Contact interaction of flexible Timoshenko beams with small deflections
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1 Jornl of Pyss: Conferene Seres PAPER OPEN ACCESS Con neron of fleble Tmosenko bems smll defleons To e s rle: I V Pkov e l 8 J. Pys.: Conf. Ser Ve e rle onlne for des nd ennemens. Ts onen s donloded from IP ddress on //8 :9
2 AMSD IOP Pblsng IOP Conf. Seres: Jornl of Pyss: Conf. Seres do :.88/7-9/9//87 Con neron of fleble Tmosenko bems smll defleons I V Pkov A V Krysko O A Slykov A A Zkrov V A Krysko Dermen of Mem nd modellng Srov Se Tenl Unversy Polenesky sr 77 Srov Rss Dermen of engneerng grs nd ndsrl desgn Tomsk Polyen Unversy Sovesky sr 8/ Tomsk Rss Absr. In s ork o dynms on neron of o fleble Tymosenko bems nder e on of rnsversl lernng lod s nvesged. Te on neron of e bems s ken no on by e Knor model. Te geomer nonlnery s ken no on by e model of T. von Krmn.Te sysem of rl dfferenl eqons of e elf order redes o e sysem of ordnry dfferenl eqons by e meod of fne dfferenes of e seond order. Te reslng sysem by meods of Rnge-K ye of e seond for nd eg orders s solved.or eorel/nmerl nlyss s sored by meods of nonlner dynms nd e qlve eory of dfferenl eqons. Co vbrons of o fleble bems of Tmosenko ere nvesged nd e oml se vles over e sl oordne nd e me ses for e nmerl eermen ere fond. Convergene for ll lble nmerl meods ve been eved nd son o sgnls re re. Keyords: Con neron Tmosenko's bem os fne dfferene meod Rnge- K ye meods geomer nonlnery. Inrodon Alog nonlner dynms nd on neron of bems bsed on yoess of dfferen romons ve been sessvely sed for mny yers ere sll re resen oen roblems orgned from bo engneerng nd sene. Severl yoeses re knon desrbe e dynms of bem elemen: frs seond rd romon eores nd oers. Te yoess of e frs romon s e Eler-Bernoll yoess [] e seond-romon yoess or e Tymosenko yoess [] llos s o ke no on e norml roon o e mdlne fer deformon. Ts eory llos more re desron of e dynms of e bem elemen. Te nonlner dynms of bems bsed on vros yoeses s nvesged n rles [-]. Ts rle llos o rove e r of e o vbrons of o bems of Tmosenko smll g nder on of e rnsverse lernng lod kng no on e on neron. In e knon lerre ere re no solons o s roblems. Ts s fndmenlly morn sse. Wen solvng s omle non-lner sysems of eqons by nmerl meods ere s robbly of obnng n norre solon de o errors n nmerl meods. In s er e sll se e defnon of os gven by Glk []. Glk beleves os ess en eer ere s sgnfn deendene on e nl ondons or e fnon s osve ynov eonen e on of e regon. In s se s no erod. As nl ondons n ddon o e ondons mosed on e fnons enerng no e sysem of dfferenl eqons e men e nmber of rons rese o e sl oordne e order of e Rnge-K meod nd e knem yoess. To rove e relbly nd r of solons n os meods of nlyss of nonlner dynms ere sed. Tese meods nlde e onsron of se orrs Forer oer ser Ponré sedo-mngs vele ser sgnls llon of e vle of e ges ynov eonen n ree dfferen lgorms Knz [7] Wolff [8] nd Rosensen [9]. If e resls obned by ll ese meods gve e sme resl en e n ssme e resls obned re re. In ddon e qeson of e onvergene of e resls deendng on e nmber of rons long e sl oordne nd on e me se for e fne-dfferene meod s been nvesged. In onrs o revos sdes [] n s er s reqred o eve onvergene of resls n os no only n Forer oer ser b lso n sgnls. A sere on of e sdy s e oe of e Rnge-K meod for solvng e on roblem. Conen from s ork my be sed nder e erms of e Creve Commons Arbon. lene. Any frer dsrbon of s ork ms mnn rbon o e ors nd e le of e ork jornl on nd DOI. Pblsed nder lene by IOP Pblsng d
3 AMSD IOP Pblsng IOP Conf. Seres: Jornl of Pyss: Conf. Seres do :.88/7-9/9//87. Meml model We sdy o-lyer bem ere e lyers n on e oer s s son n Fgre. We onsder e se en skng beeen e lyers of e bem s no ossble bese on ressre s smll. Te Cresn Coordne sysem nroded on e Fg.. Te eqons of moon of e bems s ell s e bondry nd nl ondons re obned from energy rnle of e Hmlon-Osrogrdsky / K W d ere К s e kne energy s e oenl energy / W - s e sm of elemenry ork of eernl fores. In s oordne sysem srre of o bems lke o-dmensonl domn Ω s defned s follos ; z k. Fgre. Seme of e nlyzed bem srre. To smle e on neron of bems ordng o e model of Knor B.Y. n e eqons of e defleon bems s neessry o nrode e erm K - s bem nmber fnon k s defned by sgn k ere f ere s on beeen e bems - k oerse no on [] К s oeffen of rgdy of rnsverse omresson of e srre n zone of e z z on k s g beeen e bems. Tngenl dslemen re dsrbed on kness z by lner l z z ; z ere ngle of roon of e norml o e lne z. Ten e z z ; e z z nd f denoed by - ngenl deformons of e mdlne bendng deformon rog H deformon of ser en e eressons for e deformons ll be reresened n form of lner enson by degrees z: e z zh ; z z e z ; z. We nrode ser deformon by e forml: Q f z ere Q - rnsverse fore; G - ser modls f z - fnon rerzng e l of dsrbon of ngenl sresses long e kness. Here Q Gk ere f z dz.vle k for s fnon ll be k eql o. G Tkng no on Hooke's l e eresson for e sresses n be ren n e form E E zh z G. Ten T dz s n nernl effors; Q zdz s ng fores; M zdz s bendng momens. We re don e Hmlon-Osrogrdsky rnle S K d ere
4 789 AMSD IOP Pblsng IOP Conf. Seres: Jornl of Pyss: Conf. Seres do :.88/7-9/9//87 и. Te energy of e mddle srfe s е е d T d T. Te bendng energy s eressed s follos е и d Q H M b e kne energy e d g K. From e Hmlon-Osrogrdsky rnle e re e eqons of moon of srre of o Tmosenko bems n dslemens kng energy dsson no on n e dmensonless form s follos: ; 8 ; ; q K k Here - s serl nmber bems - re e non-lner oerors - s fnon rnsverse ser re e fnons of defleons nd dslemens of bems resevely. To e sysem of dfferenl eqons e ms dd bondry ondons nd nl ondons. Te sysem of eqons e bondry nd nl ondons re reded o e dmensonless form by mens of vrbles:. Eg E q q Te reslng sysem of nonlner rl dfferenl eqons ogeer e bondry nd nl ondons redes o e sysem of ordnry dfferenl eqons by e fne dfferenes meod romon O ere с se n e sl oordne. A e on of e grd e obn sysem of ordnry dfferenl eqons. Te Cy roblem obned n me s solved by meods of e Rnge-K ye. In e ork dfferen Rnge-K meods re omred: e Rnge-K of e rk nd rk orders e Rnge-K-Felberg order rkf order Kes-Kr rkk e Rnge-K Prne- Dormnd of e eg order rk8d e ml Rnge-K meod of e seond order rkm nd e order rkm. On bss of s lgorm s reed rogrms llo o solve e roblem deendng on e onrol rmeers q. Gre enon n ork s gven o e qeson of no enerng e elemens of e srre no e oer. As noed bove e roblems nder sdy re gly nonlner so e qeson rses bo e relbly of e resls obned.. Nmerl resls Te bondry ondons for selng bo ends of e bems: Inl ondons:. On e frs bem s ed rnsverse dsrbed over e srfe sgn-vrble lod ye: sn q q
5 AMSD IOP Conf. Seres: Jornl of Pyss: Conf. Seres IOP Pblsng do:.88/7-9/9//87 ere q - mlde - freqeny of drvng vbrons. q k. /. For e nmerl eermen e onsder:. Te freqeny of e drvng vbrons s lose o e nrl freqeny of e bem. A relmnry qeson s nvesged bo onvergene of e fne dfferenes meod. In Fg. а б sgnls re resened lled for dfferen nmber of ons dvson of segmen n ; 8;; ; ; ;. For e seond bem e onvergene by e nmber of dvsons of e segmen s m orse nd does no omleely ome. Te error beeen sgnls lled n = и n = sс % Hoever e sgnls onde n se over e enre me nervl. Te resls ere obned sng e Rnge-K meod of e 8 order n e Prne-Dormnd modfon rk8d. а b... n = 8 n = n = n = n = ;. -. W W n = n = n = -. n = 8 -. n = n =. Fgre. Bem defleon of sgnls... for n ; ; а Frs bem b Seond bem. In [] s onsdered sffen o so onvergene rese o e Forer oer ser for o vbrons. By sgnl s mossble o eve onvergene. A n e onvergene of resls for o resls even sgnl ll be eved. Frer e onvergene of e sgnls s nvesged deendng on e ye of e Rnge-K meod. For bo bems e resls of e Rnge-K meods of e seond for nd eg orders onded omleely oever s deded o se e Prne-Dormnd 8- order meod rk8d n frer llons sne s meod llos om se-by me. e s nvesge e dynm rerss of bems for dfferen nmber of rons rese o e sl oordne. In Tble e gve grs of Forer oer ser D se orrs nd Ponré sedomngs for bo bems. Tble. Dynm rerss of bems n Pse orr D ' ' Poer serm Psedo Ponre m. 8 / / / '' S ' T..
6 AMSD IOP Conf. Seres: Jornl of Pyss: Conf. Seres / / IOP Pblsng do:.88/7-9/9//87. /.9 '' S ' T '' S ' T '' S... - ' T. Te dssson of e resls. Anlyzng e Forer oer ser for dfferen nmbers of rons n rese o e sl oordne for bem e n frs noe n nrese nd en redon n e nmber of freqenes nd redon n e nose omonen n. A n e oer ser of bo bems demonsre vbrons lnerly deenden freqenes / / / nd e resene of nose omonen lo freqenes. Bo bems vbre e sme freqenes.e. freqeny synronzon of vbrons ors. An nrese n n dobles e overll o edesl. A n n e oer serm of bo bems e freqeny gn ers / lso derese n e o omonen s omred o n 8. A n n e sgnl of e frs nd seond bems ere re freqenes в / и /. ри n и n oer ser re lened from e nose omonen n e sgnl of bo bems ere re freqenes /. и en e n lk bo synronzon of osllons ese freqenes. As rle D se orrs re onsdered b n s er s roosed o onsder D se orrs ' ' n s se e ll ve nformon bo ll e rerss of e dynms. Consder D se orrs ' '. A e mnmm nmber of nodes n e se orr for bem gves rng b n se s ler s rng s kness nd s non-nform. For e seond bem e se orr s sold so orresonds o o vbrons. W n nrese n e nmber of eqons e se orr nd e erne of rngs re omressed. Te sedo-ponre m for bem for ll n s e se of n ovl b smll nges n s kness. Begnnng from n = er onvergene s noed s ell s for bem ere for n = e sedo-ponré m s sered. Comle nlyss of freqeny rerss llos s o mke e onlson bo e sffeny n = for e sdy of nonlner dynms nd e on neron of o bems g en bo bems re desrbed by Tmosenko's model. In s er e vles of e ges ynov eonen for n = re lled sng e meods Wolff Rosensen nd Knz. Te vles re obned on e bss of solons of e Cy roblem by e Rnge-K meod of e 8 order rk8d. Dfferen meods of llng ynov's ndes ms be sed o deermne e re os. Wen e nmber of ons of dvson of e bem
7 AMSD IOP Pblsng IOP Conf. Seres: Jornl of Pyss: Conf. Seres do :.88/7-9/9//87 on n = ; 8 ; segmens n e meod of fne dfferenes e ynov eonens onverge ny onng meod o e seond or rd deml le. In Tble e gve e vles of e ges ynov eonen for bo bems n = lled sng e meods of Knz Wolf nd Rosensen. In order o vod obnng erroneos onlsons n e sdy of o osllons e ynov eonens re obned by severl meods. A resen ere s no relble meod for deermnng s. Tble. ynov eonen n Bem Bem meod Wolf Rosensen Knz Wolf Rosensen Knz n = Rk8d All e vles of e ges ynov eonen regrdless of e meod of solvng e Cy roblem from e nmber of nervls of e bem ron from e llon lgorm re osve. T s e re delng e re o osllons of e nvesged bem srre.. Conlsons In s rle e relbly of e nmerl resls of e solon for e roblem of e o bems on neron desrbed by e knem yoess of Tymosenko smll g beeen em s gven nd defended. A omreensve sdy of e nonlner dynms of e on neron of Tmosenko bems nder on of rnsversl lernng lod. Seleon nmber of ons of rons on e sl oordne n = nd e oe of e meod for solvng e Cy roblem e Rnge-K meod of e 8 order of Prne-Dormnd s jsfed. On bss of rred o reser of freqeny rerss e n lk bo e enomenon of o freqeny synronzon of bems vbron.. Aknoledgmens Ts ork s sored by e Mnsry of Edon nd Sene of e Rssn Federon..7 /. Referenes [] Eler 77 Sr l fore des olones Memores de Ademe de Berln Vol 8. [] Tmosenko S P 9On e orreon for ser of dfferenl eqon for rnsverse vbron of rsm br Plosol Mgzn 7-7 [] Zglov M V Pvlov S P Krysko V A Nonlner Dynms of Mllyer Bem Srres Gs beeen em kng no Aon e Geomerl nd Pysl Nonlneres Inernonl Jornl of Aled Engneerng Reser Vol 7- [] A.V. Krysko J. Arejez M.V. Zglov S.P. Pvlov V.A. Krysko 7 Nonlner bevor of dfferen fleble sze-deenden bems models bsed on e modfed ole sress eory. Pr. Governng eqons nd s nlyss of fleble bems Inernonl Jornl of Non-ner Mens 9 9- [] Krysko A V Arejez J Zglov M V Pvlov S P Krysko V A Nonlner bevor of dfferen fleble sze-deenden bems models bsed on e modfed ole sress eory. Pr. Co dynms of fleble bems Inernonl Jornl of Non-ner Mens 9 - [] Denny Glk Enoners Cos Ne York: MGr-Hll 99 [7] Knz H. 99A robs meod o esme e mmm ynov eonen of me seres Pys. e. A 8. P [8] Wolf A. Sf J.B. Snney H.. Vsno J.A. 98 Deermnng ynov Eonens from me seres Pys D 8-7 [9] Rosensen M T Collns J J Crlo J. De A rl meod for llng lrges ynov eonens from smll d ses Nero Mslr Reser Cener nd Dermen of Bomedl Engneerng Boson Unversy 99
8 AMSD IOP Pblsng IOP Conf. Seres: Jornl of Pyss: Conf. Seres do :.88/7-9/9//87 [] Arejez J. Krysko V.A. Pkov I.V. Krysko A.V. Deermns Cos n One- Dmenonl Connos Sysems Sngr: World Senf seres on Nonlner Sene Seres [] Knor B Y Bogyrenko T 98 A meod for solvng on roblems n e nonlner eory of sells Dokl. Ukrn Ademy of Senes. Ser A 8- In Rssn 7
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