CONTROL OF TANDEM-TYPE TWO-WHEEL VEHICLE AT VARIOUS NOTION MODES ALONG SPATIAL CURVED LAY OF LINE

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1 COTROL O TADEM-TYPE TWO-WHEEL EHICLE AT ARIOUS OTIO MODES ALOG SPATIAL CURED LAY O LIE АS Besha Kaves КМ Bass Т Kaves LА Toka Wheeled vehicle is cosideed as a maeial poi ude he codiios of o-uifom moveme alog cuved spaial lay of lie Hodogaph i a class of spial lies descies kiemaics of a vehicle A kieosaics polem of adem-ype wo-wheel vehicle is eig solved Equivale coac dyamics ae eig deemied Keywods: hodogaph kiemaics kieosaics Eule-Lagage equaios equivale foce sysems ivaias of saics Ioducio A polem of dyamic desig i he coex of wo-wheel vehicle coollailiy as well as dyamic ude of is desig ad oad suface is impoa i ems of vaious moio modes (acceleaed deceleaed ad seady) alog cuved spaial lay of lie i jucios a vaious gadies o saigh ad us as well as wihi ohe cuved aeas [ 5 4 7] The polem soluio will help deemie equivale coac cool foce makig aalysis of he equied cool faciliies The polem saeme Hodogaph of vehicle moio alog cuved spaial lay of lie is supposed as give A vehicle is cosideed as maeial poi of kow mass movig ude he gaviy give aeodyamic foces ad seached equivale coac movig (cool) foces esulig eacio of coac wih efeece ajecoy (lay of lie) Cuved spaial efeece ajecoy ha is lay of lie is ideified y hodogaph i moioless eah efeece Hodogaph coespode o eal ajecoy of vehicle moio [7] i a class of spial lies [5] is specified i moioless (eah) efeece as follows ( ) ρρρρ ( i cosω+ jsiω) + k hhhh whee ρ h ( i ) ae vaied paamees deemied o specified ouday i i codiios; ω is mea agula u velociy equal o ω Hee ϕ is complee u agle; ad is equied ime of u passig ollowig figue demosaes lay of lie as a ajecoy of wheeled vehicle moio i ems of cuved aea eig adequae o poposed hodogaph: ϕ

2 Hee i j k ae os of eah (moioless) efeece; ad ae os of movale aual axes I is ovious ha hodogaph is epeseed i a well-kow epeseaio fom : ( ) i+ j + k Hee veco compoes ae assumed as: ρρρρ cosω ρρρρ siω hhhh Hee you ca fid hodogaph of a vehicle moio: seady ( A B) moio wihi hoizoal ( hj ; j ) saigh ( ω ) lay of lie: ( ) ( ) i + ; A A useady: ( A < B) acceleaed; ( ) A > B deceleaed moio wihi h ; j saigh ( ω ) lay of lie hoizoal ( j ) seady ( A B A B ) moio wihi pofile-iclied lay of lie if ( < ) ise ad ( > ) iclie: A B A B A A ( ) i ( A + A ) + k B B A B A Hee usig Caesia coodiae sysem lay of lie pofile is epeseed i he fom of squae ad cuic paaolas: z x x whee

3 x ( ) A B A z ( ) B 4 Useady moio wihi hoizoal plae whee diec- agle u akes place: 6 ( ) A ( ) A ( ) cos A si A A + B A B A i + j π A π A A A Hee lay of lie pla i pola coodiae sysem is epeseed y squae ad cuic Achimedea spials: ( ϕ ) A ϕ ϕ B π π A whee pola agle is: ϕ ω ; ad pola adius is: ( ) ϕ + Moeove whe A B o A B i follows ha: ( ϕ ) A a ay ϕ iе we oai lay of lie i he fom of adial ac wihi he give ieval: π ϕ Kiemaics eco of vehicle liea velociy i he fom of maeial poi is deemied o he specified hodogaph as: d o i+ j + k d elociy value is deemied wih he help of scala poduc: υ o υ + + By defiiio velociy value is also deemied as a ime deivaive fom he pah: ds υ o υ S d The he pah of a vehicle wihi adom ime peiod is calculaed y meas of defiie iegal wih vaiale uppe limi: ( ) υ ( ) d o S( ) + + d S Whe pah is ioduced as iemediae agume velociy veco is epeseed as:

4 d ds ds d o d S ds Ad akig io accou ha: d ds we oai [6]: S I is ovious ha velociy veco pojecio o age lie o o spaial ajecoy is S Deemie velociy value as follows: S ; o picipal omal o: S ; o iomal o: S I ems of veco ad maix fom we oai: i jk o S i+ j + k I is kow ha scala ad veco poducios of vecos ae epeseed i quaeio maices The i eah efeece we defie: ( R + R ) o iе whee R R + + Similaly i aual axes we oai: o S S S iе υ S

5 Ovious equaliy follows: S + + I he coex of eah efeece liea acceleaio veco is foud ou accodig o he specified hodogaph i he fom of: W d o W i+ j + k d Acceleaio velociy is deemied wih he help of scala poducio: W W + + Liea acceleaio veco i aual axes is [6]: iе whee K is cuvaue The W S+ KS W W W W o W S W o W KS W o W 4 W W S + K S iе depedece wih specified hodogaph akes place: S + K S Hodogaph also deemies W ad W compoes of followig closed veco fom: W W Hee scala ad veco poducs ae coveie o e calculaed i quaeio maices []: ( R + R ) Cuvaue i he sysem of aual ihedal coodiaes is deemied as: K W 4 S

6 Whee W W W W Tageial acceleaio i coodiae sysem is deemied accodig o he fomula: The W S ( W) o W W ( + + ) ( + + ) + + Cosequely i eah efeece we oai: K ( + + )( + + ) ( + + ) ( + + ) Equivale veco fom povides closed cuvaue ecod [4] : ( ) ( ) ( ) K o K ( ) ( ) Usig deemia we also fid ou: K ( ) Quaeio maices povide coveie cuvaue calculaio whee Hee fis ad secod poducs of is ime compoes fo hodogaph ude cosideaio ( ) ae deemied as follows: ρρρρ cosω ω siω ρρρρ siω ω cosω + hhhh ω ω ω ω ρρρρ cosω ω siω ω ρρρρ siω+ ω cosω ω 6 ω 6 ω

7 hhhh 6 Kieics Mahemaical model of wo-wheel vehicle i ems of is spaial moio alog cuved lay of lie is developed usig o-liea diffeeial Eule- Lagage equaios i he fom of quaeio maices [] Pojecios of vehicle velociy veco o aual axes ae assumed as quasi-velociies aual ihedal is ake as a oud coodiae sysem which pole is comied wih he maeial poi assumed as a vehicle model Two-wheel vehicle is cosideed as he maeial poi of he kow mass wih he applied ieial foces gaviaio foce aeodyamic foces ad he equied coac movig (cool) foces esuig ecessay moio mode alog he specified spaial cuved lay of lie The kieosaics equaios ae as follows []: m W qs d + ga A Rd R d W m cd c c d whee m is vehicle mass; g is gaviy acceleaio; q is velociy pessue; S is chaaceisic aea; c d cd c d ae aeodyamic coefficies; W W ae quasi-acceleaios; A is quaeio maix i ems of Rodiguez-Hamilo paamees deemiig oieaio of aual ihedal i eah efeece; Rd is quaeio maix deemiig oieaio of aeodyamic axes elaive o aual oes; ad ae movig foces Kiemaic coelaios i quaeio maices closig he give kieosaics equaios ae as follows []: A A A A Saics Oaied esulig movig foce ( ) which povides moio of wowheel vehicle alog specified lay of lie i he kow mode is epeseed as a sysem of wo equivale coac cool foces ( ) o e deemied ollowig figue shows hem:

8 O ( ) M k ( ) j O ( ) O i Hee efeece pois ae give i movale aual axes: OM l The accodig o aigo heoem [6] we oai: OM l whee l + l + Hece: I paicula if he l ( l+ l) ad ( ) l + l l ad O paallelism codiio: Cosequely: l l + l + l l l l l + l If he I follows: o l l ad l l l l l l

9 l l iе pe-deemied paallelism codiio Saic ivaias ae equied o veify oaied esuls I paicula wihi omal plae of aual ihedal saic ivaia oe: + + is saisfied equally Codiio + is used o specify equied oque of acio wheel a defiie esisace of dive oe ad equied mode of vehicle moio alog he specified lay of lie Saic ivaia wo esuls i paallelism codiio ( ) ( ) + ad l l l l Thus aalyical soluio deemiig equivale coac of movig (cool) foces fo adem-ype wo-wheel vehicle a vaious moio modes alog spaial cuved lay of lie ude he effec of gaviy ad aeodyamic foces is oaied The closed veco depedeces ae show i he fom of quaeio maices povidig efficie compuaioal algoihms Refeeces Динамика системы дорога-шина-автомобиль-водитель Под ред АА Хачатурова- М: «Машиностроение» с Кравец ТВ Определение управляющих сил и моментов при движении асимметричного летательного аппарата по программной траектории сложной пространственной конфигурации // Техническая механика- - -С6-65 Kaves Usig quaeio maices o descie he kiemaics ad oliea dyamics of a asymmeic igid ody / Kaves T Kaves A Khacheko // I Applied Mechaics-9-45#-P- 4 Кравец ВВ Метод матричного представления мультипликативных композиций векторной алгебры/ ВВ Кравец ТВ Кравец АВ Харченко // Восточно- Европейский журнал передовых технологий--/6 (45) С-6 5 Кравец ТВ Об использовании кватернионных матриц в аналитической и вычислительной механике твердого тела // Техническая механика-- -С9-

10 6 Лобас ЛГ Лобас ЛюдмГ Теоретична механіка К: ДЕТУТ 9-47с 7 Мартынюк АА Лобас ЛГ Никитина НВ Динамика и устойчивость движения колесных пар транспортных машин К: Техника 98-с

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