Dynamics of two coupled 4-DOF mechanical linear sliding systems with dry friction (BIF304-15) Angelika Kosińska, Dariusz Grzelczyk, Jan Awrejcewicz

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1 Dnis o two oupd -DOF hni inr sidin ssts with dr rition BF- Ani Kosińs, Driusz Grzz, Jn Awrjwiz Abstrt: Th ppr introdus od o two idnti oupd -DOF hni inr sidin ssts with dr rition oupd with h othr b inr torsion sprin. Th pproprit oponnts bodis o th oupd ssts r ridin on two sprtd drivin bts, whih r drivn t onstnt voitis, nd sti-sip vibrtions n b obsrvd. n this s th phsi intrprttion o th onsidrd od oud b two rows o rris in on th uidws nd oupd b n sti sht, whih r ovin t onstnt voit with rspt to th uidws s oundtion. Fro thti point o viw th nzd prob is ovrnd b iht noninr ordinr sond ordr dirnti qutions o otion idd b th sond ind Lrn qutions. Nuri nsis is prord in Mthti sotwr usin th quittiv nd quntittiv thoris o dirnti qutions. So intrstin non-inr sst dnis r dttd nd rportd usin th phs portrits nd th Poinré ps. Nt, powr sptr obtind b th FFT thniqu r rportd. Th prsntd rsuts show priodi, qusi-priodi, hoti nd hprhoti orbits. Morovr, snhroniztion ts btwn th oupd ssts r so dttd nd studid.. ntrodution Th qustion o sti-sip vibrtions usd b dr rition is sti opnd. Th undnt ws o sti-sip phnon bsd on dr rition dnis hv bn proutd in th pionrin prints o Rbinoviz nd in th wors o Bubrr t []. First, onpt o noninr dr rition shoud b pind. Th or, whih is rquird to strt th ovnt o n objt, is d th stti rition or, but th inti or is ssnti to intin onstnt voit durin th ovnt o th bod. A suiint ondition or sti-sip is tht th stti oiint o rition is hihr thn th inti oiint o rition []. Sti-sip phnon r ptd durin ontt intrtion t ow-voit rition. Th onsidrd sti-sip phnonon dpnds on rqun o vibrtions, rtiv huidit nd od. Sti-sip phnon our in vrd i, or instn, ro rthqus, throuh br ssts whn r is strtd to ov ro sttionr stt [], to nno-dvis showin up in th s bov svr irons. Eps o sintii itrtur dvotd to stis-sip vibrtions in sst n b ound in th rrns [,,,, 8, 9]. Dirnt ods in iro- nd ro-s r usd or dsription o sti- sip phnon. n this wor n 8 dr-o-rdo od is usd. Th bod onsists o two idnti subssts oupd

2 b torsion sprin. Evr subsst rids on two sprtd bts whih r drivn t onstnt voit. Brin in ind prinips o rtivit on n s tht th bodis r ovin bus o iovb bts. n this s th r intrprttion o od t p in in, whr two rows o rris id to uidws r ovin t onstnt voit. As noninr in sti sip ri sst, th sprin sidr od is vr snsitiv to w trn ipts, whih on r s nists its in phnon o indud sisiit, tririn nd snhroniztion ts []. Th onsidrd in this wor hni sst n b trtd s n tnsion o th hni od prsntd in th ppr [].. Mhni Mod Th onsidrd 8-DOF od two oupd b torsion sprin -DOF hni inr sidin ssts with dr rition is shown in Fi.., φ s F r, φ F r Fiur. Th 8-DOF od with dr rition. 8

3 Th prsntd sst n b onsidrd s pnr sst in th Crtsin oordint sst in th Erth's rvittion id with th rvit oiint with horizont is nd vrti is. Dnis o th onsidrd sst n b dsribd b th oowin vribs:, v,, z,, v,,, v,, v,, z,,,. Th sss, n rott bout th pivot s S onts o inrti bout th pivot point S o th ntiond sss r,. Th ntir sst is hrtrizd b nths i, i i,,..., nd sprins with stinss oiints i, i, j, j i,,,, ; j,,,. Morovr, two ddition sss, r in on th pproprit bts s oundtion, whih r ovin with onstnt voitis v nd v, rsptiv. Btwn th ntiond sss, nd pproprit bts dr rition ors our s untions o th rtiv sidin voitis v, v, rsptiv. Equtions o otion o th onsidrd sst hv bn drivd usin th Lrnin thod th sond ind Lrn qutions [] nd th r s oows d dt T T q q q Q n, whr: q - vtor o nrizd oordints, Q - vtor o nrizd non-onsrvtiv or tin in th sst, T - tot inti nr o th sst, - tot potnti nr o th sst, t - ti. n n this s dot ns dirntition with rspt to ti t. For prsntd prvious 8-DOF od with dr rition, vtor q is rds:,,,,,,, T q. Siutnous, Q n n b dsribd b th oowin vtor Q n F r, F T r. Th rition ors F r nd F r r qu to th produt o noninr inti rition oiints v, v ssoitd with rtiv voitis o vr subssts nd th nor ors N, N, whih prss th sss nd to th irst bt nd to th sond on, rsptiv. t shoud so b notd tht in nuri utions th vus o th nor ors N nd N n b ss 9

4 thn zro, rtr thn zro or qu to zro. n th s o, N N, th rition ontt btwn sss, nd th pproprit bts ovin with voitis v, v our. n turn, th s, N N ns oss o rition ontt btwn th sss, nd th pproprit bts. This is wh in our thti od w us disontinuous stp untions dsribin ths phnon, nd dind s oow. or or N N N,. or or N N N Fin, ors r F nd r F hv th oowin or, ] [,, r v v F. ] [,, r v v F Tot inti nr T o studid od hs th oowin or: T. Sin s vus o ns nd r tn into onsidrtion, th tot potnti nr hs th oowin or. s 8

5 Coputin th prti drivtivs q q q T T dt d,,, bsd on, w obtin 9. Non-dinsion or W introdu non-dinsion ti / t, non-dinsion oordints,,,,, nd th oowin non-dinsion prtrs:,,,. ] [, ] [ s s v v

6 ,,, b, b, b, b,, 8, 9,,,,,

7 d, d, s, s,,, 8,, 9 nd th oowin non-dinsion untions,,,. n rsut, qutions o otion in th ountr prt non-dinsion or r s oows. ] [ ] [ d d d b b b b. Nuri oputtions Our nuri oputtions hv bn prord vi th ourth ordr Run-Kutt thod with onstnt ti stp. h nd zro initi onditions. W onsidr stri sst with th

8 vus o non-dinsion prtrs nd non-dinsion untions tn ro th prvious ppr [], n: 8,, 8, b b b 9, b b b, 89,, 8,, d d, 9,, 9,. Kinti rition untions nd in our od r dsribd b th Strib untions. Bus ssi sinu untion is disontinuous, w did to pproit th ntiond untions b hprboi untion with nuri ontro prtr nd in th or tnh, tnh, with id 8,, 9,, nd. Morovr, bus untions, r so disontinuous, in our oputtions w us th oowin pproitions n tnh, n tnh. 8 n rsut, in our nuri siution w onsidr th oowin qutions o otion

9 b b [ ], 9 b b [ ].. Nuri rsuts Fi. shows th phs trjoris o th sst or th voit o drivin bt. nd zro initi onditions in ti intrv [ ]. Th ti intrv ws hosn to void th trnsition stt. Obtind rsuts nd dtt n irrur dnis o th onsidrd 8 DOF sst. Th phs trjtoris, Poinré ps Fi. s w s powr sptr dnsitis Fi. indit tht th hrtr o otion is hoti. w inrs th vu o thn th hrtr o otion hns. This sitution is prsntd in th Fi., Fi. nd Fi.. Whn th dinsionss voit o drivin bts rhs th vu o. th otion hibit priodi hrtr. Fiur. Phs trjtoris o th sst or. in th ti intrv [ ].

10 Fiur. Poinré p o th sst or. in th ti intrv [ ]. Fiur. Powr sptr o th sst or. in th ti intrv [ ].

11 Fiur. Phs trjtoris o th sst or. in th ti intrv [ ].

12 Fiur. Poinré p o th sst or. in th ti intrv [ ]. Fiur. Powr sptr o th sst or. in th ti intrv [ ]. Conusions n th ppr thti od o two oupd -DOF hni inr sidin ssts with dr rition is onsidrd. Th onsidrd sst n b trtd s sst o two idnti -DOF ssts prsntd rir in [] nd oupd b torsion sprin. n this s th phsi 8

13 intrprttion o th onsidrd od oud b two rows o rris in on th uidws nd oupd b n sti sht, whih ovs t onstnt voit with rspt to th uidws s oundtion. Fro thti viwpoint th ntiond sst is prsntd s noninr qutions o otion, whih r obtind usin sond ind Lrn's qutions. Dnis o th nzd sst is rrid out or on st o sst prtrs nd vrious non-dinsion. ntrstin dnis bhviors o th onsidrd sst r rportd usin ti sris nd phs trjtoris. Th obtind rsuts indit, tht th nzd sst posssss priodi, qusi-priodi or hoti orbits, s w s id points. Morovr, th ntiond rsuts show tht snhroniztion ts btwn th oupd ssts r possib. Anowdnts Th wor hs bn supportd b th Ntion Sin Foundtion o Pond undr th rnt MAESTRO No. //A/ST8/8 or rs -. Rrns [] Abdo, J., Tht, M., Abousoud, A., nd Dnish, M. Th t o rqun o vibrtion nd huidit on th sti sip pitud. ntrntion Journ o Mhnis nd Mtris in Dsin. [] Chidz, T., Lursnshvii, O., Mthrshvii, T., rshvii, N., Zhuov, N., nd Mphridz, E. Hih-ordr snhroniztion o sti sip pross: prints on sprin sidr sst. Noninr Dnis 9. [] Fn, Q., u, C., Niu, J., Jin, G., nd Liu,. Stbiit nss nd nuri siutions o th sin dr o rdo sprin-sidr sst obin th rvisd rt- nd stt-dpndnt rition w. Journ o Sisoo 8 9. [] Grzz, D., Awrjwiz, J., Kudr, G. Dnis o Mhni Sidin Sst with Dr Frition. Mhin Dnis Rsrh 8, -. [] Mur, A.. Mirosopi sti sip in rition or irosop. Triboo Lttrs 8 9. [] Mothono-Tinu, M., Kno, G.D., nd Kon, T. C. Sti Sip Motion nd Stti Frition in Noninr Dorb Substrt Potnti. Triboo Lttrs. [] Priv,., Moshovih, A., Lpsr,., Lihtn, A., nd Rpoport. L. Disotion Strutur nd Sti Sip Phnonon. Triboo Lttrs 9-. [8] in, W.P., nd Hndris, C.P. Sidin rition dnis o hrd sin spritis on sot surs. Triboo Lttrs 9 9. [9] Zhn, J., nd Mn,. Sti Sip Frition o Stinss St in Sodiu Dod Sut Aquous Soution in th Boundr Lubrition Ri. Triboo Lttrs. [] Zhn, S.L., nd ntin, J.M. Sti sip nd tprtur t in th srthin o tris. Triboo Lttrs, 9-. 9

14 [] Zhn, J., Zhn, N., nd Crowthr, A.R.. Anti stud o br ron throuh oupd DOF br od. Jpn Journ o ndustri nd Appid Mthti -. Jn Awrjwiz, Prossor: Dprtnt o Autotion, Biohnis nd Mhtronis, Lodz Univrsit o Thnoo, / Stnowsi Str., 9-9 Lodz, Pond jn.wrjwiz@p.odz.p. Ani Kosińs, M.S. Ph.D. studnt: Dprtnt o Autotion, Biohnis nd Mhtronis, Lodz Univrsit o Thnoo, 9-9 Łódź, Pond ni.osins@dot.p.odz.p. Th uthor v prsnttion o this ppr durin on o th onrn sssions. Driusz Grzz, Ph.D.: Dprtnt o Autotion, Biohnis nd Mhtronis, Lodz Univrsit o Thnoo, 9-9 Łódź, Pond driusz.rzz@p.odz.p