Numerical Analysis of Freeway Traffic Flow Dynamics under Multiclass Drivers

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1 Zuojn Zhu, Gng-len Chng nd Tongqng Wu Numecl Anlyss of Feewy Tffc Flow Dynmcs unde Mulclss Dves Zuojn Zhu, Gng-len Chng nd Tongqng Wu Depmen of Theml Scence nd Enegy Engneeng, Unvesy of Scence nd Technology of Chn, Hefe, Anhu 006, P.R. Chn. Eml: Fx: , Phone: Pofesso, Depmen of Cvl Engneeng, Unvesy of Mylnd, USA. Eml: PhD Cndde, Depmen of Opeons Resech nd Fnncl Engneeng, Pnceon Unvesy, Pnceon, NJ, 08540, USA. Eml: Absc: Ths ppe pesens fs-ode mul-clss model fo llusng feewy ffc dynmcs, especlly egdng he empol nd spl von of boh denses nd flow e long feewy segmens whou he dsubnce fom mp flows. The poposed mcoscopc model s gounded on he ssumpon h feewy ffc my conss of mulple clsses of dves who, chezed wh he unque speed densy elon, e lely o ec dffeenly unde he sme dvng envonmen. The dsbuon of vous clsses of dves nd he dffeences n espondng o peceved dvng condons my conbue sgnfcnly o he obseved feewy ffc dynmcs h emns o be bee explned by exsng ffc flow heoes. The numecl soluon nd smulon esuls epoed n hs sudy, howeve, ndce h ou poposed fs-ode mulclss model offes he poenl o exploe he complex necons beween feewy dves nd he collecve mpc on ffc flow pens. INTRODUCTION Recognzng he lmons of he exsng ffc flow heoy, ffc eseches ove he ps sevel decdes hve devoed consdeble effos on developng elble model h cn elsclly cpue he complex ffc flow dynmcs. One of he pmy esech decons s o eplce he fs-ode hydodynmc ffc model wh hgh-ode dffeence o dffeenl sysem of equons. Fo exmple, Zhng () hs pesened new connuum ffc heoy nd nvesged s wve popees. Helbng e l. () hve developed ffc flow smulo, clled MASTER, bsed on gs-nec ffc equons. They hve lso developed new clss of molecul-dynmcs-le mcoscopc ffc models bsed on mes o collsons (Helbng, e l., ()). Some eseches n ecen yes hve emped o exend he hgh-ode mcoscopc model o ncopoe mulclss dves n ffc flow fomulons. Exmples of poneeng sudes long hs lne e due o Ngn (4) Hoogendoon nd Bovy (5). Whle he fome hve chcezed he dscepncy beween he c-followng behvo of vous ypes of dves wh he dely me, he le hs employed he gs-nec equons o model he mulclss ffc flow necons. A concse evew of esech developmens long hs lne cn be found n he wo of Kuhne nd Mchlopoulos (6). Despe he sgnfcn pogess on hgh-ode mcoscopc ffc flow models, he complex fomulons nd he numbe of pmees o be clbed my degde he poenl fo feld pplcons. Thus, nsed of developng hgh-ode mhemcl ffc elons, some eseches suggesed h he effo should be devoed o cpung he dscepncy of dve behvo n mcoscopc model fomulon (Dgnzo, (7)). A smple fs-ode mhemcl model my be suffcen fo cpung ffc flow dynmcs f he esponse of dffeen ypes of dves nd he collecve mpcs on he ffc condons hve been popely en no ccoun. An exmple of sudes long hs lne cn be found n wo-phse ffc flow model poposed ecenly by Zhu nd Wu (8) n whch he fee-flow speed n fs-ode mcoscopc ffc model s ssumed o vy coss dvng populons. In fc, ecen empcl sudy by Cssdy nd Much (9) hs lso concluded h unde congesed queued condons, hee exss well defned elon beween he flow e nd he cumuled numbe of vehcles on feewy segmen, nd he fs-ode hydodynmc ffc flow heoy s suffcen fo llusng he queue evoluon. Dgnzo (0-) hs poposed behvol heoy fo mullne feewy ffc flows, nd gued h such descpve model s suffcen fo developmen of compue pogms. Recenly, Wong e l. () hve exended he donl LWR ffc heoy (Lghhll nd Whhm, (); Rchds, (4)) wh he Lx-Fedchs scheme, nd clmed h he exended model cn expln some complce phenomenon h cnno be cpued wh he LWR model. Ths sudy pesens ou ecen wo long he sme lne, h s, fs-ode mcoscopc ffc flow model fo mulclss dves. The model pesened heefe nends o ovecome some lmons of he LWR TRB 00 Annul Meeng CD-ROM Ppe evsed fom ognl subml.

2 Zuojn Zhu, Gng-len Chng nd Tongqng Wu model whou usng complex hgh-ode fomulons. The pmy objecve of hs sudy, n fc, s sml o he ecen wo by Wong e l., bu wh dffeen modelng mehodology nd poenlly moe effecve numecl mehod o solve ou poposed mulclss ffc flow model. Wh smple fs-ode fomulon, ou poposed model s cpble of llusng some vl feewy ffc dynmcs such s he commonly obseved oscllons of flow e nd densy long feewy segmen ove me, even whou he dsubnce of mp flows nd hvng unfom nl ffc condons. The empol nd spl von of flow e nd densy bsed on ou poposed model wll vy wh dffeen dsbuons of dvng populons n he ffc sem. TRAFFIC MODEL FOR MULTI-CLASS DRIVERS Fo convenence of pesenng he coe logc, ou poposed model s developed wh he followng wo mn ssumpons: - The effecs of mp flows, vewed s necons beween he mnlne ffc sem nd exenl envonmens, e no ncluded n he fomulons. - Dvng populons of ech mnlne ffc sem cn be dvded no sevel dsnc clsses, nd he esponses o he ffc condon e govenng by he globl feewy densy, he own pefeed fee-flow speed, peceved jm densy, nd mos mponly he unque speed-densy elon. Noe h he exsence of mulclss dvng populons s evden fom he commonly seen mulple ploons n he feewy ffc sem, whee dves wh sml behvol pefeences nd vehcle condons end o ec smlly nd vel n goups unde he sme ffc condon. Thus, le u 0 be denoed s he fee-flow speed, nd m be he densy, fom he fs ode mcoscopc ffc dynmcs, he govenng equons fo he feewy ffc sysem cn be wen s ( u ) + x 0 () If nd x use he uns of x / u0, nd x. Unde he ssumpon of hvng hee clsses of dvng populons, one cn pesen he supplemeny ffc flow elons s follows: u v f, n ( ), () Whee v, nd f n e he fee-flow speed nd he ndex of speed-densy elon fo he -h clss of dves. Noe h when dves n ech clss she he sme fee-flow speed, ndces h hee exss one clss of dves who she he sme fundmenl elonshp (). When he pmee s se s, mens h lne Geensheld model s used o eflec he speed-densy elon (5). By usng he nomlzed opml densy, b wh espec o he jm densy ( b, he flow e equls he odwy cpcy), s evden h he flow devve wh espec o densy should be vnshed b. Hence, we cn hve he expesson fo b s funcon of n : n b n + () Noe h fo hee clsses of dvng populons, he govenng equons should genelly hve hee chcesc vlues coespondng o he especve chcesc decons long whch nfnesml dsubnces popge (6). To ensue h he chcesc vlues e el, he supplemeny elons should be popely seleced. Unde popely defned nl nd boundy condons, one shll be ble o hve soluons fo he govenng equon egdless of he employed numecl mehod. In hs poposed mulclss feewy ffc model, fo convenence, we ssume h he fee-flow speed hs po ssgned vlue h ves wh he ndex of speed-densy elonshp. TRB 00 Annul Meeng CD-ROM Ppe evsed fom ognl subml.

3 Zuojn Zhu, Gng-len Chng nd Tongqng Wu To evlue he popees of ou poposed model, we hve solved wh he Tol Von Dmnshng (TVD) mehod due o Yee-Roe-Dvs (8-0), nd pesened he numecl esuls of sevel expemenl ffc scenos n he ensung wo secons. NUMERICAL SOLUTIONS FOR THE PROPOSED MODEL The foemenoned govenng equons fo ou poposed mulclss ffc flow model cn be wen n he followng veco fom F( ) + 0 x (4) T whee he densy veco (,, ) s ccompned wh s coespondng flux whose componens e gven by F F F v v v f, f, f, n ( ) n ( ) n ( ) F ( F, F, F ) T (5) whee he supescp T denoes he mx nsposon. Accodng o he velocy mesue gven n he foegong secon, fo he flux componen F, we hve v f,. Thus, he Jcobn mx fo equon (4) s A F F F F F F F F F (6) fo whch he chcesc equon s λ + cλ + cλ + c 0 (7) wh he coeffcens c A, c A + A + A, c [ A + A + A] (8) Noe h A mens he ce of mx A, nd A j s he lgebc complemen of he elemens deemnn A, especvely. Expessng p nd p n ems of he coeffcens n equon j of he p c c, p c c c + c (9) fo p + p < 0, nd p < 0, fom he hndboo of mhemcs (), he hee el oos of he chcesc equon cn be gven by y / A + s cos( θ + ( ) π / ), fo,, (0) TRB 00 Annul Meeng CD-ROM Ppe evsed fom ognl subml.

4 Zuojn Zhu, Gng-len Chng nd Tongqng Wu 4 whee s nd e gven by s p, θ cos ( p / s) () Fo convenence of usng he TVD mehod descbed below, he oos e nged n gowng ode. Th mens he chcesc vlues of he ffc sysem cn be wen s λ mn( y, y, y ), λ mx( y, y, y ), λ λ λ Wh he Jcobn mx, we obn he gh chcesc mx n he followng fom: () R [,, ] () Whee he elemens cn be wen s,,, ( ( ( λ ) + λ ) + λ ) +,,, ( ( ( λ ) + λ ) + λ ) + (4) n whch ( ( ( λ )( λ )( λ )( λ ) λ ) λ ) (5) T The evese of R s denoed byl ( l, l, l ). Wh hese noons, ccodng o he TVD lgohm poposed by Yee-Roe-Dvs, nd expessng he me level s m, we hve ω( Fˆ Fˆ m+ m j j j+ j ) (6) whee ω ) s he o of me o spl nevls, nd ( x ˆ F j+ [ F( ) + ( + ) + j F j ψ, j+ ω [( ωλ, j+ ) g, j+ ψ, j+, j+ + Q ( ωλ, j+ ] )( α, j+ g, j+ )] (7) nd α g, j+, j+ l, j+ ( + j mn mod( α j, j ), α, j+, α, j+ ) (8) Q (z) s he coeffcen of vscous em, whch hs he fom TRB 00 Annul Meeng CD-ROM Ppe evsed fom ognl subml.

5 Zuojn Zhu, Gng-len Chng nd Tongqng Wu 5 Q z, fo z ε z) ( z + ε ) / ε, Ohewse ( (9) wh he pmee ε. Whle he mnmum modfcon funcon s gven by, z, z ) f sgn z sgn z sgn z sgn z mn( z, z, z mn mod( z 0. Ohewse ), (0) whee sgn z s he sgn funcon whose vlue s,0, o -, f z s posve, zeo, o negve. The mnmum modfcon funcon povdes monoonc emen fo he numecl soluon. NUMERICAL RESULTS AND DISCUSSIONS The secon pesens he numecl soluons of he bove he mul-clss model bsed on hee clsses of dvng populons. Inl pmees fo pefomng he numecl nlyses e summzed below: - Thee modes, dsngushed by he os o he opml densy, hve been en no ccoun n he numecl smulon nd e pesened n Tble. Noe h he globl densy s unfom bu defned s he sum of denses fo ll dvng clsses. The nl dsbuon of ws pefomed wh compue-bsed ndom geneo fom n Elng pocess of ode 5 s shown n Fgue. The nl dsbuons of he -nd nd he -d clsses e ssgned by usng pmee β defned n Tble. - The fee-flow speed of -h clss s gven by he expesson n Tble. The expesson mples h he od cpcy s consn fo ech clss of dves, nd he fee-flow speed coesponds o n opml densy The pmees of speed-densy elons ( n,,,) fo hese hee dvng clsses e evlued by usng equon (), whee he speed un u 0 s se s uny. - The pmees ( ε,,,) ppe n Q wee chosen s 0.05 whch he numecl vscous effec s neglgble. Noe h unde he ssumpon of hvng unfom nl globl densy, he donl LWR ffc model wll nully led o he concluson of lwys hvng me-ndependen globl densy. Ths s cenly nconssen some feld obsevons. Howeve, wh ou poposed model, s shown n he ensung pesenon of numecl esuls, seems o offe he poenl o bee expln he ffc dynmcs such s he obsevble oscllons of ffc denses nd flow es on feewy segmens even whou ncludng he dsubnce fom mp-flows (6). Tble epesens he me veged nd oo of men sque (ms) vlues of he d sequences fo flow e nd denses occued x00. The fs column ndces hose fve cses n he numecl expemen. The me veged flow e s seen o be mode dependen, bu he pe vlue of he veged flow e s found o hve pe when he nl densy (ID) s 0.5. The ms vlue s of couse mesue of oscllon mgnude fo he smuled d. The ms of flow e cn be seen fom he -d column of Tble. I ves wh he mode choce, nd hs lges vlue n n ode of bou 0 - s he ID s 0.. Howeve, he mode dependence of ms vlue s moe sgnfcn when ID vlue s less hn 0.5. Fo exmple, fo densy 0., fom he -d ow of Tble, he ms of flow e unde mode I s.8h0 - ; fo densy 0.4 equls.9h0 -. Bu unde mode III, he ms vlue s.h0 -. The ms of flow e ppoches s mnm s ID vlue s se 0.5. The ms of globl densy s, n genel, less hn h of flow e (see column 4 of Tble ). Howeve, fom he d shown n he 5-h nd 6-h columns, s cle h he ms vlue of he fconl densy s lmos mode ndependen. TRB 00 Annul Meeng CD-ROM Ppe evsed fom ognl subml.

6 Zuojn Zhu, Gng-len Chng nd Tongqng Wu 6 The evoluon of globl densy x00 fo ID beng ssgned s () 0., (b) 0.5, nd (c) 0.7 s gven n Fgue, n whch he dsh-doed, dshed, nd sold cuves coespond o modes I, II, nd III, especvely. Clely, densy oscllons hppen s ID s en s 0., whch he oscllng mgnude s compble o h of he nl wves. Fo ove sued ffc, he nl egul wves e lgely suppessed. These oscllons come fom he nl heeogeneous dsbuons of hose denses of mul-dve clsses, even hough he nl globl densy s se o be unfom long he ene feewy segmen. The evoluon of denses x00 fo he fs nd second clsses fo ID beng ssgned s () 0., (b) 0.5, nd (c) 0.7 cn be vewed fom Fgue, whee ll cuves exhb sml elons beween dffeen modes s n Fgue. The von of vehcul densy fo he second clss shows evese end o he von of he fs clss. Thus, when vlley of he -s clss densy cuve occus, hee exss pe densy vlue fo he -nd clss. Such coheen ffc ses e due o he ssumed nl condons. Fo ID beng ssgned s 0.5, he choce of dffeen modes hs lle mpc on evoluon of he densy pen. To show h he poposed model hs he popey of cpung he densy nd flow e oscllon, we hve employed powe spec ecommended n he leue () o nlyze he numecl esuls. As gven n Fgues 4 () nd (b), one cn obseve he pmy oscllng fequences fom he exhbed pens. The esuls n Fgue 4 coespond o hose n Fgues, nd, nd he d s seleced x00 fo mode III. The vlues of he wo pmy fequences e gven n Tble. Clely, he choce of dffeen modes hs some mpcs on he pmy fequences when he ID s se o be 0.. I s he mn fequency wh lge powe specum h domnes he oscllng pefomnce. Fom Tble, cn be seen h when he nl densy s less hn 0.6, he wo pmy fequences decese wh he nl densy. Fo ID 0.7, hs he sme pmy fequences s h n he cse whch hs he nl densy of 0.6. Fo exmple, fo Mode I, fom he -nd column of Tble whee ID s 0., he pmy fequences f, nd f hve he vlues of 7.58h0 -, nd.84h0 -, especvely. Noe h by seng he dmensonl vlue of x 80m, he fee-flow speed s 0m/s, nd he me un o be 6s, n he cse of ID equl o s 0., he dmensonl pmy peods of ffc oscllon e.mn, nd 5.45mn (.e. 6/f /60, nd 6/f /60). Such peod of ffc oscllon cn be found n he el-wold d fom Ps (), fo Gemny (6), nd Sn Fncsco (). To see ffc flow pen n ems of globl densy, he conous of ffc densy unde mode III fo he nl densy of () 0., (b) 0.5, nd (c) 0.7 e llused n Fgue 4. These pens ndce no only h he evoluon of sold cuves n Fgue e elly obseved x00, bu lso h he ffc wve popges ccodng o he oenon of he gven densy sucues. These conous, lbeled by he nl densy, cn bee cpue he ffc pen. One cn lso see fom Fgue 4 h he ffc wves popge downwd long he seep nd dmnshed decons (see, ps () nd (b) of Fgue 4). Howeve, he popgon decon fo he cse of densy equl o 0.7 s clely gong upwd (see p (c) of Fgue 4). These conous exhb dense spcng he ely sge of densy evoluon, nd become cose he le sge. The evden wnles nd locl sucues occu pcully fo he cses of densy equl o 0.5, nd 0.7, s shown n ps (b) nd (c) of Fgue 4. In summy, he numecl esuls shown n he bove fgues nd bles e conssen wh some elwold obsevons (, 6, ), nd offe plusble explnon fo obseved ffc densy oscllon on feewy segmens whou exenl dsubnces. Ou poposed mul-clss mcoscopc model wh s smple fs-ode elon seems cpble of llusng such vl ffc flow dynmcs. CONCLUSIONS The ppe hs pesened fs-ode mulclss mcoscopc ffc flow model nd s numecl soluons. The poposed model uses he fee-flow speed nd he speed-densy elon o chceze ech clss of dves. Numecl nlyses wh he TVD mehod hve ndced h by ng no ccoun he behvol dscepncy of vous dvng populons ou poposed model s cpble of llusng some vl ffc flow dynmcs,. e., f he densy s less hn s opml densy, feewy segmen wh n nlly unfom globl densy nd flow e my exhb ponounced oscllng pen even whou ny mp flow dsubnce. Ths s lely due o he heeogeneous dsbuons of dffeen dvng populons who my ec dffeenly unde dencl ffc condons. I, howeve, should be menoned h he mpc of mulclss dvng populons on he ffc dynmcs s que complex ssue, nd much emns o be exploed long hs esech decon. Ou on-gong wo hs focused on he neelons beween he numbe of dvng clsses nd he dynmc popees of he TRB 00 Annul Meeng CD-ROM Ppe evsed fom ognl subml.

7 Zuojn Zhu, Gng-len Chng nd Tongqng Wu 7 esulng ffc flows. The mpc of mp flows on he sbly nd he vbly of he globl ffc pens wll lso be exploed. ACKNOWLEDGEMENTS The uhos would le o hn he evewes of hs mnuscp s hey hve povded nvluble commens nd suggesons fo ou evson wo. REFERENCES. Zhng, H. M.(999). Anlyses of he sbly nd wve popees of new connuum heoy.tnsp. Res. P B: Mehodol.,, Helbng, D., Hennece, A., Shvesov, V. nd Tebe, M. (00). MASTER: mcoscopc ffc smulon bsed on gs-nec, non-locl ffc model.tnsp. Res. P B: Mehodol., 5, 8-.. Helbng, D., Bc, D. Schonholf, M nd Tebe, M. (00). Modelng wdely sceed ses n synchonzed ffc flow nd possble elevnce fo soc me dynmcs. Physc A, 0, Ngn, T. (000) Tffc behvo n mxue of dffeen vehcles. Physc A, 84, Hoogendoon, S. P. And Bovy P. H. L. (000). Connuum modelng of mulclss ffc flow. Tnsp. Res. P B: Mehodol., 4, Kuhne, R. D. nd Mchlopoulos P. (00). Connuum flow models. Chpe 5 of Tffc Flow Theoy. see, hp://www-c.onl.gov/c/esech/b/f.hml. 7. Dgnzo, C. F. (995). Requem fo second ode ppoxmon of ffc flow. Tnsp. Res. P B: Mehodol.,4(4), Zhu, Z., nd Wu, T. (00). A wo phse flud model fo feewy ffc nd s pplcon o smule he evoluon of solons n ffc. ASCE, J. Tnspoon Engneeng, o ppe. 9. Cssdy, M.J. nd Much, M. (00). An obseved ffc pen n long feewy queues. Tnsp. Res. P A 5, Dgnzo, C. F. (00). A behvol heoy of mul-ln ffc flow. P I: Long homogeneous feewy secons. Tnsp. Res. P B: Mehodol., 6, Dgnzo, C. F. (00). A behvol heoy of mul-ln ffc flow. P II: Meges nd he onse of congeson. Tnsp. Res. P B: Mehodol., 6, Wong, G. C. K., Wong, S. C. (00). A mul-clss ffc flow model-n exenson of LWR model wh heeogeneous dves. Tnsp. Res. P A, 6, Lghhll, M. J., nd Whhm G.B. (955). On nemc wves; II. heoy of ffc flow on long cowded ods. Poc., Roy. Soc. Se. 9A, Rchds, P. I. (956). Shoc wves on he feewy. Opeons Res., 4, Geenshelds, B. D. (94). A sudy of ffc cpcy. Poceedngs of he Hghwy Resech Bod 4, Wng L. X. (979). Hndboo of Mhemcs. Publshe of Hghe Educon, Bejng, Glmm, J. (965). Soluons n he lge fo nonlne hypebolc sysems of equons. Comm. Pue Appl. Mh., 8, Dvs, S. F. (988).Smplfed second- ode Godunov- ype mehods. J. Sc. Ss Compu., SIAM, No. 9, Roe, P. C. (98). Appoxme Remnn solves, pmee vecos, nd dffeence schemes. J. Compu. Phys., 4, Yee, H. C. (987). Consucon of explc nd mplc symmec TVD schemes nd he pplcons. J. Compu Phys., 68, Zhu, Z., Yng, H. (00). Dscee Hlbe nsfomon nd s pplcon o esme he wnd speed n Hong Kong. J. Wnd Eng. Ind. Aeodynmcs. 90, Ppgeogou, M., Blossevlle, J. M. nd Hdj-Slem, H., (989), Mcoscopc modelng of ffc flow on he boulevd Pepheque n Ps, Tnsp. Res. P B: Mehodol., (), NOMENCLATURE A Jcobn mx; A deemnn of mx A; A j lgebc complemens fo mx elemen j ; j elemens of A; b o of opml o jm densy; TRB 00 Annul Meeng CD-ROM Ppe evsed fom ognl subml.

8 Zuojn Zhu, Gng-len Chng nd Tongqng Wu 8 c, c, c coeffcens of chcesc equons; F flow veco; g gven by Eq(8); L lef chcesc mx; l lef chcesc veco fo oo λ ; n ndex of speed-densy elon, see Eq. (); p nemede coeffcens, see Eq. (9) ; p see Eq(9); Q coeffcens of numecl vscous em; q flow e (F + F +F ) ; q 0 od cpcy; R gh chcesc mx ; gh chcesc veco fo oo λ ; ms oo men sque; s defned n Eq.() me; ce,.e. he summon of he dgonl elemens of mx; u defned by Eq.(); v f, fee speed fo clss I; y oos of chcesc equon. x spce; z nemede vble. Gee Symbols α gven by Eq. (8); gven by Eq. (5); λ ω -h chcesc vlue; o of me o spl sep; ψ gven by Eq. (7) densy veco ; ffc densy; fconl densy fo dves clss ; m Supescps m T Ove b jm densy; me level; nsposon of mx mens me vege. Ls of bles Tble. The nl pmees used n numecl smulon Tble. The vege flow e nd ms vlues of flow e nd denses fo sevel 0 x 00 unde dffeen modes. Tble. The pmy fequency nd peod of he densy sequence x 00 TRB 00 Annul Meeng CD-ROM Ppe evsed fom ognl subml.

9 Zuojn Zhu, Gng-len Chng nd Tongqng Wu 9 Ls of Fgues Fgue. Inl dsbuon of he densy of he fs clss vehcles whch s gven by compue bsed ndom geneo egdng he Elng pocess wh ode 5. Fgue. Evoluon of globl denses fo feewy ffc fo nl densy ssgned s () 0., (b) 0.5, nd (c) 0.7 unde dffeen modes. Fgue. Evoluon of he fs nd second clsses denses fo feewy ffc fo nl densy ssgned s () 0., (b) 0.5, nd (c) 0.7 unde dffeen modes. Fgue 4. Powe spec of denses nd flow e fo he cse of nl densy 0.5 fo he hee clsses unde mode III Fgue 5. Conous of globl densy, whee he nl denses () 0., (b) 0.5, nd (c) 0.7 e used n he lbelng of conous unde mode III. Tble. The nl pmees used n numecl smulon Mode I: 0.49, 0.47, 0.45, fo,, b Mode II: 0.50, 0.48, 0.46, fo,, Mode III: 0.5, 0.50, 0.48, fo,, n v, 0.5 [ b ( )] n ε f b Evlued by Eq.() 0.05, fo,, β + ) ( 75%, fo 0. Tble. The vege flow e nd ms vlues of flow e nd denses fo sevel x 00 unde dffeen modes. 0 q σ q 0 σ σ σ Mode I h 8.05h 6.9h 6.8h h.68h.0h 8.0h h 0.70h.5h.04h h.h.66h.h h.79h.6h.9h Mode II h 8.5h 6.84h 6.h h.h.00h 8.06h h 0.45h.6h.0h h.h.66h.h h.78h.6h.0h Mode III h 0.h 6.86h 6.4h h 4.57h 0.99h 8.0h h 0.8h.6h.0h h.h.66h.h h.89h.6h.h Tble. The pmy fequency nd peod of he densy sequence x 00 TRB 00 Annul Meeng CD-ROM Ppe evsed fom ognl subml.

10 Zuojn Zhu, Gng-len Chng nd Tongqng Wu Mode I f 7.508h 6.675h 5.008h 4.75h 4.75h f.84h.50h.5h.00h.085h Mode II f 7.508h 6.675h 5.008h 4.75h 4.75h f.75h.50h.5h.00h.00h Mode III f 6.675h 6.675h 5.008h 4.75h 4.75h f.669h.50h.5h.00h.00h () x (b) x Fgue. Inl dsbuon of densy of ucs whch s gven by compue bsed ndom geneo egdng Elng pocess wh ode 5. TRB 00 Annul Meeng CD-ROM Ppe evsed fom ognl subml.

11 Zuojn Zhu, Gng-len Chng nd Tongqng Wu 0.0 Desy () Desy (b) Desy (c) Fgue. Evoluon of globl denses fo feewy ffc fo nl densy ssgned s () 0., (b) 0.5, nd (c) 0.7 unde dffen modes. TRB 00 Annul Meeng CD-ROM Ppe evsed fom ognl subml.

12 Zuojn Zhu, Gng-len Chng nd Tongqng Wu Desy 0. () Desy (b) Desy (c) Fgue. Evoluon of denses of he fs nd second clsses fo feewy ffc fo nl densy ssgned s () 0., (b) 0.5, nd (c) 0.7 unde dffeen modes. f f 0-5 f f 0-0 q Powe Spec Powe Spec () Fequency (b) Fequency Fgue 4. Powe spec of denses nd flow e fo he cse of nl densy 0.5 fo he hee clsses unde mode III. TRB 00 Annul Meeng CD-ROM Ppe evsed fom ognl subml.

13 Zuojn Zhu, Gng-len Chng nd Tongqng Wu x 50 () x 50 (b) x 50 (c) Fgue 5. Conous of globl densy, whee he nl denses () 0., (b) 0.5, nd (c) 0.7 e used n he lbelng conous unde mode III. TRB 00 Annul Meeng CD-ROM Ppe evsed fom ognl subml.