The dynamics of the wagon rolling down the hump profile under the impact of fair wind Research Paper

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1 Dic Rsch Jounl of Engining nd Infomion Tchnology (DRJEIT Vol. ( pp. 7-4 y 4 ISSN Ailbl onlin dicschpublish.og/dj 4 Dic Rsch Jounls Publish Th dynmics of h gon olling don h hump pofil und h impc of fi ind Rsch Pp Khbibull Tuno Uls S Unisiy of Rily Tnspo (USURT Russi. *Cosponding Auho E-mil: khuno@ynd.u Accpd Apil 4 Th icl gis n ccoun of h suls of h consucion of mhmicl modl of h gon olling don h hump pofil und h impc of h fi ind. Th h bn did nlyicl fomuls fo dmining gon olling spd nd disnc lld don h fis pofil hump scion. Th suls of h sch obind cn b usd fo ll h s hump scions ih h llonc md fo h spcifics of bking focs on hs scions. Ky ods: Spd nd dicion of fi ind D Almb pincipl ionl funcion ingl nd ingl conining igonomic funcion spd of gon olling don h fis pofil hump scion disnc lld. INTRODUCTION Th smn of h poblm nd is conncion o sch nd pcicl sks In Khbibull (4 conins dild ccoun of h suls of nlyicl insigion concning h dminion of h spd nd h disnc lld by h gon olling don hump pofil und h fi ind. Ho up o no scn noic hs bn cd o h dynmics of gon olling don h hump pofil und h fi ind ih sic dhnc o clssicl smns of hoicl mchnics. o his son h consucion of mhmicl modl of h gon olling don h hump pofil und h impc of h fi ind sill mins y ugn issu. Th im is o mhmiclly dscib h dynmics of h gon olling don h hump pofil und h impc of fi ind. In h long m h sch suls obind cn b usd in soling chnicl poblm of dmining hump ionl gomicl pms nd kinmic chcisics of h gon olling don hump. omulion of poblm I is ncssy o dmin h spd nd h disnc ld by h gon olling don h hump pofil und h impc of giy nd fi ind. SOLUTION ETHODS W ill mk us of h folloing clssic smns of hoicl mchnics: h hom of lociy ddiions in compound momn h pincipl of ls fom consins h fundmnl l of dynmics Loisynsky (983 nd h mjo noions of diffnil nd ingl clculus (Bonsin 98 Piskuno978. Poblm spcificion nd gd pcondiions Jus s in (Khbibull 4 nd Komo 4 gnl cs hn gon is olling pogssily don h hump spcifid iniil spd (nomlly 4 5 km/h o 38 m/s.whn singl gon (o cu is olling don h hump h gon ill minly pinc h impc of nl focs in h fom of giy focs of h gon ih cgo o ihou i G nd odynmic sisnc foc h ( y ( In (Khbibull 4 nd Komo 4 l h gon mo pogssily nsfing spd

2 Tuno 8 li spd (ind spd. (h lu o b clculd fom ind spd in lion o h hump cs (gound (h is i picl bsolu spd. (h lu o b s nd gon spd (h lu o b found. In ccodnc ih h hom of lociy ddiion (Loisynsky 983 nd Komo 4 ill i: igu.vcoil digm of gon nd fi ind spd. (h lu o b found don h hump in spc o immobil coodin sysm Oyz (igu. Wind spd in spc o h hump cs (gound (h is h bsolu spd of i picls. (h lu bing s is dicd coss long is Oyz. W ssum h mobil coodin sysm O y z is igidly bound o h gon hil i picls in hi un mo spd in spc o mobil coodin sysm O y z (h is h gon. I is ncssy o find pojcions of i picl li spd (ind spd (in spc o mobil s O y z bound o h gon.igu hs h folloing bl of symbols: O is h bginning of immobil coodin sysm Oyz igidly bound o h hump cs O is h bginning of mobil coodin sysm O y z igidly bound o h gon H V nd W hoizonl icl nd fonl plns; ψ is dscnd ngl ( in ccodnc ih hump pofil (h lu o b s; is li spd of i picls (ind spd in spc o mobil fnc fmo y z (gon (h lu o b clculd; λ is dicion ngl of h co of li i picl spd long is O (h lu o b clculd;. is i picl spd in spc o h gound (o h hump cs (ccoding o h d of Consucion Noms nd Rgulions his is h lu o b s; ξ is dicion ngl of co of i picl bsolu spd long is O (h lu o b s. Rli i picl spd (ind spd. s consid nd s locd on hoizonl pln H nd dicd ngl λ (o λ li o hoizon (is O hil nsfing spd (gon spd is locd on icl pln V nd is dicd dscnd ngl ψ (o ψ li o hoizon (O. L us sho h dpndncs of pojcions of i picl. +. h spd; (. is i picl bsolu spd (ind spd (gon spd ( is h pojcion of nsfing cos( ψ ono is O: ih llonc md fo h fc h ψ (o ψ is dscnd ngl of h hump o h hoizon (is O;. is i picl li spd (ind spd in lion o h gon. As consid h dicion of h ind o coincid ih h dicion of h gon (i.. ih h fi ind h pojcion ( ono is O hs h fom of:. cos( ξ +. Hnc cos( ξ.. h is n ngl bn suln co. (i picl bsolu spd (ind spd nd longiudinl is O d. In h pssion h modul of i picl li spd (h is ind spd in lion o h gon.в is o b found ccoding o h cosin hom (Bonsin 98 nd Vodn998:. +.. cos(ξ Dicion ngl λ of i picl li spd (ind (3

3 Dic Rs. J. Eng. Infom.Tch. 9 igu. Pofil of ious hump scions. igu 3.Simplifid modl of gon olling don h hump. spd. s in (Vodn998 is found ccoding o h hom of sins: sin(λ. sin(ξ.. In ccodnc ih (Bonsin98 odynmic sisnc foc fo fi ind on is O nd on is Oy unlik in (Khbibull 4 is found N: ( cos(ξ.5c ρ AEV. ( sin(λ. y 5c ρatv. In (Piskuno 978 nd Komo4 h is h folloing symbol bl: c is infini cofficin of i sisnc dpndn on h fom of h solid nd h y i is oind in h pocss of moion (nomlly i is ccpd ccoding o h fom of h solid sufc o y fom.55 o. fo mpl cylindicl solids ih cicl s coss scion (pip c.6 fo fl sufcs c : c.; ρ is g i dnsiy (kg/m 3 (nomlly ccpd o b.6.9; A EV is h of h nd sufc of h gon ih cgo m ; A EV B H (h B и H idh nd high of indd sufcs of h gon ih cgo m; A TV is h of nd sufcs of h gon ih cgo: A TV L H (h L is h lngh of nd indd sufcs of h gon ih cgo m m. L us ssum h hump s in (Khbibull4 consiss of h scions nd 3 linkd by o bking poins nd. L h fis bking posiion ( s BP ih coodins nd b b locd on h scond scion hil on scion 3 h is poin sich (PS ih coodin c (igu. igu hs h folloing bl of symbols: HH is hump high ; H (4 (5 nd L is dsign high of h hump m; h h h 3 nd l l l 3 high nd lngh of h cosponding hump scions; l b is lngh of h fis bking posiion m; ψ ψ и ψ 3 slop ngls of h cosponding hump scions d.; DТ is dsign poin. omion of olling gon dsign modl Th poblm s sold s gnl cs: h impcs of fi ind on h gon (igu pssing hough il joins impc scnd up h poin scnd up h sid ck chng of i dnsiy c. Th modl shon in (igu 3 is kn s simplifid modl of h gon olling don h hump ih llonc md fo olling ficion of gon hls ih sliding hil h modl psnd in igu 4 is kn s dsign modl (Khbibull4.igu 3 hs h folloing bl of { } symbols: fa ( A fa fa fa fa f { nd fb ( B fb fb fb fb innl focs in h fom of olling ficionl oqu in bings of jounl bo unis of h lding uck A nd uck B f fa + fb ; P A P A P B P B bing spd momny cns (Khbibull4. In igu 4 llonc is md fo h fc h f } { N A N A} N A { N A N A } N A { N B N B} N B { N B N B } N B { A τ τa } τa { τ B τb} τb { B τ τb } τb noml nd ngn componns of il hd cions. In so doing llonc is md fo h fc h τ A τ A τ B τ B τ B W cohsion ficion focs cing bn concing hl sufcs nd il hds h is τ A f A τ A f A τ B f B

4 Tuno igu 4. Dsign modl of gon olling don h hump sid i; b op i; c nd i. τ B B f τ A W f A W τ B W τ B W. I mns h τ dicd long h sufc of il hds is opposi in dicion o gon olling nd is sliding ficion foc τ f : f f. A + f. A + f. B + f. B + f. AW + f. A W + f. BW + f. B W icion foc duing hl olling moion ih sliding sl f ( s bking foc of h gon bk. ( dicd opposily gon olling don h hump is o b psnd in h fom (Khbibull4: bk. ( f ( + fsl ( f h is symbolic sliding ficion duing idl hl olling nd olling lmns in bing bos: y ( G + ( sin(ψ f ( f.5. 5 H f is som symbolic (o ducd fco of sliding ficion (Khbibull4: f n f n f + k n n b m ; (6 (7 (7. f sl is cofficin of sliding ficion of hl flngs don y h il (nomlly kn s f sl 5; is pojcions of odynmic sisnc foc ono h gon nss is (ccoding o (5 his lu is clculd.in (7 s in (Vodn 998 h folloing symbols ccpd: n к is h numb of hls in bogis ims (n 8; f is olling ficion cofficin s his cofficin is omnipon o n m of coupl of olling ficion (hl ccoding o il f 5-6 hdnd sl ccoding o sl f -6 ; is hl dius qul fo figh c.475 m; n b 8 is h numb of bo unis in bogis ims is cofficin of olling ficion ccoding o c ings (nomlly is kn s. 3 m; n q is ol numb of olling lmns pciing lod in ch bing unis; k is pmnn cofficin kn ccoding o o lyou nd yp of olling bings (fo clculion i is kn k 4 6 (Khbibull4; is h numb of bings in bogi bo unis unis (n 6; is ou dius of innl olling bing ing m (79 m. Inoducing h noions of shing sh. х nd ining. х focs ih considion fo ci nd ll ci focs ill g: sh. х Gsin( ψ + ( ; (8 (. х( bk. W i h bo pssion ih considion fo (6 nd (7

5 Dic Rs. J. Eng. Infom.Tch. ( G + ( sin(ψ + f.. х ( f sl Th condiion of gon olling don h fis pofil hump scion ih gdin no sp hn 5 scion lngh up o 5 m is (Khbibull 4: >> sh. х. х ( y (9 ( ( I follos h css of focs.5 sh. х. х occuing on h fis hump scion is h moi foc cusing gon olling of gin foc G nd fi ind foc ( spd.5 ( nd cclion dpnding mosly on olling ngl of h hump ψ.5 nd o som dg of sliding ficion of hl flngs don h il nd lso on h s of olling bing in bogi jounl bo unis. Th is hy in od o nsu h gon moion h nd of h fis pofil hump scion spd.5 ( lss hn spd х ( of nnc ono h fis bking posiion ( s BP h is ( < х ( i is nough o find ionl lu ψ.5 s hump mjo gomicl pm. RESULTS O SOLUTION hmicl dscipion of h dynmics of gon olling don h hump W ill k ino ccoun h fc h gon is olling don h hump pogssily h is hy gon nsfing cclion is qul o bsolu d / d cclion bs bs (Loisynsky983. Th fundmnl l of dynmics fo gon nsfing momn (o pincipl of D Almb in coodin fom (Loisynsky983 s usd: d d. n k k + n k R k ( h is mss of h gon ih cgo kg; sh. х is pojcions of ll ci (shing focs ono h dicion gon olling (is N; R. х ( is pojcions of ll ci (ining focs ono is N. Subsiuing ( fo (8 nd (9 ill g: Tnsfoming h bo pssion ih considion fo (8 nd (9 nd h fc h G g fo h fis pofil hump scion ih gdin no sp hn 5 scion lngh up o 5 m ill g: d gsin(ψ + ( f d f ( g + ( sin(ψ. sl y ( Puing (4 in ( fo fi ind psn h fundmnl l of dynmics in h fom d d + b ( c (3 h is h diffnc bn knon lus of moi focs nd sisnc focs pplid o h sysm gon-cgo N; ( sin(ψ f f g.5.5 sl y b is pmnn cofficin nd h lu of hich is knon hing dimnsion N/(m/s : b ( sin(ψ.5cρ AEV.5. 5 f c is pmnn cofficin h lu of hich is knon hing dimnsion of spd m/s: c cos(ξ. cos( ψ Dsigning hough nd sping boh ps (3 by b ill h: b d d h + ( c ; (4 is consn lu hing dimnsion of spd (m/s b :. Sping ibls (4 nd fulfilling nsfomions ill g (Piskuno978: d d sh. х. х (. b d ( c d + ( c

6 Tuno Tking ingls fom ionl funcions of boh ps of h bo quions ill h: b d( c + ( c Th igh p of h bo quion is buld ingl fom ionl funcion in h fom (Bonsin98 nd Vodn998: d + cg In ccodnc ih his smn psn h igh p of h quion in h fom: b c cg Hnc: puing ingion limis nd f nsfomion ill g: b c cg + C h C is consn numb c C cg. (5 Tnsfoming (5 ill g h folloing igonomic quion (Ilin967: cg c b C uliplying boh ps of h bo quion by ngn: g cg b g C c Tnsfoming h bo pssion f lmny compuion ill h: c b - g C Bcus ngn (g is n odd funcion (Ilin967 fcoing ou minus bhind g h bo pssion ill b in in h fom: c b + g C Thn ill fin h pojcion of gon spd on h longiudinl is O duing gon olling fom h hump und h impc of pojcion of giy focs nd fi ind ono is O: c + gα ( C. (6 h α (lph is consn lu hing dimnsion /s: b α. Tking ino considion h fc h piously cos( ψ s dsignd hough fom (6 ill find gon nsfing spd ods h dicion gon olling don h hump O ( ( c + g(α C (7 I follos h gon spd dpnding on spd duing olling don h hump pofil is dscibd ccoding o h l (7: h gon cn quickly gin spd nd in h follos in h bsnc of focd sisnc ill mo pciclly unifomly. d Tking ino ccoun h fc h d i h bo pssion: d d ( c + g(α - C uliplying boh ps of h bo quion by d ill h: d ( c + g(α C d Inging h obind quion h: sin(α -C d c + d cos(α -C (8

7 Dic Rs. J. Eng. Infom.Tch. 3 Thn subsiu h ibl in h scondly summnd (8 α C z fom h α d dz hnc d α dz In ccodnc ih his smn l us psn (8 f nsfomions in h fom: d c d cos z α cos( z (9 Th scond mmb und h bck of h bo quion is buld ingl conining igonomic funcion in h fom (Bonsin98; Vodn998; Ilin997: dz z cos z ln g( + π 4 Accoding o his smn ill i (9 in h fom: π ln g(. α 4 z c + Tking ino ccoun h fc h i h bo pssion α π ln g(. α 4 C c + z α C Subsiuing ingion limis of h bo quion f lmny mhmicl compuion ill g: α C g( + ln c α π C g( 4 α α π Dsigning nd 4 bo pssion in h fom: π 4. g(α + β ( ln. α g(β c C β psn h ( I follos h h disnc (y ld dpnding on im is dscibd ccoding o h l (: ih incsing olling im lds o lin incsing h disnc ld. As in (Khbibull4 h mhod of djusmn of h soluions of picis-lin quions s usd on h bsis of Hisid dimnsionlss dd uni funcion (Lsyn4 h spd of gon (cu momn nd h disnc lld on ny hump scion ihin h limis of h gin pofil up o h bking momn cn b psnd in h fom: ( c + g(α C σ ( ( i g(α + β c ln σ (. α g(β ( i ( ( σ ( h i ( (h i is Hisid dimnsionlss dd uni funcion mking i possibl o psn im s on nlyicl pssion fi ny lu of coodin i in h inl i h σ ( i i <. To sum up by mns of using D Almb pincipl of mchnics ibls spion mhod ionl funcions buld ingl nd unlik in [5] using h ingl conining igonomic funcion s ll s h mhod of picis-lin quions h h bn did nlyicl fomuls fo dfining h spd of gon olling don h hump pofil ( nd disnc ld (ih considion fo im. CONCLUSION. Obind on h bsis of clssicl concps of hoicl mchnics clculd nd mhmicl modls of h gon olling don h hump pofil und h impc of giy focs pojcion nd unlik in (Khbibull4 fi ind ono h longiudinl is mk i possibl o dfin h spd of olling gon ( nd h disnc ld ( don h fis hump pofil ccoding o im. In picul cs h obind nlyicl pssions of gon olling dynmics nbl spciliss o find fini fomuls fo dfining h spd nd h disnc ld ih und h impc of only giy focs pojcion ono h longiudinl is o und h impc of only fi ind.. Th suls of nlyicl insigions of h gon olling don h fis high spd hump scion cn b usd fo ll h s hump scions ih considion fo spcifics of bking focs on hs scions.th disinci fu (noly of h did nlyicl fomuls of gon spd

8 Tuno 4 olling don h high spd hump scion consiss in psning h foc unlik in (Komo4 of fi ind dpnding on gon olling spd ( nd coc considion fo sisnc focs occuing h moion of h gon. Th obind sch suls ilbl fo hump dsigns n sg in h soluion of his picul poblm. Th dng (significnc of his mhodology is h possibiliy of h consucion of mhmicl modl of gon olling don h hump ih considion fo h dpndnc of fi ind foc on gon olling spd ( spd (ξ.. в nd i flo dicion Bonsin IN (98. Hndbook on mhmics fo ngins nd sudns of chnicl insiuions / I.N. Bonsin K.A. Smndy. : Nuk In Russin. pp Piskuno NS(978. Diffnil nd ingl clculus fo high chnicl schools. V... Scinc. pp [In Russin]. Komo KL(4. Thoicl mchnics in poblms of ily nspo / K.L. Komo A. Ysin. Noosiisk: Nuk. pp. 96 [In Russin]. Vodn VT (998.undmnl mhmicl fomuls / V.T. Vodn A.. Numoich N.. Numoich. insk High School pp. 69 [In Blus]. Ilin VA(967. hnicl nlysis fundmnls / V.A. Ilin E.G. Poznyk. М.: Scinc pp. 57 [In Russin]. Lsyn VA(4. Gnlizd funcions in mchnics poblms / V.A. Lsyn S.I. Konshnko. Dnposk: DIIT pp. 87 [In Ukin]. REERENCES Khbibull T (4. Anlyicl insigion of gon spd nd sd disnc duing gon hump olling und h impc of giy focs nd fi ind / Khbibull Tuno // Globl Jounl of Rschs in Engining: A. chnicl nd chnics Engining. Volum XIV Jssu I. Vsion.. N Yok pp. 9. Loisynsky LG (983. Cous of hoicl mchnics. V. II. Dynmics / L.G. Loisynsky A. I. Lui. : Nuk In Russin. pp