVLGHZD\VDVDUHVXOWRIWKHSUHVHQFHRIDFDPEHUDQJOH $QRWKHUIHDWXUHWKDWGLIIHUVIURPWKHRULJLQDOPRGHOFRQFHUQVWKHFUHDWLRQRI WKHDOLJQLQJWRUTXH7KHSDUW
0¶
] LVQRZREWDLQHGE\PXOWLSO\LQJWKHSQHXPDWLF
WUDLOZLWKWKHVLGHIRUFHWKDWLVDWWULEXWHGWRWKHVLGHVOLSDQGQRWWRWKHFDPEHU DQJOH LQGLUHFWO\WKURXJKFDPEHULQGXFHGVLGHVOLS 7KLVPD\HVSHFLDOO\DWODUJH FDPEHUDQJOHVEHFORVHUWRWKHDFWXDOPHFKDQLVP7KLVSDUWRIWKHVLGHIRUFH
)
GHQRWHGZLWK
\
LVREWDLQHGIURPWKHVDPH(TV ( E\VHWWLQJ
([DPSOHV RI FRPSXWHG FKDUDFWHULVWLFV FRPSDUHG ZLWK H[SHULPHQWDOO\ DVVHVVHGFXUYHVIRUERWKSXUHVOLSDWYDULRXVFDPEHUDQJOHVDQGFRPELQHGVOLS FRQGLWLRQVDUHSUHVHQWHGLQ6HFWLRQ)LJV
02725&<&/('<1$0,&6
)\ )\
α\
0
πβγ
βα
βγ
'\ sinβ
γ
βγ =
γ arctan{% γγ
&
)LJ
βα =
β =βα +βγ
...}
\ arctan{%\α\ ...}
&
α\
7KHVWUXFWXUHRIWKHIRUPRWRUF\FOHW\UHVDGDSWHG0DJLF)RUPXODUHQGHULQJWKH VLGHIRUFHYHUVXVVOLSDQJOHFKDUDFWHULVWLFIRUDJLYHQFDPEHUDQJOH
)XOO6HWRI7\UH0DJLF)RUPXOD(TXDWLRQV
/RQJLWXGLQDO)RUFH SXUHORQJLWXGLQDOVOLS
)
''
[R
[
'
S
[
'
[
'
S
[
'
S
'
(
.
VLQ>&
[
DUFWDQ ^%
[
&(
[
[
%
[
&DUFWDQ %
[
[
[
`@
%6 @VJQ 9 9[
F[
' %
[
&
[
[
&[
@)
[
(
@
(
&[
(
]
'[
([
%S %S
'[
([
GI ]
GI
]
@
!
[
%S
([
GI
]
@
^
&S
(
([
VJQ
[
`
@
.[
.[
[
]
[
.[
[
[R
]
[
#
([
' ) @ S %S GI @ H[S S GI @ ' % & ' ' M) M DW ' ' & ]
(
.[
[
(
)
(
% 6 6
02725&<&/('<1$0,&6
'
[
.
& '
[
[
+[
' & T
9[
'
)
]
)
V\
%
[
]
0\
(
[
%6
@ S % S 9[
FI (
@
GI ^_9
9[
]
_
F[
9[
% _9
_ `
F[
@
(
@ N
9[
(
[
/DWHUDO)RUFH SXUHVLGHVOLS )
'
\R
'
\
S
\
'
\
'
S
\
'
S
\
'
@
@)
\
'\
\ R
DUFWDQ ^%
%
6
\
.
@ H[S S
(\
.\
%S
)
'\
(\
N VLQ> S
]R
'
\
.
' '
\
S
S
& ' \
+\
@
\
\
(
@
&\
S
.\
@ &S
GI ]
% S % S (\
.\
DUFWDQ ^)
%
'
\
%
\
\ DW
(\
]
@
VJQ
S
!
\
\
%S
.\
\' LI
`
@
#
(\
.\
\
)
N `@
]R
'
(
.\
' &)
(
(
(
\
(
(
(
%S
(
.
.\
'\
&
(\
(
+\
'%& 'S @
\
(
\ R
'
&
\
]
' . @ & S
+\
\
!
&\
' % \&\'\ ' M)\R M
\
& ( % & DUFWDQ % ` % & ( % & DUFWDQ % `@
.
\
(
&\
'S
.
%&
\
(%
'
&
(
'
DUFWDQ ^%
\
&\ % &
VLQ>&
\
.\
'
\
@ @
GI ) ]
' M)
]
\R
.\
M
\
' ' ' &
)
(
(
(
& '
DW
\
%
\
(
02725&<&/('<1$0,&6
$OLJQLQJ7RUTXH SXUHVLGHVOLS
0
R
\R
' W
W
R
W
'
0
'
'
%
'
'
(
'
@
T
6
'
W
& ( % & DUFWDQ % W
W
W
W
W
@
`@ FRV
N
(
' FRV >DUFWDQ % U
U
U
(
@
(
+U
(
+]
%T
]
WR
GI
+]
GI
%]
]
% T
%T
]
+]
GI
%]
%T
+]
GI ]
@ %T
]
(
]
%T
%]
]
%]
_
]
_
@
.\
!
\
( (
!
@ 5
R
)
(]
N @ T
]R
@ % T
' T %T
W
'
&]
)
'
W
W
]
T
'
WR
W
%]
W
FRV > & DUFWDQ ^ %
U
' T % T
W
&
]U
(%
'
+U
(
'
(
0
]'
6
''
(
]UR
(
]UR
U
N %0
]R
]R
W
0
N ' &W @)
0 W
'
]R
(]
_
_
']
GI
]
%T
]
']
%T
%T
GI ]
@
(
W
']
(]
']
GI ]
]
(
@ % T %T (]
(]
]
DUFWDQ % & W
W
W
#
( %
'
'
U
')
U
'
.
] R
.
]
%]
@
.\
5 ^ T R
'
')
] R
T
]
.
R
%T GI % T %T GI % %T GI _ _` FRVN @ 'a &M0 M DW ' ' ' & ']
']
\ R
5 T
(
\
']
% T
WR
]
']
0U
]
']
]
]
]R
%T
']
GI
']
]
\
]
.]
\
.]
]R
]
(
\
' a M0 M
']
DW
0
' '
'&
0
(
(
02725&<&/('<1$0,&6
/RQJLWXGLQDO)RUFH FRPELQHGVOLS )
[
*
'
*
@)
[
' FRV>&
[
6
%
[
6
DUFWDQ %
6
[
' (%6 ' U % U
+[
!
[ R
(
@
( (
+[
%[
'
&
[
6
[
( @*
[
' FRV>&
[ R
DUFWDQ %
[
*
[R
U
'
+[
FRV> DUFWDQ U
%[
@
%[
@
!
[
(
&[
U
(
(
+
/DWHUDO)RUFH FRPELQHGVOLS )
\
* *
\ R
\R
DUFWDQ %
@*
\
' FRV>&
\
6
DUFWDQ %
6
\
(
!
\ R
(
@
+\
(
+\
\
'
U
'
U
%\
'
U
9\
'
'
9\
'
%U
FRV> DUFWDQ ^ U
%\
%\
(& U
`@
%\
@
! (
\
(
&\
+\
'
(
9\
' %6
\
6
@) % 6
\
\
&
6
*
' FRV>&
\
6
%
'
(
+\
VLQ> U
9\
) \
]
9\
DUFWDQ U
@ U % U 9\
9\
GI
]
%U
(
9\
@
FRV > DUFWDQ U
9\
9\
( @
(
1RUPDO/RDG )
]
'
S
]
&
@ )N 5 @ PD[ ' M) M ' S ]R
)]
R
]
]
]
]
&]
@
$
&]
)
N 5
]R
R
(
2YHUWXUQLQJ&RXSOH 0
[
'
) 5 ]
R
@ T & T V[
V[
%T
V[
) ) \
N @
]R
0[
(
02725&<&/('<1$0,&6
5ROOLQJ5HVLVWDQFH0RPHQW
0
\
' &)
]
5
R
@ T % T V\
V\
) ) [
N @
]R
(
0\
$OLJQLQJ7RUTXH FRPELQHGVOLS 0 0
]
'
V
(
[
)
'
' WHT
UHT
5
*
0
R
' '
W
W
W
&( %
WHT
W
W
&DUFWDQ %
WHT
' & 69\
\
' '
]U
'' FRV>& DUFWDQ^%
WHT
N'
\
(
\
\
0
]U
N ' & W @ )N
'W
)
N % 0 % V@)
]
]
W
)
0
]U
@)
\
V]
W
U
% %
N
(
(
'
@ ^V % V
@
`@ FRV
WHT
(
\R
UHT
W
'
U
) )
V]
.
' FRV > DUFWDQ %
\
N
.
N % V % V
]R
[
U
V]
UHT
V]
(
@
GI ` ]
@
V
(
@ VJQ
W
@ VJQ
U
(
(
\
.
[
N
.
\
0HDVXUHGDQG&RPSXWHG0RWRUF\FOH7\UH&KDUDFWHULVWLFV
$V DQ H[DPSOH DQG VLPLODU WR WKH SUHVHQWDWLRQ RI WKH FDU DQG WKH WUXFN W\UH FKDUDFWHULVWLFVLQ&KDSWHUWKHFKDUDFWHULVWLFVRID=5PRWRUF\FOH W\UH KDYH EHHQ FROOHFWHG LQ WKH GLDJUDPV RI )LJ DQG $JDLQ DW EUDNLQJWKHPRPHQWLVGLIILFXOWRULPSRVVLEOHWRPRGHOPRUHDFFXUDWHO\
)\
4
02725&<&/('<1$0,&6
γ =0
)\
] 1000N 2000N 3000N
)
2
[kN]
γ
] 3000 N
)
=5
2
o
-30
-2
test data MF-fit
-4 -10 -8
-6
-4
-2
0
α
2
4 6 [deg.]
γ = -30,-20,-15,-10,-5,0,5 o
8
-4 -10 -8
10
γ =0
0
] 1000N, 2000N, 3000N
)
-6
-4
-2
0
α
2
4 6 [deg.]
8
-1
[kN]
[kN]
-2
-2
)[
γ =0 ] 3000 N
α = -8
o
)[
-3
-5
-3
-2 2
-4 -1
-0.8
-0.6
κ
-0.4
10
)
-1
)\
0
0
-2
1
o
-20
[kN]
0
0
4
[-] -0.2
0
-4 -1
α=2
-0.8
-0.6
κ
0
-0.4 [-] -0.2
0
o
γ =0
)\
] 3000 N
)
0
0
[kN]
α=2
o
1
0
0
[kN]
-2
-1
-1
-2
-5 -2
-2
-3 -1
-0.6
κ
-0.4
[-] -0.2
-5
] 3000 N
)
-8 -0.8
γ=0
-3 0
-3
)[
-8 -2
[kN]
-1
0
)LJ )RUFHFKDUDFWHULVWLFVRID=5PRWRUF\FOHW\UH0RGLILHG0DJLF)RUPXOD FRPSXWHGUHVXOWVFRPSDUHGZLWKGDWDIURPPHDVXUHPHQWVFRQGXFWHGZLWKWKH'HOIW 7\UH7HVW7UDLOHU
02725&<&/('<1$0,&6
γ =0
40
0]
20
] 3000 N
)
40
] 1000N 2000N 3000N
)
0]
γ
0
[Nm]
[Nm] 0
-40
o
=50 -5 -10 -15 -20 o
-20
-30 -80
-40 -10 -8
60
0] 40
-6
-4
-2
0
α
2
4 6 [deg.]
8
-10
10
-8
-6
-4
-2
0
α
2
4 6 [deg.]
8
60
γ =0 ] 3000 N
0]
-2
)
-2
40
[Nm]
[Nm]
0
20
-5
20
0
-5 0
0
-8 o
-0.8
-0.6
κ
-0.4
[-] -0.2
-8
γ =0
α=2
-20
-1
10
-20
0
α=2
o
] 3000 N
)
-3
)[
-2
[kN]
-1
0
)LJ $OLJQLQJWRUTXHFKDUDFWHULVWLFVRID=5PRWRUF\FOHW\UH0RGLILHG0DJLF )RUPXODFRPSXWHGUHVXOWVFRPSDUHGZLWKGDWDIURPPHDVXUHPHQWVFRQGXFWHGZLWK WKH'HOIW7\UH7HVW7UDLOHU
5HIHUHQFHV $EUDPRZLW] 0 DQG 6WHJXQ ,$ +DQGERRN RI PDWKHPDWLFDO IXQFWLRQV 'RYHU1HZ
WK
PXOWLERG\PRGHOOLQJDQGVLPXODWLRQ,Q 3URFHHGLQJVRI ,QW&RQIHUHQFHRQ+LJK 7HFK(QJLQHVDQG&DUV0RGHQD %HVVHOLQN,-0 6KLPP\RI$LUFUDIW0DLQ/DQGLQJ*HDUV'LVVHUWDWLRQ78 'HOIW %LUDO ) DQG 'D /LR 0 0RGHOOLQJ GULYHUV ZLWK WKH RSWLPDO PDQHXYUH PHWKRG,Q 3URFHHGLQJVRI$7$7KH5ROHRI([SHULPHQWVLQWKH$XWRPRWLYH 3URGXFW'HYHORSPHQW3URFHVV)ORUHQFH %|KP) 'HU5ROOYRUJDQJGHV$XWRPRELO5DGHV=$0077 %RUJPDQQ: 7KHRUHWLVFKHXQGH[SHULPHQWHOOH8QWHUVXFKXQJHQDQ/XIWUHLIHQ EHL6FKUlJODXI'LVVHUWDWLRQ%UDXQVFKZHLJ %UHXHU7DQG3UXFNQHU$ $GYDQFHGG\QDPLFPRWRUELNHDQDO\VLVDQGGULYHU
WK
VLPXODWLRQ,Q (XURSHDQ$'$068VHUV¶&RQIHUHQFH3DULV
5()(5(1&(6
%URFNPDQ5$DQG%UDLVWHG:5 &ULWLFDO6SHHG(VWLPDWLRQRI$LUFUDIW 7LUHV7LUH6FLHQFHDQG7HFKQRORJ\ %UXQL6&KHOL)DQG5HVWD) 2QWKHLGHQWLILFDWLRQLQWLPHGRPDLQRIWKH SDUDPHWHUV RI D W\UH PRGHO IRU WKH VWXG\ RI LQSODQH G\QDPLFV ,Q 3URFHHGLQJV RI 3URFHHGLQJVRI
QG
&ROORTXLXPRQ7\UH0RGHOVIRU9HKLFOH$QDO\VLVHGV)%|KPDQG
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lIWHDPUROOHQGHQ/XIWUHLIHQ9',=HLWVFKULIW )UDQN ) D *UXQGODJHQ ]XU %HUHFKQXQJ GHU 6HLWHQIKUXQJVNHQQOLQLHQ YRQ 5HLIHQ.DXWFKXNXQG*XPPL )UDQN) E 7KHRULHGHV5HLIHQVFKUlJODXIV'LVVHUWDWLRQ%UDXQVFKZHLJ )UHXGHQVWHLQ * /XIWUHLIHQ EHL 6FKUlJ XQG .XUYHQODXI 'HXWVFKH .UDIWIDKU]HXJIRUVFKXQJXQG6WU9HUNWHFKQ )ULW] : )HGHUKlUWH YRQ 5HLIHQ XQG )UHTXHQ]JDQJ GHU 5HLIHQNUlIWH EHL SHULRGLVFKHU9HUWLNDOEHZHJXQJGHU)HOJH'LVVHUWDWLRQ.DUOVUXKH )URPP+ .XU]HU%HULFKWEHUGLHJHVFKLFKWHGHU7KHRULHGHV5DGIODWWHUQV %HULFKWGHU/LOLHQWKDO*HVHOOVFKDIW 1$&$70 *LOOHVSLH7' )XQGDPHQWDOVRI9HKLFOH'\QDPLFV6$( *LSVHU 0 +RIHU 5 DQG /XJQHU 3 '\QDPLFDO 7LUH )RUFHV 5HVSRQVH WR 5RDG 8QHYHQQHVVHV ,Q 3URFHHGLQJV RI 3URFHHGLQJV RI
QG
&ROORTXLXP RQ 7\UH
0RGHOVIRU9HKLFOH$QDO\VLVHGV)%|KPDQG+3:LOOXPHLW%HUOLQ 6XSSO 9HKLFOH6\VWHP'\QDPLFV *RQFKDUHQNR9,/REDV/6DQG1LNLWLQD19 :REEOHLQJXLGHZKHHOV 6RYLHW$SSOLHG0HFKDQLFV *RQJ6 $6WXG\RI,Q3ODQH'\QDPLFVRI7LUHV'LVVHUWDWLRQ78'HOIW
5()(5(1&(6
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
QG
&ROORTXLXPRQ7\UH0RGHOVIRU9HKLFOH$QDO\VLV%HUOLQ HGV
)%|KPDQG+3:LOOXPHLW6XSSO9HKLFOH6\VWHP'\QDPLFV +DGHNHO5 7KHPHFKDQLFDOFKDUDFWHULVWLFVRISQHXPDWLFW\UHV6 70HPR %ULWLVK0LQLVWU\RI6XSSO\73$7,% +DUWRJ-3GHQ 0HFKDQLFDO9LEUDWLRQV1HZ
WK
$QDO\VLVRIVWDELOLW\RIKLJKVSHHGGULYLQJ,Q 3URFHHGLQJVRI ,QW7HFK&RQIRQ ([SHULPHQWDO6DIHW\9HKLFOHV +HQNHU ( '\QDPLVFKH .HQQOLQLHQ YRQ 3.: 5HLIHQ :LVVHQVFKDIWOLFK 7HFKQLVFKH9HU|IIHQWOLFKXQJHQDXVGHP$XWRPRELOEDX ,)$''5 +HIW +LJXFKL $ DQG 3DFHMND +% 7KH 5HOD[DWLRQ /HQJWK &RQFHSW DW /DUJH :KHHO 6OLS DQG &DPEHU ,Q 3URFHHGLQJV RI
QG
&ROORTXLXP RQ 7\UH 0RGHOV IRU
9HKLFOH$QDO\VLV%HUOLQ HGV)%|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
5()(5(1&(6
.DWD\DPD 7 $RNL $ DQG 1LVKLPL 7 &RQWURO EHKDYLRXU RI PRWRUF\FOH ULGHUV9HKLFOH6\VWHP'\QDPLFV .DWD\DPD71LVKLPL72NR\DPD7DQG$RNL$ $VLPXODWLRQPRGHOIRU PRWRUF\FOHULGHU¶VEHKDYLRXUV,Q 3URFHHGLQJVRI6(7&¶
8QWHUVXFKXQJ QLFKWOLQHDUHU
5()(5(1&(6
6FKZLQJXQJVXQG5HJHOXQJV6\VWHPH9',)RUVFKXQJVKHIW% 0DVWLQX* $6HPL$QDO\WLFDO7\UH0RGHOIRU6WHDG\DQG7UDQVLHQW6WDWH 6LPXODWLRQV,Q 3URFHHGLQJVRI
QG
&ROORTXLXPRQ7\UH0RGHOVIRU9HKLFOH$QDO\VLV
%HUOLQ HGV)%|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
WK
RI PRWRUF\FOHV ,Q 3URFHHGLQJV RI ,QW 7HFK &RQI RQ ([SHULPHQWDO 6DIHW\ 9HKLFOHV 1RUGHHQ'/DQG&RUWHVH$' )RUFHDQGPRPHQWFKDUDFWHULVWLFVRIUROOLQJ WLUHV6$(SDSHU$ 6$(7UDQVDFWLRQV 2HUWHO&K 2QPRGHOOLQJFRQWDFWDQGIULFWLRQFDOFXODWLRQRIW\UHUHVSRQVHRQ XQHYHQURDGV,Q 3URFHHGLQJVRI3URFHHGLQJVRI
QG
&ROORTXLXPRQ7\UH0RGHOVIRU
9HKLFOH$QDO\VLVHGV)%|KPDQG+3:LOOXPHLW6XSSO9HKLFOH6\VWHP'\QDPLFV 2OOH\0 5RDGPDQQHUVRIWKHPRGHUQFDU-,QVW$XWR(QJQUV 2RVWHQ --0 YDQ DQG %DNNHU ( 'HWHUPLQDWLRQ RI 0DJLF 7\UH 0RGHO
VW
3DUDPHWHUV,Q 3URFHHGLQJVRI &ROORTXLXPRQ7\UH0RGHOVIRU9HKLFOH$QDO\VLV HG+%3DFHMND'HOIW 6XSSO9HKLFOH6\VWHP'\QDPLFV 2RVWHQ --0 YDQ 8QUDX -+ 5LHGHO * DQG %DNNHU ( 7<'(;
5()(5(1&(6
:RUNVKRS 6WDQGDUGLVDWLRQRI'DWD([FKDQJHLQ7\UH7HVWLQJDQG7\UH0RGHOOLQJ ,Q 3URFHHGLQJVRI
QG
&ROORTXLXPRQ7\UH0RGHOVIRU9HKLFOH$QDO\VLVHGV)%|KP
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
WK
3URFHHGLQJV RI 96'
QG
,87$0 6\PSRVLXP HGV $ 6OLEDU DQG + 6SULQJHU
9LHQQD 6ZHWVDQG=HLWOLQJHU/LVVH 3DFHMND+% $QDO\VLVRI7LUH3URSHUWLHV&KDSWHU 0HFKDQLFVRI3QHXPDWLF 7LUHVHG6.&ODUN'27+6 3DFHMND +% D ,Q3ODQH DQG 2XWRI3ODQH '\QDPLFV RI 3QHXPDWLF 7\UHV 9HKLFOH6\VWHP'\QDPLFV 3DFHMND+% 1RQ/LQHDULWLHVLQ5RDG9HKLFOH'\QDPLFV9HKLFOH6\VWHP '\QDPLFV
3DFHMND+%DQG6KDUS56 6KHDU)RUFH'HYHORSPHQWE\3QHXPDWLF7\UHV LQ 6WHDG\ 6WDWH &RQGLWLRQV $ 5HYLHZ RI 0RGHOOLQJ $VSHFWV 9HKLFOH 6\VWHP '\QDPLFV 3DFHMND+%DQG7DNDKDVKL7 3XUH6OLS&KDUDFWHULVWLFVRI7\UHVRQ)ODW DQGRQ8QGXODWHG5RDG6XUIDFHV,Q 3URFHHGLQJVRI$9(&¶
5()(5(1&(6
-DSDQ 3DFHMND+%DQG%DNNHU( 7KH0DJLF)RUPXODW\UHPRGHO,Q 3URFHHGLQJV
VW
RI &ROORTXLXPRQ7\UH0RGHOVIRU9HKLFOH$QDO\VLV'HOIW HG+%3DFHMND 6XSSO9HKLFOH6\VWHP'\QDPLFV 3DFHMND+% 7KH7\UHDVD9HKLFOH&RPSRQHQW,Q 3URFHHGLQJVRI;;9, ),6,7$&RQJUHVVHG0$SHWDXU3UDJXH 3DFHMND +% DQG %HVVHOLQN ,-0 0DJLF )RUPXOD 7\UH 0RGHO ZLWK 7UDQVLHQW3URSHUWLHV,Q 3URFHHGLQJVRI
QG
&ROORTXLXPRQ7\UH0RGHOVIRU9HKLFOH
$QDO\VLV HGV )%|KP DQG +3:LOOXPHLW %HUOLQ 6XSSO 9HKLFOH 6\VWHP '\QDPLFV 3HYVQHU-D0 7KHRU\RIWKHVWDELOLW\RIDXWRPRELOHPRWLRQV ,Q5XVVLDQ 0DVMJL]/HQLQJUDG 5DGW+6DQG3DFHMND+% $QDO\VLVRIWKH6WHDG\6WDWH7XUQLQJ%HKDYLRU RIDQ$XWRPRELOH,Q 3URFHHGLQJVRIWKH6\PSRVLXPRQ&RQWURORI9HKLFOHV,QVWRI 0HFK(QJQUV/RQGRQ 5DGW+6DQG0LOOLNHQ:) 1RQGLPHQVLRQDOL]LQJ7\UH'DWDIRU9HKLFOH 6LPXODWLRQ5RDG9HKLFOH+DQGOLQJ,QVWRI0HFK(QJQUV & 5HLPSHOO - DQG 6SRQDJHO 3 )DKUZHUNWHFKQLN 5HLIHQ XQG 5lGHU 9RJHO %XFKYHUODJ:U]EXUJ 5LHNHUW 3 DQG 6FKXQFN 7( =XU )DKUPHFKDQLN GHV *XPPLEHUHLIWHQ .UDIWIDKU]HXJV,QJHQLHXU$UFKLY 5RFDUG< '\QDPLTXHJpQpUDOGHVYLEUDWLRQV3DULV 5RJHUV/&DQG%UHZHU+. 6\QWKHVLVRIWLUHHTXDWLRQVIRUXVHLQVKLPP\ DQGRWKHUG\QDPLFVWXGLHV-RI$LUFUDIW 5RJHUV/& 7KHRUHWLFDOWLUHHTXDWLRQVIRUVKLPP\DQGRWKHUG\QDPLFVWXGLHV $,$$-RI$LUFUDIW 5XLMV3$-DQG3DFHMND+% 5HVHDUFKLQ/DWHUDO'\QDPLFVRI0RWRUF\FOHV
WK
,Q 3URFHHGLQJVRI ,$96'6\PSRVLXPRQWKH'\QDPLFVRI9HKLFOHVRQ5RDGVDQG 7UDFNV HG 2 1RUGVWU|P /LQN|SLQJ 6XSSO 9HKLFOH 6\VWHP '\QDPLFV 6$(-H 9HKLFOH'\QDPLFV7HUPLQRORJ\6$(-H6RFLHW\RI$XWRPRWLYH (QJLQHHUV,QF:DUUHQGDOH3$-XO\ 6DLWR< $VWXG\RIWKHG\QDPLFVWHHULQJSURSHUWLHVRIW\UHV,Q 3URFHHGLQJV ,;),6,7$&RQJUHVV/RQGRQ 6DNDL + 7KHRUHWLFDO DQG H[SHULPHQWDO VWXGLHV RQ WKH G\QDPLF FRUQHULQJ SURSHUWLHVRIW\UHV,QW-RI9HKLFOH'HVLJQ
WK
6DNDL+ 6WXG\RQ&RUQHULQJ3URSHUWLHVIRU7LUHDQG9HKLFOH7KH DQQXDO PHHWLQJRIWKH7LUH6RFLHW\$NURQ
5()(5(1&(6
6DNDL+ 6WXG\RQ&RUQHULQJ3URSHUWLHVRI7LUHDQG9HKLFOH7LUH6FLHQFHDQG 7HFKQRORJ\7&7&$ 6DYNRRU$5 7KHODWHUDOIOH[LELOLW\RIDSQHXPDWLFW\UHDQGLWVDSSOLFDWLRQWR WKHODWHUDOFRQWDFWSUREOHP,Q 3URFHHGLQJV),6,7$,QW$XWR6DIHW\&RQIHUHQFH6$( 3DSHU 6FKOLSSH % YRQ DQG 'LHWULFK 5 'DV )ODWWHUQ HLQHV EHSQHXWHQ 5DGHV %HULFKWGHU/LOLHQWKDO*HVHOOVFKDIW 1$&$70 6FKOLSSH%YRQDQG'LHWULFK5 =XU0HFKDQLNGHV/XIWUHLIHQV=HQWUDOHIU ZLVVHQVFKDIWOLFKHV%HULFKWZHVHQ%HUOLQ$GOHUVKRI 6FKOLSSH % YRQ DQG 'LHWULFK 5 'DV )ODWWHUQ HLQHV PLW /XIWUHLIHQ YHUVHKHQHQ5DGHV,Q -DKUEXFKGHUGHXWVFKH/XIWIDKUWIRUVFKXQJ 6FKPHLW] $-& DQG 3DXZHOXVVHQ -3 $Q (IILFLHQW '\QDPLF 5LGH DQG +DQGOLQJ7\UH0RGHOIRU$UELWUDU\5RDG8QHYHQQHVVHV9',%HULFKWH
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
5()(5(1&(6
9HKLFOH6\VWHP'\QDPLFV 6KDUS56 6WDELOLW\&RQWURODQG6WHHULQJ5HVSRQVHVRI0RWRUF\FOHV9HKLFOH 6\VWHP'\QDPLFV 6KDUS 56 DQG /LPHEHHU '-1 D $ PRWRUF\FOH PRGHO IRU VWDELOLW\ DQG FRQWURODQDO\VLV0XOWLERG\6\VWHP'\QDPLFV 6PLOH\5) &RUUHODWLRQDQGH[WHQVLRQRIOLQHDUL]HGWKHRULHVIRUWLUHPRWLRQ DQGZKHHOVKLPP\1$&$5HSRUW 6PLOH\5) &RUUHODWLRQHYDOXDWLRQDQGH[WHQVLRQRIOLQHDUL]HGWKHRULHVIRU WLUHPRWLRQDQGZKHHOVKLPP\1$&$ 1$6$ 7HFK1RWH 6SHUOLQJ ( =XU .LQHPDWLN XQG .LQHWLN HODVWLVFKHU 5lGHU DXV GHU 6LFKW YHUVFKLHGHQHU7KHRULHQ'LVVHUWDWLRQ780XQLFK 6SLHULQJV 37- 7KH (IIHFWV RI /DWHUDO )URQW )RUN )OH[LELOLW\ RQ WKH 9LEUDWLRQDO 0RGHV RI 6WUDLJKW 5XQQLQJ 6LQJOH 7UDFN 9HKLFOHV 9HKLFOH 6\VWHP '\QDPLFV 6WRNHU-- 1RQ/LQHDU9LEUDWLRQV1HZ
WK
RI ,$96'6\PSRVLXPRQWKH'\QDPLFVRI9HKLFOHVRQ5RDGVDQG7UDFNVHG0 $SHWDXU3UDJXH 6XSSO9HKLFOH6\VWHP'\QDPLFV 7DQJX\* $QWLVNLG6\VWHPVDQG9HKLFOH6XVSHQVLRQ6$(3DSHU 7URJHU+DQG=HPDQ. $QRQOLQHDUDQDO\VLVRIWKHJHQHULFW\SHVRIORVV RI VWDELOLW\ RI WKH VWHDG\VWDWH PRWLRQ RI D WUDFWRUVHPLWUDLOHU 9HKLFOH 6\VWHP '\QDPLFV 9DON 5 YDQ GHU DQG 3DFHMND +% $Q DQDO\VLV RI D FLYLO DLUFUDIW PDLQ ODQGLQJJHDUVKLPP\IDLOXUH9HKLFOH6\VWHP'\QDPLFV 9ULHV (-+ GH DQG 3DFHMND +% D 0RWRUF\FOH W\UH PHDVXUHPHQWV DQG PRGHOV,Q 3URFHHGLQJVRI
WK
,$96'6\PSRVLXPRQWKH'\QDPLFVRI9HKLFOHVRQ
5RDGVDQG7UDFNVHG/3DONRYLFV%XGDSHVW 6XSSO9HKLFOH6\VWHP'\QDPLFV 9ULHV (-+ GH DQG 3DFHMND +% E 7KH HIIHFW RI WLUH PRGHOOLQJ RQ WKH VWDELOLW\DQDO\VLVRIDPRWRUF\FOH,Q 3URFHHGLQJV$9(&¶ 1DJR\D6$(RI-DSDQ :HLU'+ 0RWRUF\FOHKDQGOLQJG\QDPLFVDQGULGHUFRQWURODQGWKHHIIHFWRI
5()(5(1&(6
GHVLJQFRQILJXUDWLRQRQUHVSRQVHDQGSHUIRUPDQFH'LVVHUWDWLRQ8&/$ :KLSSOH)-: 7KH6WDELOLW\RIWKH0RWLRQRID%LF\FOH4XDUW-RI3XUHDQG $SSOLHG0DWKHPDWLFV :KLWFRPE ': DQG 0LOOLNHQ :) 'HVLJQ ,PSOLFDWLRQV RI D *HQHUDO 7KHRU\ RI $XWRPRELOH 6WDELOLW\ DQG &RQWURO ,Q 3URFHHGLQJV RI $XWR 'LYLVLRQ RI ,0HFK( :LOOXPHLW+3 7KHRUHWLVFK8QWHUVXFKXQJHQDQHLQHP0RGHOOGHV/XIWUHLIHQV 'LVVHUWDWLRQ%HUOLQ :LQNOHU&% 6LPSOLILHG$QDO\VLVRIWKH6WHDG\6WDWH7XUQLQJRI&RPSOH[ 9HKLFOHV9HKLFOH6\VWHP'\QDPLFV :LVVHOPDQ ' ,IIHOVEHUJHU ' DQG %UDQGOKXEHU % (LQVDW] HLQHV )DKUG\QDPLN6LPXODWLRQVPRGHOOVLQGHU0RWRUUDGHQWZLFNOXQJEHL%0:$7= =HJHODDU3:$*RQJ6DQG3DFHMND+% 7\UH0RGHOVIRUWKH6WXG\RI
WK
,Q3ODQH'\QDPLFV,Q 3URFHHGLQJVRI ,$96'6\PSRVLXPRQWKH'\QDPLFVRI 9HKLFOHVRQ5RDGVDQG7UDFNVHG=6KHQ&KHQJGX 6XSSO9HKLFOH6\VWHP '\QDPLFV =HJHODDU 3:$ DQG 3DFHMND +% 7KH ,Q3ODQH '\QDPLFV RI 7\UHV RQ
WK
8QHYHQ 5RDGV ,Q 3URFHHGLQJV RI ,$96' 6\PSRVLXP RQ WKH '\QDPLFV RI 9HKLFOHVRQ5RDGVDQG7UDFNVHG/6HJHO$QQ$UERU 6XSSO9HKLFOH6\VWHP '\QDPLFV =HJHODDU 3:$ DQG 3DFHMND +% '\QDPLF 7\UH 5HVSRQVHV WR %UDNH 7RUTXH9DULDWLRQV,Q 3URFHHGLQJVRI
QG
&ROORTXLXPRQ7\UH0RGHOVIRU9HKLFOH
$QDO\VLV HGV )%|KP DQG +3:LOOXPHLW %HUOLQ 6XSSO 9HKLFOH 6\VWHP '\QDPLFV =HJHODDU3:$ 7KHG\QDPLFUHVSRQVHRIW\UHVWREUDNHWRUTXHYDULDWLRQV DQGURDGXQHYHQQHVVHV'LVVHUWDWLRQ78'HOIW
/LVWRI6\PEROV D
GLVWDQFHIURQWD[OHWRFJ KDOIRIFRQWDFWOHQJWK
D[
ORQJLWXGLQDODFFHOHUDWLRQ
D\
ODWHUDODFFHOHUDWLRQ
D
VOLSYHORFLW\GHSHQGHQF\FRHIILFLHQWIRUIULFWLRQ
$U
UROOLQJUHVLVWDQFHFRHIILFLHQW
E
GLVWDQFHUHDUD[OHWRFJ KDOIFRQWDFWZLGWK
%
VWLIIQHVVIDFWRULQ 0DJLF)RUPXOD
%
EUDNHIRUFHRIUROOLQJZKHHO
F
VWLIIQHVV IDFWRU
FF
ODWHUDOFDUFDVVVWLIIQHVVSHUXQLWOHQJWK
FJ\U
QRQGLPHQVLRQDOJ\URVFRSLFFRHIILFLHQW
FS[\
WUHDGHOHPHQWVWLIIQHVVSHUXQLWOHQJWKRIFLUFXPIHUHQFH
F¶ S[
WUHDGHOHPHQWORQJLWXGLQDOVWLIIQHVVSHUXQLWDUHD
&
FRUQHULQJVWLIIQHVV VXPIURQWDQGUHDU
&L
FRUQHULQJVWLIIQHVVVXPOHIWDQGULJKW
&
FRQWDFWFHQWUH SRLQWRILQWHUVHFWLRQ
&
VKDSHIDFWRULQ 0DJLF)RUPXOD
&G$
DLUGUDJFRHIILFLHQW
&)\
ODWHUDOVWLIIQHVVRIVWDQGLQJW\UH
&)[
ORQJLWXGLQDOVWLIIQHVVRIVWDQGLQJW\UH
&)
FRUQHULQJVWLIIQHVV
&)
ORQJLWXGLQDOVOLSVWLIIQHVV
&)
FDPEHUVWLIIQHVVIRUVLGHIRUFH
&) n
VSLQVWLIIQHVVIRUVLGHIRUFH
&J\U
W\UHJ\URVFRSLFFRHIILFLHQW
&0
DOLJQLQJWRUTXHVWLIIQHVV
&0
FDPEHUVWLIIQHVVIRUDOLJQLQJWRUTXH
&0 n
VSLQVWLIIQHVVIRUDOLJQLQJWRUTXH
&0
WRUVLRQDO\DZVWLIIQHVVRIVWDQGLQJW\UH
&0[
RYHUWXUQLQJFRXSOHVWLIIQHVVDJDLQVWFDPEHU
&F[\
FDUFDVVKRUL]RQWDOVWLIIQHVVRIVWDQGLQJW\UH
&J\U
J\URVFRSLFFRHIILFLHQW
GI]
QRUPDOL]HGFKDQJHLQQRUPDOORDG(T (
GW
WUHDGGHSWK
'
SHDNIDFWRULQ 0DJLF)RUPXOD GLVVLSDWLRQIXQFWLRQ
(
FXUYDWXUHIDFWRULQ 0DJLF)RUPXOD
/,672)6<0%2/6
H
FDVWHUOHQJWK WUHDGHOHPHQWGHIOHFWLRQ
I
WUDLORIFJ
IU
UROOLQJUHVLVWDQFHFRHIILFLHQW
)D[
IRUFHIRUIRUZDUGDFFHOHUDWLRQ
)G
DLUGUDJIRUFH
)[WRW
VXPRIORQJLWXGLQDOW\UHIRUFHV
)[
ORQJLWXGLQDOW\UHIRUFH
)\
ODWHUDOW\UHIRUFH
)]
YHUWLFDO QRUPDO W\UHIRUFH ORDG ! LQ&KDS )]
)U
UROOLQJUHVLVWDQFHIRUFH !
)1
W\UHQRUPDOIRUFH !
)1R
UHIHUHQFHYHUWLFDOORDGQRPLQDOORDG
)9
W\UHYHUWLFDOIRUFH
_)]R_
)+
W\UHORQJLWXGLQDOKRUL]RQWDOIRUFH
J
DFFHOHUDWLRQGXHWRJUDYLW\ IHHGEDFNULGHUFRQWUROJDLQ
*
ZHLJKWLQJIDFWRU
K
KHLJKW
L
%
L]
UDGLXVRILQHUWLD
+
KHLJKW VKDUSQHVVIDFWRULQ 0DJLF)RUPXOD
+
WUDQVIRUP +XUZLW]GHWHUPLQDQW
M
%
,
PRPHQWRILQHUWLD
,Z
ZKHHOSRODUPRPHQWRILQHUWLD
,S
ZKHHOSRODUPRPHQWRILQHUWLD
N
UDGLXVRILQHUWLD YLVFRXVGDPSLQJFRHIILFLHQW
.
FHQWULIXJDOIRUFH IRUFHDFWLQJLQEHOWZKHHOFHQWUH
O
ZKHHOEDVH
OV
VKLIW WZRSRLQWIROORZHUOHQJWK
OE
OHQJWKRIEDVLFFXUYH
OI
RIIVHW
O
XQLWYHFWRUDORQJOLQHRILQWHUVHFWLRQ
P
PDVV IUDFWLRQRIDZKHUHDGKHVLRQRFFXUV
PF
FRQWDFWSDWFKPDVV GXPP\
PW
W\UHPDVV
PP
PDVVRIPDLQIUDPH LQFOXGLQJORZHUSDUWRIULGHU
PPU
PDVVRIPDLQIUDPHSOXVULGHU
PU
PDVVRIXSSHUWRUVR
0%'
EUDNHGULYHWRUTXH
0[
RYHUWXUQLQJFRXSOH
/,672)6<0%2/6
0\
UROOLQJUHVLVWDQFHPRPHQW
0]
VHOI DOLJQLQJWRUTXH
0¶ ]
VHOI DOLJQLQJWRUTXHGXHWRODWHUDOGHIOHFWLRQV
0]
DOLJQLQJWRUTXHGXHWRORQJLWXGLQDOGHIOHFWLRQV
0]J\U
J\URVFRSLFFRXSOH
0
VWHHUWRUTXH
Q
QXPEHURIHOHPHQWV IUHTXHQF\>+]@
Q
XQLWYHFWRUQRUPDOWRWKHURDG
7
QVW
VWHHUV\VWHPUDWLR
S
/DSODFHYDULDEOH>P@
SL
LQIODWLRQSUHVVXUH
T
DYHUDJHYHKLFOH\DZUHVLVWDQFHDUP JHQHUDOL]HGFRRUGLQDWH
T
FRQWDFWIRUFHSHUXQLWOHQJWKRIFLUFXPIHUHQFHYHFWRU
4
JHQHUDOL]HGIRUFH
U
\DZUDWH W\UH ORDGHG UDGLXV
UF
UDGLXVRIFDUFDVV EHOW XQORDGHG FURVVVHFWLRQFURZQUDGLXV
U\R
IUHHW\UHUDGLXVYDU\LQJDORQJFURVVVHFWLRQFRQWRXUU\R
UH
HIIHFWLYHUROOLQJUDGLXVRIIUHHO\UROOLQJZKHHO
UI
IUHHXQORDGHGW\UHUDGLXV
UO
ORDGHGUDGLXV
UR
IUHHXQORDGHGW\UHUDGLXV
5
UDGLXVRIFXUYDWXUH
5R
IUHHXQORDGHGW\UHUDGLXV
V
IRUZDUGSRVLWLRQRIQHXWUDOVWHHUSRLQW KDOIWUDFNZLGWK
V
/DSODFHYDULDEOH WUDYHOOHGGLVWDQFH
VV[
5R
UR
U\R \FR
SUDFWLFDOVOLSFRPSRQHQW
VV\
WDQ
SUDFWLFDOVOLSFRPSRQHQW
V
XQLWYHFWRUDORQJZKHHOVSLQD[LV
6
ZKHHOVOLSSRLQW LPSXOVH VWULQJWHQVLRQIRUFH
69+
YHUWLFDOKRUL]RQWDOVKLIW
W
SQHXPDWLFWUDLO WLPH
WF
FDVWHUOHQJWK
WU
ULVHWLPH
W
XQLWYHFWRULQURDGSODQHSHUSHQGLFXODUWROLQHRILQWHUVHFWLRQO
7
NLQHWLFHQHUJ\ PRPHQWDFWLQJDURXQGEHOWZKHHOFHQWUH
X
IRUZDUGYHORFLW\RIFJ ORQJLWXGLQDOGHIOHFWLRQ
Y
ODWHUDOYHORFLW\RIFJ ODWHUDOGHIOHFWLRQ
8
SRWHQWLDOHQHUJ\
9
VSHHGRIWUDYHORIFJ ZLWK[\FRPSRQHQWV
9
VSHHGRIWUDYHORIZKHHOFHQWUH ZLWK[\FRPSRQHQWV
/,672)6<0%2/6
9F
VSHHGRIFRQWDFWFHQWUH& ZLWK[\FRPSRQHQWV
9J
VSHHGRIVOLGLQJ ZLWK[\FRPSRQHQWV
9R
UHIHUHQFHYHORFLW\
9U
ZKHHOOLQHDUVSHHGRIUROOLQJ
9V
ZKHHOVOLSYHORFLW\RIVOLSSRLQW6 ZLWK[\FRPSRQHQWV
9[
ORQJLWXGLQDOVSHHGFRPSRQHQWRIZKHHOFHQWUH
%J5
R
9F[9V[
YHORFLW\RIFRQWDFWSDWFKPDVV ZLWK[\FRPSRQHQWV
9V Z
YHUWLFDOURDG HIIHFWLYH SURILOH SRVLWLYHGRZQZDUGV
:
ZRUN
[\]
ORQJLWXGLQDOODWHUDOYHUWLFDOGLVSODFHPHQW
[\] R
R
[ \ ]
FRRUGLQDWHVZLWKUHVSHFWWRPRYLQJD[HVV\VWHP]D[LVYHUWLFDO R
JOREDOFRRUGLQDWHV
[\]
222
JOREDOFRRUGLQDWHV
;
ORQJLWXGLQDOKRUL]RQWDOW\UHIRUFH
;<=
JOREDOFRRUGLQDWHV
\FR
GLVWDQFHIURPZKHHOFHQWUHSODQH
\PU
ODWHUDORIIVHWRIPPUFJ
ZKHHO VLGH VOLSDQJOH D[OH VLGH VOLSDQJOH
N D
URDGWUDQVYHUVHVORSHDQJOH WUDQVLHQWW\UHVOLSDQJOH YLUWXDOD[OHVOLSDQJOH YHKLFOHVLGHVOLSDQJOH W\UH\DZWRUVLRQDQJOH URDGIRUZDUG HII VORSHDQJOH
J\U
N
J\URVFRSLFZKHHOFRXSOLQJFRHIILFLHQW(T FDPEHU ZKHHOLQFOLQDWLRQ DQJOH WUDQVLHQWW\UHFDPEHUDQJOH XQLWVWHSUHVSRQVHIXQFWLRQ VWHHUDQJOHRIIURQWZKHHOV
R
.O5VWHHUDQJOHDW96 LQFUHPHQW
g g g g g
1/
UROOVWHHUFRHIILFLHQW UDNHDQJOHRIVWHHULQJD[LV
FDPEHUVWLIIQHVVUHGXFWLRQIDFWRU QRQODJJLQJSDUW VPDOOTXDQWLW\WRDYRLGVLQJXODULW\ GDPSLQJUDWLR VSLQIDFWRU
K
M M
VWULQJOHQJWKUDWLR(T HIIUROOUDGLXVJUDGLHQW UH GW
LIVSLQLQIOXHQFHLVGLVUHJDUGHG
KHLJKWUDWLR(T FRUQHULQJVWLIIQHVVORDGWUDQVIHUFRHIILFLHQW FDPEHUVWLIIQHVVORDGWUDQVIHUFRHIILFLHQW
/,672)6<0%2/6
M M
XQGHUVWHHUFRHIILFLHQW HIIHFWLYHUROOLQJUDGLXVJUDGLHQW UH
]
FJRIIVHWVWHHUFRHIILFLHQW
\
W\UHPRGHOSDUDPHWHU(TV DQJXODUGLVSODFHPHQWDERXW D[LV SLWFKDQJOH VWULQJPRGHOFRPSRVLWHSDUDPHWHU(T
F
ORQJLWXGLQDOZKHHOVOLS
N
WUDQVLHQWORQJLWXGLQDOW\UHVOLS
GDPSLQJFRHIILFLHQWGXHWRWUHDGZLGWK ZDYHOHQJWK URRWFKDUDFWHULVWLFHTXDWLRQ IUDFWLRQRIDZKHUHDGKHVLRQRFFXUV XVHUVFDOLQJIDFWRU
FRHIILFLHQWRIIULFWLRQ W\UHUDGLDO YHUWLFDO GHIOHFWLRQ
[\]
W\UHORQJLWXGLQDOODWHUDOQRUPDOGHIOHFWLRQ UHOD[DWLRQOHQJWK ORDGWUDQVIHUFRHIILFLHQW WKHRUHWLFDOVOLSYHFWRU(T
LQWHUVHFWLRQOHQJWKLQVWULQJPRGHOZLWKWUHDGHOHPHQWV
F
VWULQJPRGHOOHQJWKSDUDPHWHU(T
F
FRQWDFWSDWFKUHOD[DWLRQOHQJWK UROOFDPEHUFRHIILFLHQW
n nN n
k
W
ERG\UROODQJOH VSLQVOLS WUDQVLHQWVSLQVOLS WXUQVOLS SKDVHDQJOH \DZDQJOH VWHHUDQJOH
F
LR
FRPSOLDQFHVWHHUDQJOH WRHDQJOH IUHTXHQF\>UDGV@
R
XQGDPSHGQDWXUDOIUHTXHQF\
QDWXUDOIUHTXHQFLHV
Q
GDPSHGQDWXUDOIUHTXHQF\
V
SDWKIUHTXHQF\>UDGP@ ZKHHOVSHHGRIUHYROXWLRQ
PRYLQJD[HVV\VWHP D[LVDORQJVSLQD[LV KRUL]RQWDO
6XEVFULSWVDQGVXSHUVFULSWV D
D[OH IURPEHOWWRZKHHOULPFHQWUH
E
EHOW IURPEHOWFHQWUHWRULP
F
FRPSOLDQFH VWHHU
F
FRQWDFWSDWFK IURPFRQWDFWSDWFKFHQWUHWREHOW FURZQ FRQWRXU
/,672)6<0%2/6
'
GUDJ
H
HIIHFWLYH
HII
HIIHFWLYH FRUQHULQJVWLIIQHVV
HT
HTXLYDOHQW
I
IUHHXQORDGHG RIIURQWIUDPH
J
JOREDO
L
IURQW UHDU
/5
OHIWULJKW
P
RIPDLQIUDPH
PU
RIPDLQIUDPHSOXVULGHU
1/
QRQODJJLQJ
R
RULJLQDO LQLWLDO DYHUDJH
R
XQORDGHG QRPLQDO DWYDQLVKLQJVSHHG QDWXUDO
U
UROO UROOLQJ UROOLQJUHVLVWDQFH RIUHVLGXDOVSULQJ RIULGHU
V
VOLS IURPURDGVXUIDFHWRFRQWDFWSDWFK RIIURQWVXEIUDPH
VO
DWYHUJHRIWRWDOVOLGLQJ
VI
VLGHIRUFH VWHHU
VV
VWHDG\VWDWH
VW
VWDWLF
VWZ
VWHHULQJZKHHO
W
WUDQVLWLRQIURPDGKHVLRQWRVOLGLQJ
Z
ZKHHO
[\]
IRUZDUG ORQJLWXGLQDO ODWHUDO WRWKHULJKW GRZQZDUG
]U
UHVLGXDO WRUTXH
DORQJDURXQG D[HV IURQWUHDU OHDGLQJWUDLOLQJHGJH
$SSHQGL[ 6LJQ&RQYHQWLRQVIRU)RUFHDQG0RPHQWDQG:KHHO6OLS
adapted SAE (Pacejka, this book)
SAE [
)[
[
α
9
[
α )[
9
. ψ
0]
. ψ
γ )] inclination/ camber angle (rear view) 0
)\
turn slip
tan α =
κ =-
tan α = -
9V[
κ =-
9 [
9V\
. ψ
wheel load
overturning moment
rolling resistance moment self aligning moment
)\
)] < 0 0[
9 [
.
ψ
9 [
)[
α >0
α >0
0[ α
α <0
0] α
α >0
<0
0]
κ
)\
0[
α
*except for Chapter 9 where )1 = -)] > 0
ϕ=-
9V[
κ
α
0[
α
κ =-
9 [
)\
> 0*
0\ > 0 0]
9[
9V[
κ
α
0[
9 tan α = - V\
9[
)[
κ
side force
9V\
γ>0
)[
)\
0[
QRWGHILQHG
γ=0
longitudinal force
]
)\ \
κ =-
9 [
9 [
)[
γ
)]
tan α =
9[
9V[
ϕ=-
QRWGHILQHG
\
9
. ψ
0]
γ
)\ \
]
9[
. ψ ]
)\
0[
9V\
\
)]
\
α
)\
0]
γ
]
longitudinal slip
\
) 1 )]
[
side slip
\
0]
)[
)\
)\ \
[
α )[
9
)\
side angle (top view)
adapted ISO (Besselink, 2000)
ISO
0] α
α
$SSHQGL[
7UHDG6LP $0DWODESURJUDPIRUWKHFDOFXODWLRQRIVWHDG\VWDWHIRUFHDQGPRPHQWUHVSRQVH WRVLGHVOLSORQJLWXGLQDOVOLSFDPEHUDQGWXUQVOLSXVLQJDEUXVKPRGHOZLWKXS WRWKUHHURZVRIWUHDGHOHPHQWVDQGDIOH[LEOHFDUFDVV&I&KDSWHU6HFWLRQ DQG)LJV
7UHDG6LPP
WUHDGHOHPHQWIROORZLQJPRGHO
RQHWZRRUWKUHHURZV
VWDWLRQDU\UHVSRQVHWRODWHUDOORQJLWXGLQDOFDPEHUDQGWXUQVOLS
FDUFDVVULJLGZLWKRQO\ODWHUDOFRPSOLDQFHHIIHFWLQ0]
RU
FDUFDVVODWHUDO\DZDQGEHQGLQJFRPSOLDQFH
ZLWKDSSUR[LPDWHVOLSVSHHGGHSHQGHQWIULFWLRQFRHIILFLHQW
FRQLFLW\SO\VWHHUFRUUHFWLRQIDFWRUVIRUODWHUDOUROODQGFDPEHUGLVWRUWLRQ
FDOFXODWLRQRIIRUFHV)[)\DQGPRPHQW0] DQGW
DVIXQFWLRQRINDSSDDOSKDRUD5
ZLWKNDSSDD5
RSWLRQIRUPDNLQJGHIRUPDWLRQSLFWXUHIRURQHVHWRIPRWLRQYDULDEOHV
DSKLWJDPPDRUDOSKDDVFKDQJLQJPRWLRQSDUDPHWHU
FOI SLFWXUH
LI
WKHQVLQJOHVHWRIYDOXHVRIPRWLRQYDULDEOHVLVJLYHQDQGGLDJUDPVRI
IRUFHGLVWULEXWLRQDQGRIFDUFDVVDQGWUHDGHOHPHQWVGHIRUPDWLRQDUHSURGXFHG ULJLG
LIULJLG LLWHQG
LIFDUFDVVLVULJLGHOVH
QRLWHUDWLRQVDWULJLGFDUFDVV
HOVH LLWHQG
QXPEHURILWHUDWLRQVGHSHQGVRQFDUFDVVFRPSOLDQFH
HQG Q
QURZ D
QXPEHURIHOHPHQWV
QXPEHURIURZV RURU
KDOIFRQWDFWOHQJWK>P@
LISLFWXUH
DOSKDSLFW
JDPPDSLFW 5SLFW
VOLSDQJOHLQGHJUHHV
FDPEHUDQJOHLQGHJUHHV
NDSSDSLFW Q
VLQJOHVHWRIYDOXHVRIPRWLRQYDULDEOHV
WXUQVOLS
D5SLFWSDWKFXUYDWXUHUDGLXV5SLFWLQP
ORQJVOLS IURPWR
HQG DOSKDUXQV
LIUXQQLQJYDULDEOH DEVFLVVD WKHQ
VOLSDQJOH
HOVH
$33(1',;
NDSSDUXQV
DSKLWUXQV
QRQGLPHQVLRQDOWXUQVOLS DSKLW
D5
LIQRWFKRVHQDVUXQQLQJYDULDEOHEXWDVFKDQJLQJSDUDPHWHU WKHQ
JDPPDSDU DSKLWSDU
ORQJLWXGLQDOVOLSUDWLR
75($'6,0
HOVH
DOSKDSDU
NDSSDSDU
VLGHVOLSDQJOH DOSKDUDQJHDQGVWHS>GHJ@ LIFKRVHQDVUXQQLQJYDULDEOH DOSKGHJPLQU
DOSKGHJPD[U
JD
JD
GHOWDOSKGHJU
JD DEV DOSKDGHJ JD YDU\LQJDOSKDLQFUHPHQW
ORQJLGLQDOVOLSUDWLR NDSSDUDQJHDQGVWHS>@ LIFKRVHQDVUXQQLQJYDULDEOH NDSSDPLQU
NDSSDPD[U
JN
JN
QHJDWLYH EUDNLQJ SRVLWLYH GULYLQJ
GHOWDNDSSDU
QRQGLPWXUQVOLS DSKLW DSKLWPLQU
DSKLWPD[U
JW
JW
JN DEV NDSSD JN YDU\LQJNDSSDLQFUHPHQW D5 UDQJHDQGVWHS>@ LIFKRVHQDVUXQQLQJSDUDPHWHU
GHOWDSKLWU
JW DEV DSKLW JW YDU\LQJDSKLWLQFUHPHQW
FRQVWDQWPRWLRQSDUDPHWHUV RYHUUXOHGLIFKRVHQDVFKDQJHGSDUDPHWHURUDVUXQQLQJYDULDEOH
DOSKDGHJ
JDPPDGHJ 5
>GHJ@ VOLSDQJOH
>GHJ@
NDSSD
FDPEHUDQJOH
>P@ DSKLW
D5 QRQGLPHQVLRQDOWXUQVOLS
>@ ORQJVOLSUDWLR
DSKLW
D5
MPD[
GHOWJDPGHJ
JDPGHJPLQ
JDPPDGHJ
GHOWDOSKGHJ
DOSKGHJPLQ
DOSKDGHJ
GHOWDSKLW
DSKLWPLQ
GHOWDNDSSD
DSKLW
NDSSDPLQ
NDSSD
FKDQJHGPRWLRQSDUDPHWHUV QXPEHUPLQLPXPVWHS
LIJDPPDSDU MPD[
JDPPD
DVFKDQJHGPRWLRQSDUDPHWHU>GHJ@
JDPGHJPLQ
GHOWJDPGHJ
HQG LIDSKLWSDU MPD[
DSKLWPLQ
DSKLW GHOWDSKLW
D5
DVFKDQJHGPRWLRQSDUDPHWHU>@
HQG LIDOSKDSDU MPD[
DOSKD
DVFKDQJHGPRWLRQSDUDPHWHU>GHJ@
DOSKGHJPLQ
GHOWDOSKGHJ
HQG LINDSSDSDU MPD[
NDSSDPLQ
NDSSD
DVFKDQJHGPRWLRQSDUDPHWHU>@ GHOWDNDSSD
$33(1',;
75($'6,0
HQG
V\VWHPSDUDPHWHUV
UH
HIIHFWLYHUROOLQJUDGLXV>P@
)]
QRUPDOORDG>1@
9F
VSHHGRIFRQWDFWFHQWUH>PV@
PX
IULFWLRQFRHIILFLHQW>@
DPX
VSHHGGHSHQGHQF\FRHIILFLHQWIRUPX>VP@ PX
&)NDSSD)] &)NDSSD
PX DPX 9EL
ORQJVOLSVWLIIQHVVRYHU)]>@
&)NDSSD)] )]
>1@
IURPWKLV FS
QURZ D D &)NDSSD
FRUQHULQJVWLIIQHVVFRHII&)DOSKD)]>@LVFDOFXODWHGLIDOSKDUXQV
IRUODVWSDUDPFDVH
D KDOIFRQWDFWOHQJWK GHILQHGDERYH E
EURZ
KDOIFRQWDFWZLGWK>P@
LIQURZ EURZ
KDOIHIIHFWLYHFRQWDFWZLGWK>P@
HQG DOSKDGHJSO\
JDPPDGHJFRQ
SO\VWHHUHTXLYDOHQWVOLSDQJOH>GHJ@
FRQLFLW\HTXLYDOHQWFDPEHUDQJOH>GHJ@
VWLIIQHVVHVRIW\UHFDUFDVVDWFRQWDFWSDWFKDWEDVHRIWUHDGHOHPHQWV DVLIPHDVXUHGRQEDUHW\UH \
LQLWLDOODWHUDORIIVHW YR LIULJLG 0]
0]F )[ )\\ )[
LIULJLG F\DZ FEHQG FODW
>1PUDG@ >P1@
>1P@VXJJHVWHG FODW
)] D
HOVH F\DZ
FEHQG FODW F
>1PUDG@
>P1@
>1P@ F
HSV\)\ FODW 0]
0]F )[ )\\ )[
FLVXVHGLIµULJLG¶
HQG FS
UHVXOWLQJVWLIIQHVVRIWUHDGHOHPHQWVSHUURZSHUXQLWRIFLUFXPIHUQFH QURZ D D &)NDSSD
WKHWD
&)NDSSD PX )] QRWXVHG
FRUUHFWLRQSDUDPHWHUV
HSV\SULPH
UHGXFWLRQIDFWRUIRUPRPHQWDUPRI)[FDXVLQJ\DZGLVWRUWLRQ VORSH
DUP
a HSV\SULPH \PHII
ZKHUH\PHIIUHSUHVHQWVHIIHFWLYHODWHUDOGLVSODFHPHQWRIEHOWLQFRQWDFW]RQH HSV\UJDPPD
PRPHQW 0] DUPIRU)[GXHWRVLGHZD\VUROOLQJFDXVHGE\FDPEHU
\JDPUROO HSV\)\ )\
HSV\UJDPPD JDPPD E LIDEV \JDPUROO !E \JDPUROO
E VLJQ JDPPD
UHGXFWLRQIDFWRUPRPHQW 0] DUPIRU)[GXHWRVLGHZD\VUROOLQJFDXVHGE\
$33(1',;
75($'6,0
DQGWKHFRXQWHUHIIHFWRIORQJLWXGLQDOGHIOHFWLRQ WRWDOPRPHQW^0] DUP HSVGUJDP
\PHII
\P\GHIO HSV\)\ \JDPUROO
FRHIILFLHQWIRUUHGXFHGFKDQJHHIIUROOLQJUDGLXVOHIWULJKW
FDXVHGE\FDPEHU GHOWDBUHB5LJKW
HSV[JDPPD
HSVJDPPD
a HSVGUJDP JDPPD EURZ VXJJHVWLRQ HSVGUJDP
HSVJDPPD
UHGXFHGFDPEHUFXUYDWXUHFRHIILFLHQW ZLOOEH! GXHWRGLVWRUVLRQ
HSV\JDPPD
UHVXOWLQJFDPEHUFRQWDFWOLQHFXUYDWXUH DWPX 5JDPPD
UH HSVJDPPD VLQ JDPPD
HQGRISDUDPHWHUVHWWLQJV
VWDUWRIFRPSXWDWLRQ D
D D D
Q&)D
D D
&)DOSKD)]
LISLFWXUH MPD[
"
DOSKDGHJPLQ JDPGHJPLQ DSKLWPLQ
DOSKDSLFW JDPPDSLFW
D5SLFW
NDSSDPLQ
NDSSDSLFW
HQG
EHJLQJDPPDDSKLW IRUM
D5 DOSKDRUNDSSDORRS DVPRWLRQSDUDPHWHU
MPD[
JDPPDGHJ DSKLW
DOSKDGHJ NDSSD
DOSKD
JDPGHJPLQ M GHOWJDPGHJ
DSKLWPLQ M GHOWDSKLW DOSKGHJPLQ M GHOWDOSKGHJ
NDSSDPLQ M GHOWDNDSSD
DOSKDGHJ SL
JDPPD
JDPPDGHJJDPPDGHJFRQ SL
VLQJDP
VLQ JDPPD
FV
DOSKDGHJSO\ SL
FF
UH HSVJDPPD VLQJDP LQLWLDOEHOWODWHUDOFXUYDWXUHGXHWRFDPEHUDQGFRQLFLW\
\P
LQLWLDOEHOWVORSHGXHWRSO\VWHHU
FF D LQLWLDOEHOWODWHUDOSRVLWLRQDWFRQWDFWFHQWUHUHVXOWLQJIURPFDPEHU
\JDPUROO
DWPX
7KLVLVDJXHVV WKHQ\L
LIDEV \JDPUROO !E \JDPUROO
E VLJQ JDPPD
HQG
FV
FV
FF
FF
\P )\
FV DDWOHDGLQJHGJH [
D
HSV\UJDPPD VLQJDP E PRPHQWDUPRI)[GXHWRVLGHZD\VUROOLQJDWFDPEHU
\P 0]SULPH
LIDOSKDUXQV
)\P
0]P
$33(1',;
DOSKDGHJ LPD[
75($'6,0
DOSKGHJPLQU
URXQG ORJ DEV JD DOSKGHJPLQUJD ORJ DEV JD DOSKGHJPD[UJD JD
HQG LINDSSDUXQV NDSSD LPD[
NDSSDPLQU URXQG ORJ DEV JN NDSSDPLQUJN ORJ DEV JN NDSSDPD[UJN JN
HQG LIDSKLWUXQV
DSKLW
DSKLWPLQU
LPD[
URXQG ORJ DEV JW DSKLWPLQUJW ORJ DEV JW DSKLWPD[UJW JW
HQG
LISLFWXUH
DOSKDGHJ DOSKD NDSSD DSKLW
DOSKDSLFW
DOSKDGHJ SL NDSSDSLFW D5SLFW
DOSKDGHJPLQU NDSSDPLQU DSKLWPLQU LPD[
DOSKDSLFW
NDSSDSLFW D5SLFW
HQG
IRULD
LPD[
LIDOSKDUXQV
EHJLQNDSSDRUDOSKDRUD5 LIUXQQLQJYDULDEOH LDORRS
DOSKD
SL DOSKDGHJ
)\P
)\
QHHGHGIRUFDOFXODWLRQRIFRUQHULQJVWLIIQHVV
HQG 5
D DSKLW
)\
0]SULPH
LQLWLDOYDOXHIRULWHUDWLRQ
EHJLQFDUFDVVGHIOHFWLRQLWHUDWLRQLLWORRS
IRULLW
LLWHQG
VORSH
0]SULPHF\DZ
FV
FVVORSH
FXUYH FF
FFFXUYH
\GHIO \P
EHOWVORSHDWFRQWDFWFHQWUHZUWZKHHOSODQH
)\FEHQG EHOWODWHUDOFXUYDWXUHLQFRQWDFW]RQH
)\FODW
\P\GHIO EHOWODWHUDOSRVLWLRQDWFRQWDFWFHQWUHZUWZKHHOSODQH
\)\UROO \PHII
HSV\)\ \GHIO
\P\JDPUROO\)\UROO HIIHFWLYHPRPHQW 0] DUPIRU)[
IRUL/5
QURZ
EHJLQOHIW L/5
LIQURZ LE/5
E/5
HQG LIQURZ LE/5
L/5 E/5
LE/5 EURZ
DQGULJKWURZ L/5
ORRS LIRQHURZ QURZ
$33(1',;
75($'6,0
HQG LIQURZ
LE/5
L/5 E/5
LE/5 EURZ OHIW E/5
EURZPLG E/5
ULJKW E/5
HQG SVLGRW
9F5
9F[
9F FRV DOSKD
9V[
9F[ NDSSD
9V\
9F[ WDQ DOSKD
9U
9F[9V[
RPHJD
9UUH
9F[/5
9F[E/5 SVLGRW
9V[/5
9V[E/5 SVLGRWE/5 RPHJD HSVGUJDP VLQJDP
9V\/5
9V\
9U/5
9U
GHOWDW
D Q 9U/5
GHOWD[
DQ
[L
D [
HL
H[L
S[
D H\L
S\
VOLGLQJ
)[LP S]
Q
HLP
HL H[LP
[LGHOWD[ [L
H[L H\LP
H\L
[L [L
[L
G\EG[LPHDQ \EL
FVFF [L GHOWD[
\PFV [L FF [L [LE/5
\ELHII 9E[L
\PHIIFV [L FF [L [LE/5 9V[/5
9E\LPHDQ 9EL
9V\/5 [L GHOWD[ SVLGRW9U/5 G\EG[LPHDQ
VTUW 9E[L 9E[L9E\LPHDQ 9E\LPHDQ 9E L
PX
PX DPX 9EL
S]L
)] D[L D QURZ S] L
GHOWDV[L
GHOWDW 9E[L
GHOWDV\L
GHOWDW 9E\LPHDQ
LIVOLGLQJ! HL
EHJLQLORRS SDVVDJHWKURXJKFRQWDFWOHQJWK
IRUL
[ L
0]LP
[L
)\LP
S
9EL EHOWSRLQWYHORFLW\
S]L YHUWLFDOSUHVVXUHGLVWULEXWLRQ T]
HOHPHQWLVVOLGLQJ
PX S]LFS
LIL
HLP
H[LP
HLP 9E[L9EL
H\LP
HLP 9E\LPHDQ9EL
HQG JL
HLP H[LPGHOWDV[L H[LPGHOWDV[L H\LPGHOWDV\L H\LPGHOWDV\L HL HL H[LP H[LPGHOWDV[L H\LP H\LPGHOWDV\L
EURZ
$33(1',;
H[L
JLHLP H[LPGHOWDV[L
H\L
JLHLP H\LPGHOWDV\L
HLQ
VTUW H[L H[LH\L H\L
H[L
HLHLQ H[L FRUUHFWLRQ
H\L
HLHLQ H\L FRUUHFWLRQ
S[L
H[LHL PX S]L
S\L
H\LHL PX S]L
LIJL VOLGLQJ
HOHPHQWVWDUWVWRDGKHUHWRWKHURDG
H[L
GHOWDV[LH[LP
H\L
GHOWDV\LH\LP
HL
VTUW H[L H[LH\L H\L H L
SLL
75($'6,0
HL
FS HL
S[L
FS H[L
S\L
FS H\L
LISLL!PX S]L S[L
H[LHL PX S]L
S\L
H\LHL PX S]L
SLL
PX S]L
VOLGLQJ
HQG HQG HOVH H[L H\L HL
HOHPHQWLVDGKHULQJWRWKHURDG
GHOWDV[LH[LP GHOWDV\LH\LP VTUW H[L H[LH\L H\L H L
SLL
HL
FS HL
S[L
FS H[L
S\L
FS H\L
LISLL!PX S]L HOHPHQWVWDUWVWRVOLGH VOLGLQJ LIL
HLP
H[LP
HLP 9E[L9EL
H\LP
HLP 9E\LPHDQ9EL
HQG HL
PX S]LFS
JL
HLP H[LPGHOWDV[L H[LPGHOWDV[L H\LPGHOWDV\L H\LPGHOWDV\L HL HL H[LP H[LPGHOWDV[L H\LP H\LPGHOWDV\L
H[L
JLHLP H[LPGHOWDV[L
H\L
JLHLP H\LPGHOWDV\L
HLQ
VTUW H[L H[LH\L H\L
H[L
HLHLQ H[L FRUUHFWLRQ
H\L
HLHLQ H\L FRUUHFWLRQ
S[L
H[LHL PX S]L
S\L
H\LHL PX S]L
HQG HQG
SLL
$33(1',;
75($'6,0
VTUW S[L S[LS\L S\L
)[L
)[LPS[L DQ
)\L
)\LPS\L DQ
0]L
0]LP [L S\L\ELHII S[L DQ
S\ L
S\L
)[LP
)[L
)\LP
)\L
0]LP
0]L
LISLFWXUH
LILE/5 \/ L
PX S]L \/ L
S[L \/ L
[HO/ L
[L [HO/ L
\HO/ L
\ELE/5 \HO/ L
H\ L
S\L \/ L
[LH[L [HO/ L
SLL \/ L
[L
\ELE/5H\L \HO/ L
\ELE/5
H\L
HOVH LILE/5
\0 L \0 L
PX S]L \0 L
S[L \0 L
[HO0 L
[L [HO0 L
\HO0 L
\ELE/5 \HO0 L
H\ L
S\L \0 L
SLL \0 L
\0 L
9EL [LH[L [HO0 L
[L
\ELE/5H\L \HO0 L
\ELE/5
H\L
HOVH \5 L
PX S]L \5 L
S[L \5 L
[HO5 L
[L [HO5 L
\HO5 L
\ELE/5 \HO5 L
H\5 L
S\L \5 L
[LH[L [HO5 L
SLL \5 L
[L
\ELE/5H\L \HO5 L
\ELE/5
H\L
HQG HQG HQG
HQG
LILE/5 )[/
HQGLORRS SDVVDJHWKURXJKFRQWDFWOHQJWK
)[L )\/
)\L 0]/
0]L
HOVH LILE/5 )[0
)[L )\0
)\L 0]0
0]L
HOVH )[5
)[L )\5
)\L 0]5
0]L
HQG HQG HQG
LIQURZ
)[
)[0
)\
)\0
0] HOVH
0]0
HQGL/5ORRS RQHWZRRUWKUHHURZVRIHOHPHQWV
S]L
$33(1',;
LIQURZ
)[
)[/)[5
)\
)\/)\5
0]
75($'6,0
0]/0]5
HOVH )[
)[/)[5)[0
)\
)\/)\5)\0
0]
0]/0]50]0
HQG HQG
0]SULPH
0])[ \PHII HSV\SULPH IRUWRUVLRQ VORSH FDOFXODWLRQ
HQG
LIULJLG 0]
HQGLLWORRS FDUFDVVGHIOHFWLRQLWHUDWLRQ
0]F )[ )\\ )[
HOVH 0]
0]\ )[
HQG
W
0])\
SQHXPDWLFWUDLO
\I[ LDM
)[
\I\ LDM
)\
\P] LDM \W LDM
0]
W
LINDSSDUXQV [D LDM
NDSSD
HQG LIDOSKDUXQV [D LDM
DOSKDGHJ
HQG LIDSKLWUXQV [D LDM
DSKLW
HQG LIDOSKDUXQV
LIQ&)D
FDOFXODWLRQRI&)DOSKD)] WDNHGHOWDOSKDGHJUVPDOO
LILD!
LIDOSKD!
LIJDPPD
LIDEV DSKLW
LINDSSD
&)DOSKD)]
Q&)D
HQG HQG HQG
)\)\P )] SL GHOWDOSKGHJU
$33(1',;
75($'6,0
HQG
HQG HQG
HOVH &)DOSKD)]
"
HQG
LIDOSKDUXQV
GHOWDOSKGHJU
JD DEV DOSKDGHJ JD
DOSKDGHJ
DOSKDGHJGHOWDOSKGHJU
HQG LINDSSDUXQV GHOWDNDSSDU NDSSD
JN DEV NDSSD JN
NDSSDGHOWDNDSSDU
HQG LIDSKLWUXQV GHOWDSKLWU DSKLW
JW DEV DSKLW JW
DSKLWGHOWDSKLWU
HQG
HQG
HQGUXQQLQJYDULDEOHLDORRS
HQG
HQGPRWLRQSDUDPHWHUMORRS
LISLFWXUH
[HO/
D [HO/
D [HO/
D
\HO/
\HO/
\HO/
\PFV D FF D D
[HO/ Q
D \HO/ Q
[HO/ Q
D \HO/ Q
[HO0
D [HO0
D [HO0
D
\HO0
\HO0
\HO0
\PFV D FF D D
[HO0 Q
D \HO0 Q
[HO0 Q
D \HO0 Q
[HO5
D [HO5
D [HO5
D
\HO5
\HO5
\HO5
\PFV D FF D D
[HO5 Q
D \HO5 Q
[HO5 Q
D \HO5 Q
LIULJLG \F
)\ F IRUL
\HO/ L \HO0 L \HO5 L
Q \HO/ L \F \HO0 L \F \HO5 L \F
HQG HQG SORWVSLFWXUHRIIRUFHGLVWULEXWLRQDQGGHIRUPDWLRQV LIQURZa ILJXUH
$33(1',;
75($'6,0
SORW [\0 [ODEHO [>P@ \ODEHO GEOXH PX S]OEOXH SJUHHQ S[UHG S\VHSLD S]>1P@EODFN 9E>PPV@ WH[W > DOSKD
QXPVWU DOSKDSLFW GHJJDPPD
QXPVWU NDSSDSLFW 5
QXPVWU JDPPDSLFW GHJNDSSD
QXPVWU 5SLFW P @
WH[W > F\DZ
QXPVWU F\DZ FEHQG
QXPVWU FODW FS WH[W > WKHWD
QXPVWU HSV\SULPH
QXPVWU HSV\UJDPPD @
WH[W > HSV\)\
QXPVWU HSV\)\ HSVJDPPD
WH[W > HSVGUJDP LIQURZ
QXPVWU WKHWD @
WH[W > HSV\SULPH
HSV\UJDPPD
QXPVWU FEHQG FODW
QXPVWU FS @
QXPVWU HSVGUJDP \
QXPVWU HSVJDPPD @
QXPVWU \ PP @
WH[W > 6,1*/(52: @ HOVH WH[W > 0,''/(52: @ HQG D[LV >@ JULG
ILJXUH SORW [HO0\HO0 [ODEHO [DQGORQJLWXGLQDOGHIOHFWLRQV>P@ \ODEHO ODWHUDOGHIOHFWLRQV>P@ WH[W > DOSKD
GHJNDSSD LIULJLG
QXPVWU DOSKDSLFW GHJJDPPD
QXPVWU NDSSDSLFW 5
QXPVWU JDPPDSLFW
QXPVWU 5SLFW P @
WH[W >
5,*,'
F
QXPVWU F P1 @
HOVH WH[W > F\DZ
FODW
QXPVWU F\DZ FEHQG
QXPVWU FODW FS
QXPVWU FEHQG
QXPVWU FS @
HQG WH[W > )\ LIQURZ
QXPVWU URXQG )\ 10]
QXPVWU URXQG 0] 1P @
WH[W > 6,1*/(52: @ HOVH WH[W > 0,''/(52: @ HQG D[LV >@ HQG LIQURZa
ILJXUH SORW [\5 [ODEHO [>P@ \ODEHO GEOXH PX S]OEOXH SJUHHQ S[UHG S\>1P@ WH[W > DOSKD
>GHJ@NDSSD
QXPVWU DOSKDSLFW >GHJ@JDPPD
QXPVWU NDSSDSLFW >@5
QXPVWU JDPPDSLFW
QXPVWU 5SLFW >P@ @
$33(1',;
WH[W > F\DZ
FODW
75($'6,0
QXPVWU F\DZ FEHQG
QXPVWU FODW FS
QXPVWU FEHQG
QXPVWU FS @
WH[W > 5,*+752: @ D[LV >@ JULG ILJXUH SORW [HO5\HO5 [ODEHO [DQGORQJLWXGLQDOGHIOHFWLRQV>P@ \ODEHO ODWHUDOGHIOHFWLRQV>P@ WH[W > DOSKD
>GHJ@NDSSD LIULJLG
QXPVWU DOSKDSLFW >GHJ@JDPPD
QXPVWU NDSSDSLFW >@5
QXPVWU JDPPDSLFW
QXPVWU 5SLFW >P@ @
WH[W >
5,*,'
F
QXPVWU F P1 @
HOVH WH[W > F\DZ
FS
QXPVWU F\DZ FEHQG
QXPVWU FEHQG FODW
QXPVWU FODW
QXPVWU FS @
HQG WH[W > 5,*+752:EURZD
QXPVWU EURZD @
D[LV >@
ILJXUH SORW [\/ [ODEHO [>P@ \ODEHO GEOXH PX S]OEOXH SJUHHQ S[UHG S\>1P@ WH[W > DOSKD
GHJNDSSD
WH[W > F\DZ
FODW
QXPVWU DOSKDSLFW GHJJDPPD
QXPVWU NDSSDSLFW 5
QXPVWU F\DZ FEHQG
QXPVWU FODW FS
WH[W > WKHWD
QXPVWU FEHQG
QXPVWU FS @
QXPVWU WKHWD @
WH[W > HSV\SULPH
HSV\UJDPPD
QXPVWU JDPPDSLFW
QXPVWU 5SLFW P @
QXPVWU HSV\SULPH
QXPVWU HSV\UJDPPD @
WH[W > HSV\)\
QXPVWU HSV\)\ HSVJDPPD
WH[W > HSVGUJDP
QXPVWU HSVGUJDP \
QXPVWU HSVJDPPD @
QXPVWU \ PP @
WH[W > /()752: @ D[LV >@ JULG
ILJXUH SORW [HO/\HO/ [ODEHO [DQGORQJLWXGLQDOGHIOHFWLRQV>P@ \ODEHO ODWHUDOGHIOHFWLRQV>P@ WH[W > DOSKD
GHJNDSSD LIULJLG
QXPVWU DOSKDSLFW GHJJDPPD
QXPVWU NDSSDSLFW 5
QXPVWU JDPPDSLFW
QXPVWU 5SLFW P @
WH[W >
5,*,'
F
QXPVWU F P1 @
HOVH WH[W > F\DZ
FS
QXPVWU F\DZ FEHQG
QXPVWU FEHQG FODW
QXPVWU FODW
QXPVWU FS @
HQG WH[W > )\
QXPVWU URXQG )\ 10]
QXPVWU URXQG 0] 1P @
$33(1',;
75($'6,0
WH[W > /()752: @ D[LV >@ HQG
HQGLISLFWXUH
HOVH SORWVZLWKUXQQLQJYDULDEOH LIDOSKDUXQV [PLQ
DOSKGHJPLQU [PD[
DOSKGHJPD[U [WH[W
)[PLQ
PX )] )[PD[
0PLQ
D )\PLQ 0PD[
WPLQ
D WPD[
SDUWH[W
> &)NDS
PX )] )\PLQ
D D[
VOLSDQJOHDOSKD>GHJ@
PX )] )\PD[
PX )]
D )\PD[
D\
QXPVWU &)NDSSD)]
)]&)DOI
QXPVWU &)DOSKD)]
)] @
HQG LINDSSDUXQV [PLQ
NDSSDPLQU [PD[
NDSSDPD[U [WH[W
)[PLQ
PX )] )[PD[
0PLQ
D )\PLQ 0PD[
WPLQ
D WPD[
SDUWH[W
> &)NDS
PX )] )\PLQ
D D[
ORQJLWXGLQDOVOLSUDWLRNDSSD>@ PX )] )\PD[
PX )]
D )\PD[
D\
QXPVWU &)NDSSD)]
)] @
HQG LIDSKLWUXQV [PLQ
[PD[
DSKLWPD[U [WH[W
)[PLQ
PX )] )[PD[
0PLQ
D )\PLQ 0PD[
WPLQ
D WPD[
SDUWH[W
D D[
> &)NDS
QRQGLPWXUQVOLSDSKLW
PX )] )\PLQ
D5>@
PX )] )\PD[
PX )]
D )\PD[
D\
QXPVWU &)NDSSD)]
)] @
HQG
[O
[PD[[PLQ )[O
LINDSSDSDU
)\PD[)\PLQ 0O
0PD[0PLQ WO
WPD[WPLQ
LIDOSKDUXQV
)[PD[)[PLQ )\O
SDUWH[W
> D5
QXPVWU DSKLW JDPPD
QXPVWU JDPPDGHJ >GHJ@ @
HOVH
SDUWH[W
> DOSKD
QXPVWU DOSKDGHJ JDPPD
QXPVWU JDPPDGHJ >GHJ@ @
HQG
LIMPD[
SDUWH[W
> NDSSD
QXPVWU NDSSDPLQ @
HOVH SDUWH[W
> NDSSDPLQ
QXPVWU NDSSDPLQ GHOWDNDSSD
QXPVWU GHOWDNDSSD >@ @
HQG HOVHLIJDPPDSDU
LIDOSKDUXQV
SDUWH[W
> D5
HOVHLINDSSDUXQV
QXPVWU DSKLW NDSSD
QXPVWU NDSSD @
SDUWH[W
> DOSKD
QXPVWU DOSKDGHJ >GHJ@ D5
QXPVWU DSKLW @
SDUWH[W
> DOSKD
QXPVWU DOSKDGHJ >GHJ@ NDSSD
HOVH QXPVWU NDSSD @
$33(1',;
HQG
LIMPD[
75($'6,0
SDUWH[W
> JDPPD
QXPVWU JDPGHJPLQ @ >GHJ@
HOVH SDUWH[W
> JDPPDPLQ
QXPVWU JDPGHJPLQ GHOWDJDPPD
QXPVWU GHOWJDPGHJ >GHJ@ @ HQG HOVHLIDOSKDSDU
LINDSSDUXQV SDUWH[W
> D5
QXPVWU DSKLW JDPPD
QXPVWU JDPPDGHJ >GHJ@ @
HOVH
SDUWH[W
> NDSSD
QXPVWU NDSSD JDPPD
QXPVWU JDPPDGHJ >GHJ@ @
HQG
LIMPD[
SDUWH[W
> DOSKD
QXPVWU DOSKGHJPLQ >GHJ@ @
HOVH SDUWH[W
> DOSKDPLQ
QXPVWU DOSKGHJPLQ GHOWDDOSKD
QXPVWU GHOWDOSKGHJ >GHJ@ @ HQG HOVHLIDSKLWSDU
LIDOSKDUXQV
SDUWH[W
HOVH
SDUWH[W
> NDSSD
QXPVWU NDSSD JDPPD
QXPVWU JDPPDGHJ >GHJ@ @
> DOSKD
QXPVWU DOSKDGHJ >GHJ@ JDPPD
QXPVWU JDPPDGHJ >GHJ@ @
HQG
LIMPD[
SDUWH[W
> DSKLW
QXPVWU DSKLWPLQ @
HOVH SDUWH[W
> DSKLWPLQ
QXPVWU DSKLWPLQ GHOWDDSKLW
QXPVWU GHOWDSKLW >@ @
HQG HOVH LIDOSKDUXQV
> D5
HOVHLINDSSDUXQV
SDUWH[W
QXPVWU DSKLW NDSSD
QXPVWU NDSSD @
SDUWH[W
> DOSKD
QXPVWU DOSKDGHJ >GHJ@ D5
QXPVWU DSKLW @
SDUWH[W
> DOSKD
QXPVWU DOSKDGHJ >GHJ@ NDSSD
HOVH
HQG
SDUWH[W
> JDPPD
QXPVWU NDSSD @
QXPVWU JDPGHJPLQ >GHJ@ @
HQG LIULJLG
WH[WWLWOH
> 9F
F\DZ
QXPVWU 9F PVPX
QXPVWU F\DZ FEHQG
QXPVWU PX DPX
QXPVWU FEHQG FODW
QXPVWU DPX
QXPVWU FODW @
HOVH WH[WWLWOH F
> 9F
QXPVWU 9F PVPX
QXPVWU F \
QXPVWU PX DPX
QXPVWU DPX
QXPVWU \ ULJLGFDUFDVV @
HQG SDUWH[W SO\VW
> D
QXPVWU D E
QXPVWU E EURZ
QXPVWU DOSKDGHJSO\ FRQLF
QXPVWU EURZ )]
QXPVWU JDPPDGHJFRQ LLWHQG
QXPVWU )]
QXPVWU LLWHQG @
$33(1',;
75($'6,0
ILJXUH SORW [D\I\ \ODEHO VLGHIRUFH)\>1@ [ODEHO [WH[W WLWOH WH[WWLWOH WH[W [PLQ [O)\PLQ D\ )\O> QUURZV WH[W [PLQ [O)\PLQ D\ )\O> WKHWD
QXPVWU QURZ @
QXPVWU WKHWD @
WH[W [PLQ D[ [O)\PLQ )\OSDUWH[W WH[W [PLQ D[ [O)\PLQ )\OSDUWH[W WH[W [PLQ D[ [O)\PLQ )\OSDUWH[W WH[W [PLQ D[ [O)\PLQ )\OSDUWH[W D[LV >[PLQ[PD[)\PLQ)\PD[@ JULG
ILJXUH SORW [D\I[ \ODEHO ORQJLWXGLQDOIRUFH)[>1@ [ODEHO [WH[W WLWOH WH[WWLWOH WH[W [PLQ [O)\PLQ )\O> QUURZV WH[W [PLQ [O)\PLQ )\O> WKHWD
QXPVWU QURZ @
QXPVWU WKHWD @
WH[W [PLQ D[ [O)\PLQ )\OSDUWH[W WH[W [PLQ D[ [O)\PLQ )\OSDUWH[W WH[W [PLQ D[ [O)\PLQ )\OSDUWH[W D[LV >[PLQ[PD[)[PLQ)[PD[@ JULG
ILJXUH SORW [D\P] \ODEHO DOLJQWRUTXH0]>@ [ODEHO [WH[W WLWOH WH[WWLWOH WH[W [PLQ [O0PLQ 0O> QUURZV WH[W [PLQ [O0PLQ 0O> WKHWD
QXPVWU QURZ QUHOHP
QXPVWU WKHWD @
WH[W [PLQ [O0PLQ 0O> HSV\SULPH QXPVWU HSV\SULPH HSV\UJDPPD
WH[W [PLQ [O0PLQ 0O> HSV\)\ QXPVWU HSV\)\ HSVJDPPD
QXPVWU HSVJDPPD @
WH[W [PLQ [O0PLQ 0O> HSVGUJDP QXPVWU HSVGUJDP \
QXPVWU HSV\UJDPPD @
QXPVWU \ PP @
WH[W [PLQ [O0PLQ D\ 0OSDUWH[W WH[W [PLQ [O0PLQ D\ 0OSDUWH[W WH[W [PLQ [O0PLQ D\ 0OSDUWH[W WH[W [PLQ [O0PLQ D\ 0OSDUWH[W D[LV >[PLQ[PD[0PLQ0PD[@ JULG
ILJXUH SORW [D\W \ODEHO SQHXPWUDLO>P@ [ODEHO [WH[W [DPHDQ\WGLII WLWOH WH[WWLWOH WH[W [PLQ [OWPLQ WO> QUURZV
QXPVWU QURZ @
WH[W [PLQ [OWPLQ WOSDUWH[W WH[W [PLQ [OWPLQ WOSDUWH[W WH[W [PLQ [OWPLQ WOSDUWH[W D[LV >[PLQ[PD[WPLQWPD[@
QXPVWU Q @
$33(1',;
75($'6,0
JULG
ILJXUH SORW \I[\I\ \ODEHO VLGHIRUFH>1@ [ODEHO ORQJLWXGLQDOIRUFH>1@ WLWOH WH[WWLWOH WH[W )[PLQ )[O)\PLQ )\O> QUURZV
QXPVWU QURZ @
WH[W )[PLQ )[O)\PLQ )\OSDUWH[W WH[W )[PLQ )[O)\PLQ )\OSDUWH[W WH[W )[PLQ )[O)\PLQ )\OSDUWH[W WH[W )[PLQ )[O)\PLQ )\OSDUWH[W D[LV > )[PLQ )[PD[)\PLQ)\PD[@ JULG
ILJXUH SORW \I[\P] \ODEHO DOLJQLQJWRUTXH>1P@ [ODEHO ORQJLWXGLQDOIRUFH>1@ WLWOH WH[WWLWOH WH[W )[PLQ )[O0PLQ 0O> QUURZV
QXPVWU QURZ @
WH[W )[PLQ )[O0PLQ 0O> HSV\SULPH
HSV\UJDPPD
WH[W )[PLQ )[O0PLQ 0O> HSV\)\
HSVJDPPD
QXPVWU HSV\)\
QXPVWU HSVJDPPD @
WH[W )[PLQ )[O0PLQ 0O> HSVGUJDP
\
QXPVWU \ PP @
WH[W )[PLQ )[O0PLQ 0OSDUWH[W WH[W )[PLQ )[O0PLQ 0OSDUWH[W WH[W )[PLQ )[O0PLQ 0OSDUWH[W WH[W )[PLQ )[O0PLQ 0OSDUWH[W D[LV > )[PLQ )[PD[0PLQ0PD[@ JULG
HQG FOHDU
QXPVWU HSV\SULPH
QXPVWU HSV\UJDPPD @
QXPVWU HSVGUJDP
$SSHQGL[
6:,)7
SDUDPHWHUV
$SS3DUDPHWHUYDOXHVRI0DJLF)RUPXODDQG6:,)7PRGHO
7DEOH$ 5 9EDU,62VLJQGHILQLWLRQ$OVRFI$SS
5R
UR
P)]R
)1R
1PR
NJ9R
PV
S&[
S'[
S'[
S([
S([
S([
S([
S.[
S.[
S.[
S+[
S+[
S9[
S9[
U%[
U%[
U&[
U+[
TV[
TV[
TV[
S&\
S'\
S'\
S'\
S(\
S(\
S(\
S(\
S.\
S.\
S.\
S+\
S+\
S+\
S9\
S9\
S9\
S9\
U%\
U%\
U%\
U&\
U+\
U9\
U9\
U9\
U9\
U9\
U9\
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+]
VV]
VV]
VV]
VV]
T,D\
TPD
TFE[]
TNE[]
TFE
T,D[]
TPE
TFE\
TNE\
TFE
T,E\
TPF
TFF[
TNF[
TFF
T,E[]
TFF\
TNF\
TNE
T,F
TNE
TNF
TV\
TD
%UHII
T)F[
T9
TV\
TD
'UHII
T)F\
T)]
TV\
TEY[]
)UHII
T)F[
T)]
TEY
T)F\
T9
î
$33(1',;
6:,)73$5$0(7(56
$SS1RQ'LPHQVLRQDOL]DWLRQ ZLWKVRPHUHPDUNV
7KHTXDQWLWLHVOLVWHGLQ7DEOH$DUHQRQGLPHQVLRQDO7KH\DUHXVHGLQWKH IROORZLQJHTXDWLRQV 0DJLF)RUPXOD (TV (( H[FHSW (( 6:,)7 (TV H[FHSW 7KH LQHUWLD TXDQWLWLHV KDYH EHHQ PDGH QRQGLPHQVLRQDO E\ GLYLGLQJ E\ WKH UHIHUHQFHPDVVPRRUWKHUHIHUHQFHPRPHQWRILQHUWLD,R)RUPRWKHWRWDO W\UH PDVVKDVEHHQWDNHQ7KHTXDQWLW\,RLVGHILQHGDV
,R ' PR5R
>NJP @
$
&RQVHTXHQWO\ZHKDYHWKHQRQGLPHQVLRQDOSDUDPHWHUVVKRZQLQWKHOLVWDERYH
PD PR
T PD '
HWFDQG
T ,D\ '
,D\
PR 5R
HWF
$
$VKDVEHHQLQWURGXFHGDOUHDG\LQFRQQHFWLRQZLWK(TV WKHQRQ GLPHQVLRQDOL]DWLRQ RI WKH VWLIIQHVVHV DQG GDPSLQJ FRHIILFLHQWV LV UHDOL]HG E\ GLYLGLQJWKHOLQHDU WUDQVODWLRQDO VWLIIQHVVHVDQGGDPSLQJFRHIILFLHQWVE\
)]R 5R
FWR '
>1P@DQG
NWR '
PR ) ]R 5R
>1VP@
$
DQGWKHURWDWLRQDOVWLIIQHVVHVDQGGDPSLQJFRHIILFLHQWVE\
FUR ' )]R5R
>1PUDG@DQG
NUR ' P R )]R 5R
>1PVUDG@
$
$VDFRQVHTXHQFHZHKDYHIRUH[DPSOH
FE[ FWR F ' E FUR
T FE[ '
TFE
HWFDQG
HWFDQG
NE[ NWR N ' E NUR
T NE[ '
HWF
$
TNE
HWF
$
5HPDUNV ,Q WKH IDFWRUV NE[]
DUH VHW HTXDO WR ]HUR DV UROOLQJ
UHVLVWDQFHKDVEHHQDFFRXQWHGIRUDOUHDG\E\WKHPRPHQW0\ 7KHPDVVPDLVWKHPDVVRIWKDWSDUWRI WKHW\UHWKDWDFFRUGLQJWRWKHPRGHO PRYHVZLWKWKHZKHHOULPRUD[OH7KHTXDQWLW\,D\LVWKHPRPHQWRILQHUWLDRI WKDWSDUWRIWKHW\UHWKDWURWDWHVDERXWWKH D[LVDVLILWZHUHDWWDFKHGDVDULJLG ERG\WRWKHZKHHOULP7KHW\UHLQHUWLDWKDWLVDOORFDWHGWRWKHEHOWLVGHQRWHGZLWK VXEVFULSW E7KHLQHUWLDGHQRWHGZLWKVXEVFULSW FLVDVVXPHGWRPRYH ZLWKWKH FRQWDFWSDWFK7KHLQWHUDFWLRQSDUDPHWHUVT)F[DQGT)F\KDYHEHHQFKRVHQHTXDO WR]HUR7KH\ZRXOGKDYHWDNHQYDOXHV DQGLIT)F[
T)F\
$33(1',;
6:,)73$5$0(7(56
$SS(VWLPDWLRQRI6:,)73DUDPHWHU9DOXHV XVLQJDOLPLWHGVHWRI VHPLVWDWLF PHDVXUHPHQWV &RQWULEXWHGE\,-0%HVVHOLQN712$XWRPRWLYH
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
)RUFHDQGPRPHQWWHVWLQJ )LWWLQJRIWKH0DJLF)RUPXOD )[)\0[0\0]
I )]
9[
7KHVRIWZDUHSURJUDP0)7RRODOVRGHYHORSHGE\712$XWRPRWLYHPD\ EHXVHGWRILWWKH0DJLF)RUPXODSDUDPHWHUV7KLVSURJUDP UHTXLUHVWKDWWKH PHDVXUHPHQWGDWDLVSURYLGHGLQWKHIRUPRI7<'(;ILOHVDVWDQGDUGLVHGIRUPDW DVGHYHORSHGE\WKHLQWHUQDWLRQDO7<'(; 7\UH'DWD([FKDQJH ZRUNJURXSFI 2RVWHQHWDO 1RWH WKHFRQWULEXWLRQRIWKHIRUZDUGYHORFLW\9[LVSUHVHQWLQWKHUROOLQJ UHVLVWDQFHIRUPXOD
I U ' TV\ % TV\ _9[9R_ % TV\ 9[9R
WKDW LV XVHG LQ FRQQHFWLRQ ZLWK (TV 2I FRXUVH LQ FDVHV VXFK DV PRYLQJRQZHWURDGVWKHIULFWLRQFRHIILFLHQWPD\EHIRUPXODWHGDVIXQFWLRQVRI WKHVSHHGFI(T (
/RDGHGUDGLXVHIIHFWLYHUROOLQJUDGLXV 7KHW\UHQRUPDOGHIOHFWLRQ
]
LVWKHGLIIHUHQFHEHWZHHQWKHIUHHW\UHUDGLXVRIWKH
URWDWLQJW\UHU DQGWKHORDGHGW\UHUDGLXVUO
U
] ' R%
U & UO ' U & U O
7KHIROORZLQJVHWRIHTXDWLRQVQHHGVWREHILWWHG
$33(1',;
6:,)73$5$0(7(56
D FHQWULIXJDOJURZWKRIWKHW\UHIUHHUDGLXV
U FRHIILFLHQWT9
U ' T9 UR UR 9R
E YHUWLFDOIRUFHGHIOHFWLRQHTXDWLRQ FRHIILFLHQWV TYT)F[T)F\T)]DQGT)]
)]' % T9 _
UR ) ) & T)F[ [ & T)F\ \ 9R )]R )]R
_
]
T)]
UR
]
% T)]
UR
)]R
F HIIHFWLYHUROOLQJUDGLXVVLPLODUWR FRHIILFLHQWV %')UHII
) ]R & 'UHII DUFWDQ %UHII )] &)] )]R
UH ' UR% U &
]
% )UHII
&)] )]R
]
$
ZLWKWKHYHUWLFDOVWLIIQHVVRIWKHVWDQGLQJW\UHDWQRPLQDOORDG)]RDVGHULYHGIURP (T
& )] '
)]R T % T)] UR )]
1RWHV )]R9RDQGUR
E
5RDUHUHIHUHQFHSDUDPHWHUV WKH\DUHMXVWXVHGWRPDNH
WKH FRHIILFLHQWV GLPHQVLRQOHVV $Q DGGLWLRQDO SDUDPHWHU TUH PD\ EH LQWURGXFHGWKDWPXOWLSOLHGZLWKURLQ(TV JLYHVDQLPSURYHGYDOXH IRUWKHIUHHXQORDGHGUDGLXV 7KH PHDVXUHPHQW GDWD IRU DVVHVVLQJ WKH ORDGHG DQG WKH HIIHFWLYH UROOLQJ UDGLXVW\SLFDOO\FRQVLVWVRIYDOXHVIRUUOUH9)[)\)]FDUULHGRXWIRUDQXPEHU RIGLIIHUHQWYHUWLFDOORDGVIRUZDUGYHORFLWLHVDQGSRVVLEO\ORQJLWXGLQDODQGRU ODWHUDOIRUFHV%DVHGRQWKHUHODWLRQ9
UHLWLVVXIILFLHQWWRVSHFLI\WZRRXWRI
WKUHH YDULDEOHV IRUZDUG YHORFLW\ 9 DQJXODU YHORFLW\
DQG HIIHFWLYH UROOLQJ
UDGLXVUH7KHGDWDIRU)[DQG) \ FDQEHFRQVLGHUHGRSWLRQDOIRUSDVVHQJHUFDU W\UHVEXWJHQHUDOO\VKRXOGEHLQFOXGHGIRUUDFLQJW\UHV )LWWLQJRIWKHSDUDPHWHUVLVJHQHUDOO\GRQHLQWZRVWHSV,QRUGHUWRREWDLQ PD[LPXPDFFXUDF\IRUWKHORDGHGUDGLXVWKHHTXDWLRQV DQG DUH ILWWHGILUVW,QDVHFRQGVWHSWKHFRHIILFLHQWVIRUWKHHIIHFWLYHUROOLQJUDGLXVDV JLYHQE\HTXDWLRQ D DUHGHWHUPLQHG
&RQWDFWOHQJWK ,Q 6:,)7 WKH IROORZLQJ IRUPXOD LV XVHG IRU WKH FRQWDFW OHQJWK IRUPXOD D UHSUHVHQWVKDOIRIWKHFRQWDFWOHQJWK FRHIILFLHQWV TDDQGTD
$33(1',;
6:,)73$5$0(7(56
)] ) % TD ] U R )]R )]R
D ' TD
&RQVHTXHQWO\ PHDVXUHPHQW GDWD VKRXOG SURYLGH KDOI WKH FRQWDFW OHQJWK DV D IXQFWLRQRIWKHYHUWLFDOORDG)],IQRRURQO\YHU\OLPLWHGGDWDLVDYDLODEOHWKH IROORZLQJIRUPXODPD\JLYHDJRRGHVWLPDWH ZKHUH
D ' SD SD
]
UR
%
]
UR
]
LVWKHW\UHGHIOHFWLRQ
UR
$
7KLVIRUPXODZDVSURSRVHGE\%HVVHOLQN DQGEHWWHUDFNQRZOHGJHVWKH IDFWWKDWWKHFRQWDFWOHQJWKLVDJHRPHWULFDOSURSHUW\DQGSULPDULO\DIXQFWLRQRI WKH W\UHGHIOHFWLRQ7KHVXJJHVWHGSDUDPHWHUYDOXHVIRUDSDVVHQJHUDQGOLJKW WUXFNW\UHVDUHUHVSHFWLYHO\
SD
DQGSD
RQDIODWVXUIDFH
SD
DQGSD
RQDQH[WHUQDOGUXPZLWKDGLDPHWHURIP
,WKDVEHHQIRXQGWKDWWKLVIRUPXODLVYDOLGIRUDIDLUO\ZLGHUDQJHRIW\UHVDQG FDQDOVREHXVHGIRUGLIIHUHQWW\UHSUHVVXUHVZLWKRXWFKDQJLQJWKHFRHIILFLHQWV VHH)LJ$7KLVILJXUHDOVRFOHDUO\LOOXVWUDWHVWKDWRQ DQ H[WHUQDOGUXPWKH FRQWDFWOHQJWKLVVKRUWHUFRPSDUHGWRWKHVLWXDWLRQRQDIODWURDGVXUIDFH
SD = 0.35, SD = 2.25 Eq.(A3.8) SD = 0.45, SD = 1.25
0.4
half of contact length /
UR D
drum 2.4 bar drum 3.1 bar flat plate 2.4 bar flat plate (different tyre, Fig.9.30)
0.3
0.2
plate drum
0.1
0 0
0.02
0.04
0.06
tyre deflection
0.08
ρ] /UR
0.1
0.12
0.14
)LJ$ 0HDVXUHPHQWDQGHPSLULFDOIRUPXODUHVXOWVIRUKDOIWKHFRQWDFWOHQJWKDVDIXQFWLRQ RIW\UHGHIOHFWLRQSDVVHQJHUFDUW\UHV
$33(1',;
6:,)73$5$0(7(56
5HOD[DWLRQOHQJWKVRYHUDOOORQJLWXGLQDODQGODWHUDOVWLIIQHVV $SDUW IURP DQ DFFXUDWH UHSUHVHQWDWLRQ RI WKH KLJK IUHTXHQF\ UHVSRQVHV LW LV LPSRUWDQWWKDWWKHORZIUHTXHQF\WUDQVLHQWEHKDYLRXU LHUHOD[DWLRQHIIHFWVRI WKH W\UH LVLQFOXGHGFRUUHFWO\0HDVXUHPHQWVRIWKHUHOD[DWLRQOHQJWKVFDQEH XVHGWRDVVHVVWKHUHVXOWLQJORQJLWXGLQDODQGODWHUDOVWLIIQHVVRIWKHW\UHFDUFDVV WKDWLVH[FOXGLQJWKHWUHDG DWJURXQGOHYHO)RU6:,)7WKHIROORZLQJHTXDWLRQV KROGIRUWKHORQJLWXGLQDODQGODWHUDOUHOD[DWLRQOHQJWKVUHVSHFWLYHO\FI(TV DQG ZLWKP
DW YDQLVKLQJVOLS DQGWKHUHODWLYHO\VPDOOHIIHFWRIWKH
WRUVLRQDOFRPSOLDQFHF QHJOHFWHG
&) %D F[ &) \' F %D \ ['
$
$
ZKHUH
[ &
)
F
[
\ &
ORQJLWXGLQDOVOLSVWLIIQHVVRIWKHW\UH ORQJLWXGLQDOFDUFDVVVWLIIQHVVDWJURXQGOHYHO ODWHUDOUHOD[DWLRQOHQJWK
)
FRUQHULQJVWLIIQHVVRIWK\HW\UH
\
ODWHUDOFDUFDVVVWLIIQHVVDWJURXQGOHYHO
F
ORQJLWXGLQDOUHOD[DWLRQOHQJWK
D
KDOIRIWKHFRQWDFWOHQJWK
,QWKHODWHUDOGLUHFWLRQDOOSDUDPHWHUVPD\EHNQRZQ
\ &)
F D EXWXVXDOO\
\
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
&DUFDVVFRPSOLDQFHV ,Q6:,)7WKHULJLGEHOWULQJLVHODVWLFDOO\VXVSHQGHGZLWKUHVSHFWWRWKHULPLQDOO GLUHFWLRQV 7KH FRQWDFW SDWFK LV HODVWLFDOO\ DWWDFKHG WR WKH ULQJ 1RUPDOO\ WKH
$33(1',;
6:,)73$5$0(7(56
YDULRXVVWLIIQHVVYDOXHVDUHGHWHUPLQHGIURPG\QDPLFH[SHULPHQWDOUHVXOWV%\ XVLQJWKHVHYDOXHVLWLVSRVVLEOHWRFDOFXODWHWKHLUUHVSHFWLYHFRQWULEXWLRQWRWKH RYHUDOOFDUFDVVGHIOHFWLRQDWJURXQGOHYHOZKHQLWLVDVVXPHGWKDWDORQJLWXGLQDO RUODWHUDOIRUFHLVDSSOLHGDQGWKDWWKHULPLVKHOGIL[HGFI)LJ$ 7DEOH$'LVWULEXWLRQRIORQJLWXGLQDODQGODWHUDOFDUFDVVFRPSOLDQFHFRPSRQHQWV
ORQJLWXGLQDODWJURXQGOHYHO
W\UH
W\UH
W\UH
UXOHRIWKXPE
ULJLGULQJWUDQVODWLRQ
D
ULJLGULQJURWDWLRQ
E
FRQWDFWSDWFKWUDQVODWLRQ F
ODWHUDODWJURXQGOHYHO
W\UH
W\UH
W\UH
UXOHRIWKXPE
ULJLGULQJWUDQVODWLRQ
D
ULJLGULQJURWDWLRQ
E
FRQWDFWSDWFKWUDQVODWLRQ F
7DEOH $ VKRZV WKH UHODWLYH FRQWULEXWLRQV D E DQG F WKDW KDYH EHHQ DVVHVVHGIRUWKUHHGLIIHUHQWW\UHV%DVHGRQWKHVHGDWDLWPD\EHFRQFOXGHGWKDW DOWKRXJK WKH FRPSOLDQFH YDOXHV WKHPVHOYHV DUH GLIIHUHQW WKH UHODWLYH FRQWULEXWLRQVWRWKHRYHUDOOFDUFDVVGHIOHFWLRQDUHIDLUO\FRQVWDQW6LQFHXQGHU
VLGHYLHZ
rim fixed
UHDUYLHZ
belt deflected (rigid ring)
belt deflected (rigid ring) rim fixed
FE[
FE\
FEθ
FEγ FF[
D
E
F
total circumferential carcass deflection
FF\
contact patch deflected
)[
D
E
contact patch deflected
F
total lateral carcass deflection
)LJ$/RQJLWXGLQDODQGODWHUDOGHIOHFWLRQVRIWKHULJLGULQJDQGFRQWDFWSDWFK
)\
$33(1',;
6:,)73$5$0(7(56
VWHS WKH RYHUDOO ORQJLWXGLQDO DQG ODWHUDO FDUFDVV VWLIIQHVVHV KDYH EHHQ GHWHUPLQHGLWLVQRZSRVVLEOHWRDVVHVVDSSUR[LPDWHLQGLYLGXDOVWLIIQHVVYDOXHV XVLQJWKHVXJJHVWHGUXOHRIWKXPE $GGLWLRQDOPHDVXUHPHQWGDWD HJ \DZ VWLIIQHVVRU)(0PRGHOUHVXOWV PD\DOVREHXVHGWRHQKDQFHWKHHVWLPDWHVRUWRJDLQDGGLWLRQDOFRQILGHQFHLQWKH VWLIIQHVVGHIOHFWLRQGLVWULEXWLRQ
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
&DUFDVVGDPSLQJ *HQHUDOO\WKHH[DFWDPRXQWRIGDPSLQJRI WKH W\UHLVYHU\GLIILFXOWWRDVVHVV H[SHULPHQWDOO\ )RU LQVWDQFH LW LV REVHUYHG WKDW D ODUJH GLIIHUHQFH LQ YHUWLFDO GDPSLQJH[LVWVEHWZHHQDW\UHWKDWVWDQGVVWLOODQGDW\UHWKDWUROOV XQGHUUROOLQJ FRQGLWLRQVWKHDSSDUHQWGDPSLQJPD\EH D IDFWRUVPDOOHUFRPSDUHGWRWKH GDPSLQJRIDW\UHVWDQGLQJVWLOODOVRFI-LDQPLQHWDO $VLPSOHPRGHO ZLWK ILQLWH FRQWDFW OHQJWK DQG SURYLGHG ZLWK UDGLDO GU\ IULFWLRQ GDPSHUV 3DFHMND D PD\H[SODLQWKLVSKHQRPHQRQ %DVHGRQH[SHULHQFHWKHIROORZLQJJXLGHOLQHPD\EHSURYLGHG ZKHQWKHW\UH LV QRW LQ FRQWDFW ZLWK WKH JURXQG DQG WKH ULP LV IL[HG WKH GDPSLQJ ZLOO EH UHODWLYHO\ORZ7\SLFDOYDOXHVRIWKHGDPSLQJFRHIILFLHQWVOLHLQWKHUDQJHRIWR RIWKHFRUUHVSRQGLQJFULWLFDOGDPSLQJFRHIILFLHQWV*HQHUDOO\WKHPRGHVWKDW FRQWDLQDODUJHWUDQVODWLRQDOFRPSRQHQWDUHPRUHKHDYLO\GDPSHGWKDQWKHPRGHV ZLWKDODUJHURWDWLRQDOFRPSRQHQW
