254. Chain type system with wave excitation

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1 54 CHAIN TYPE SYSTEM WITH WAVE EXCITATION KAZIMIERAS RAGULSKIS MINVYDAS RAGULSKIS 54 Ca tpe sste wt wave ectato Kazeras Raglss Mvdas Raglss Ltaa Acade o Sceces Departet o Teccal Sceces Gedo 3 Vls LT-3 Ltaa e-al: Kazeras3@otalco Kaas Uverst o Tecolog Departet o Mateatcal Researc Sstes Stdet 5- Kaas LT-5368 Ltaa e-al: MvdasRaglss@tlt (Receved 7 Jaar 7 accepted 5 Marc 7 Abstract Varos sstes based o waves ad vbratos are sed or dsplacg gropg classcato o ltdesoal eda ad bodes Ts paper deals wt sc wave operato based sstes ageted b sel-stoppg eleets wc esre oe-drectoal oto o te drve sb-sstes Sc eaceets ca prove soe dacal caracterstcs o te aalsed sstes Te objectve o ts paper s to develop odels o sstes ad etods o aalss wc wold elp to reveal olear dacal propertes ad peoea o tose sstes Te codtos o soltos estece ad stablt bas bodares are detered Spler cases are aalsed aaltcall Te obtaed relatosps provde sgt to olear dacs o cople sstes wc are aalsed ercall Kewords: stadg ad propagatg waves sel-stoppg eleet olear dacs Itrodcto No-cotrolled or cotrolled sel-stoppg ecass pla portat role deret sstes ad elp to prove dacal caracterstcs o tose sstes ad to spl ter strctre Ca tpe sstes wt sel-stoppg eleets geeralse soe spled odels o lds ad gralar tpe aterals Ts paper s te rter vestgato o vbratoal ad wave trasportato [ ] Despte o portat scetc aceveets [3 4] t s portat to aalse sc ew tpe o trasportato sstes ad detere attractors ter bas bodares develop oto cotrol strateges [5] Model o te sste Te aalzed sste (Fg cossts ro te pt eber wt te worg prole ad te otpt eber te ca cosstg ro blocs Te -t eber s te cotact wt te worg prole at te cotact pot A Te worg prole o te pt eber s deed accordg to - ad -aes b: η η( t ξ( t ξ ( Te -t eber wt te ass ca ove wt respect to te eber (wt ass accordg to - as Te eber ca ove te drectoal gdes alog -as Te sel-stoppg devce s attaced to te eber It lets te -t eleet to ove ol oe (postve drecto Te sel stoppg devces ca be attaced also to te -t eber ad ca lt te drecto o oto wt respect to te worg prole Te separate ebers o te otpt sste are coected b te elastc-dsspatve eleets Te orce o rcto betwee te -t eber ad te worg prole at te cotact pot A s actg te tagetal drecto: F N sg s s ( N s te oral reacto orce actg to te -t eber at pot A ad are te coecets o dr ad vscos rcto respectvel ṡ s te slppage Vbroecaa Joral o Vbroegeerg 7 Jaar/Marc Vol 9 No ISSN

2 54 CHAIN TYPE SYSTEM WITH WAVE EXCITATION KAZIMIERAS RAGULSKIS MINVYDAS RAGULSKIS veloct o te -t eber wt respect to te worg prole at pot A top dot stads or d dt F ξ s cosα ( ξ ( F s sα sgs sgs (7 ( ( ( F ξ F ξ F η F ( ξ (8 sgs Fg Te sceatc dagra o te sste Accordg to Fg : η ξ (3 Fro b tag to accot eq (: η t η η (4 t Slar relatosps are vald or ξ ξ Te agle α betwee te -as ad te taget at te pot A s detered b te ollowg eqato: ξ ta α (5 Te veloct o slppage o te -t eber wt respect to te worg prole at te pot A s: ( ηt cos α ( ξ t sα s or ( η ξ ( η s s ξ s η ξ (6 O te bass o te eqatos o eqlbr wt respect to te -t eber te deretal eqatos o oto tae te ollowg or: tt ( F F H F F ξ F F F F H ξ C ξ ( l ( C ( l ( ξ ξ C ( ξ l F H F H ξ (9 Eqatos (8 are vald we tere are o sel-stoppg devces I case we sel-stoppg eleets are attaced betwee te eber ad te otoless bass eqatos (8 are vald we η > ( ad do ot old tre we η ( I case we sel-stoppg eleets are attaced betwee ebers ad eqatos (8 wll be vald we ṡ > ( ad wll ot old tre we ṡ (3 Eqatos (8 ca be spled we paraeters η ad ξ oscllate wt g reqec ad sall apltdes ad assg tat: η ( ξ ad at te sae te: t tt η (4 (5 Te eqatos (8 are spled to te ollowg or: Vbroecaa Joral o Vbroegeerg 7 Jaar/Marc Vol 9 No ISSN

3 54 CHAIN TYPE SYSTEM WITH WAVE EXCITATION KAZIMIERAS RAGULSKIS MINVYDAS RAGULSKIS F ( ξ F F s ξ F sg (6 Case we I ts case ad accordg to eqatos (9: ( F q ( F q F ξ p ξ l q ( F ξ p ξ l q ( ( ( l ξ ξ ξ ξ l H C p q ( ( H q H (7 C (8 Eqato (6 s trasored to: ξ C ( ξ sg (9 ( q ξ ξ p ξ l q F F ( ( q q are detered aalogosl to eq (8 Eq (6 s trasored to: ( ξ ξ sg ( 3 Ectato b travellg waves Accordg to eq ( η η( t ξ( t ξ (3 η ad ξ are perodc ctos o ter argets I ts case te dervatves eq (3-6 tae te ollowg or: ( t η η η η tt η t η η η η t η (4 η η η η (5 ( t t Eq (4 sples to: ( ( η ( η η η (6 Aalogos relatosps ca be obtaed or ξ I tat case eq (5 6 are trasored to: ξ ta α s ( η ξ s ( η ξ s η ξ (7 Case we ( I ts case te eqato (8 or (6 olds tre We ad accordg to eq (9 ad tag to accot η η ξ ξ te ollowg relatosps are obtaed: Net we asse aroc waves η Acos ( t B ( t For eaple eq (8 at becoes: ( ( ξ η ξ ( sgs Vbroecaa Joral o Vbroegeerg 7 Jaar/Marc Vol 9 No ISSN ξ s (8 ( ξ ( (9 are detered b eq ( 3

4 54 CHAIN TYPE SYSTEM WITH WAVE EXCITATION KAZIMIERAS RAGULSKIS MINVYDAS RAGULSKIS B tag to accot tat te apltdes o ectato are sall ad te reqec s g also l q eq (9 tag to accot (4 ad (5 becoes: ( ξ ( ξ ξ p tt t tt t ξ (3 dervatves o η ad ξ are calclated ro eq (8: ( t A s (3 t Eq (3 ca be epressed te ollowg or: ( t a (3 Case Te otpt sste oves wt te veloct o te propagatg wave I tat case t (33 cost I tat case η η ξ ξ ad η η η( ( ( η η (34 I case o aroc waves ad tag to accot ( : η A cos η As η Acos ξ B s ξ B cos ξ B s (35 Eq (8 b tag to accot (33 34 becoes: ( ( ( η ξ F ξ η ( ξ φ ( F F F H F F F F F (36 C( ξ l F C ( l C ( ξ ξ l (37 F F It s possble to d ro eq (36 Te estece o ter real vales s te ecessar codto o estece o otos Te stablt o tose otos ca be detered ro lear varatoal deretal eqatos costrcted or eq (8 ad 36: δ (38 δ s te varato I case we t s assed tat (39 ad ro eq (36 b tag to accot (35 t s obtaed: ( As B cos φ (4 q H q p B s l q ( C p Eq (4 s trasored to: q (4 φ a cos b s a cos b s a (4 5 p AB 5 ( A a B b a 5 p B ( ( q p l B ( p B ( q p l A A A ( q p l 5 p AB ( A B b a ( 5 Te codto o stablt s: (43 φ a s b cos a s b cos > (44 4 Vbroecaa Joral o Vbroegeerg 7 Jaar/Marc Vol 9 No ISSN

5 54 CHAIN TYPE SYSTEM WITH WAVE EXCITATION KAZIMIERAS RAGULSKIS MINVYDAS RAGULSKIS I case we B te codtos o estece ad stablt are: ( 5 A ( A < (45 ( ξ ( ( ξ η ( ξ ( (5 cos > (46 It ollows ro (45 tat te estg stable otos are te terval π ad stable π (47 I case we A q (48 te codtos o estece ad stablt are: 5 p B < cos > (49 Stable otos est tervals 3 7 π π ad stable oes 4 3 π ad π π ad π π (5 4 4 I case o logtdal travellg waves detered b (45 47 two tpes o otos ca coest ad case o trasverse travellg waves detered b (48 5 or tpes o otos ca coest Tose are te ltg cases geeral case or 4 tpes o otos ca coest ro wc al are stable ad al are stable Case we I ts case ro eq (8 7 8 t s obtaed: ( ξ ( ( ξ η ( ξ ( ( η l q η ( η η l q ( ξ ξ l p ( l q ( ξ ξ l p( l q ξ ξ ( H C p q c q (5 c Codtos o estece ad stablt are detered ro eq (5 Case : Te otpt sste oves wt a sall veloct wt respect to te veloct o te propagatg wave: vt (53 t T v<< dt (54 T t T s perod o I a separate case t s sow tat at ad te deretal eqatos o oto becoes eq (3 For te deterato o stead state odes o oto ts eqato b tag to accot (53 ad 54 s rearraged to te ollowg or: ( δ t ε( ( δt ( t v δ (55 ε s a sall paraeter δ v Te stead state oto accordg to (55 s sogt wt te elp o te power seres: ε ε (56 Vbroecaa Joral o Vbroegeerg 7 Jaar/Marc Vol 9 No ISSN

6 54 CHAIN TYPE SYSTEM WITH WAVE EXCITATION KAZIMIERAS RAGULSKIS MINVYDAS RAGULSKIS ad te eqatos or deterato o becoe: ( δt ( t ( t v δ (57 δ j j Te codto o perodct o b tag to accot ol te lear part o ( δ t b power seres ters o s obtaed te ollowg or: ( δt v (58 δt ro v s od I case o aroc waves ad eq (3 taes te ollowg or: ( Acos( t As( t 5 B s ( t Aalogosl: (59 ( Acos( t As( t 5 B s ( t ( ( v ( Asδt Acosδt B cos δt (6 ro : δ 4 A ( δ ( B δ ( ( 4 Te codto o perodct o v q 4 4 ( δ ( ( δ ( cosδt sδt ( cos δt δ( s δt (6 ro ( elds: ( ( δ( δ ( 3 4 B ( A ( δ (6 Te coecet o o-ort: ( a δ δ δ B q v A 4 ( 4 δ ( ( p B δ δ A q 4 Ectato b stadg waves I ts case eq ( ave te ollowg or: η ( η( t ξ ( ξ( t η ξ (63 (64 ctos η ξ η ξ are perodc ctos I case o aroc waves: η A cos cost ξ B cos cost (65 Case : I ts case te vestgato o te sste dacs s cocetrated o a sc tpe o eqato: ( t a (66 s a perodc cto Te oto o te sste s dvded to te slow ad ast otos: (67 ad eq (66 s rearraged te ollowg wa: ( t ε( a ( t ( t (68 Te solto o eq (63 s sogt aalog wt (55-57: ε ε (69 wat prodces a ( t ( ( t ( t (7 Tag to accot te codto o perodct o ad lear part o te epaso o ( ( t respect o eld te ollowg relatosp: 6 Vbroecaa Joral o Vbroegeerg 7 Jaar/Marc Vol 9 No ISSN

7 54 CHAIN TYPE SYSTEM WITH WAVE EXCITATION KAZIMIERAS RAGULSKIS MINVYDAS RAGULSKIS a ( ( t (7 Te estece o real vales o s te ecessar codto o estece o te aalzed ode o oto Its stablt ca be detered ro φ ( ( t < (7 I case o aroc waves accordg to eq (9 ad (8 at ad b tag to accot ( ad l q te eqato o oto elds: ( 5 Acos cost Acos st I case we B : B s ( cost (73 ( Acos ( cost st ε ( A( cos cos ( cost st Aalogosl to (69 7: Acos 4 ( ( ( ( ( st As Te codtos o estece ad stablt eld: 4q ( ( A (74 < cos < (75 cost ad stable reges o oto are te tervals π ad 5 π 3 π Ustable otos are 4 4 at π 5π ad π ( We q stable pots are odes ad stable saddle pots Case we A ( q 5 B s ( cos t ( 5 B ( s s ( cos t ε Zero approato elds B s 5 (77 Te codtos o estece ad stablt tae or: 4q ( p B < ( cos < p (78 I case o logtdal ad trasverse travellg waves te codtos o estece ad stablt (75 78 are slar bt tere s a essetal derece I case o trasverse waves te stable ad stable codtos we p > are qte slar to te case o logtdal waves bt we p < (79 te cage essetall tat s stable reges are te 3π 7π tervals π ad π ad stable 4 8 te tervals 3π ad 3 π 7 π (8 4 6 We q ad p > stable pots are te odes ad stable saddles We p < stable pots becoe saddles ad stable odal pots Case : trasverse stadg waves Accordg to te spled eqatos (6 case o trasverse stadg waves η ad deretal eqato o oto elds: ξ ξ ( ( l ξ p ( ξ l ξ ( q tt t ( ( ( l ξ p ( ξ l ξ q q ( tt t q (8 Vbroecaa Joral o Vbroegeerg 7 Jaar/Marc Vol 9 No ISSN

8 54 CHAIN TYPE SYSTEM WITH WAVE EXCITATION KAZIMIERAS RAGULSKIS MINVYDAS RAGULSKIS t s assed tat l l I case o aroc stadg trasverse waves detered b eq (65 we l q te stead state rege elds ro eq (8: ( l q 5 B s ( l q 5 B s Eq (8 leads to: ( p (8 s 5 q q B s (83 I case we λ (84 4 π λ s te wavelegt eq (83 elds: s 5B ( q or b tag to accot (8 : s 5B q ( ( c c Te codto o estece o sc reges s: ( ( c c 5B (85 < (86 ad ter codto o stablt o te bass o eq ( s: ( ( c c cos < (87 π 5π ad π ( I case we ( ( c c (9 < stable reges are located at 7π π ad stable at 4 3π π ad 4 3π 3π 7π ad (9 4 4 Case 3: logtdal stadg waves Accordg to te spled eqatos (6 ad at ξ te deretal eqatos o oto eld: ( ( q t ( ( t q ( η η l t tt ( η η l tt (9 I case we te ectato s te or o logtdal stadg aroc wave te accordg to (65 ad gorg dsspatve eleets prodces: ( l q 5A ( s ( l q 5A ( s Fro (93 t s obtaed ( s q q (93 5A ad te ollowg codtos o estece ad stablt are obtaed: t I case we ( ( c c 8 (88 > π stable reges are located at ad 4 5π 3π ad stable at 4 5B ( < Vbroecaa Joral o Vbroegeerg 7 Jaar/Marc Vol 9 No ISSN ( cos > (94 Stable pots are te odes wc te larger ass s located

9 54 CHAIN TYPE SYSTEM WITH WAVE EXCITATION KAZIMIERAS RAGULSKIS MINVYDAS RAGULSKIS Coclsos A seqetal ca coprsed ro asses led wt elastc eleets s aalsed Te asses o te ca are ected b propagatg wave prole Several partclar cases are aalsed detal We te propagatg wave s perodc te stead state oto ode s aalsed at te codto tat te average veloct o te ca eleets s eqal to te veloct o wave propagato Codtos o estece ad stablt o sc otos are derved We te average veloct o te ca eleets s sall copared to te veloct o te propagatg wave te stead state oto odes are aalsed eplotg approate aaltcal tecqes Average veloct oscllatos arod te average veloct ad te propertes o soltos are aalsed We te ectato wave s a stadg wave te otos are aalsed te odal ad a veloct odes Moto caracterstcs codtos o estece ad stablt are vestgated Reereces K Raglss Mecass o Vbratg Fodato (Dacs ad Stablt Kaas Isttte o Eerg 963 K Raglss I Vts V Raglsee Sel Scrosato o Mecacal Sstes Mts Vls K Raglss R Josas A Bass M Rodoasas J Taosas R Desevsj Vbrato o Rotor Sstes Moslas Vls R Barasas G Klvets K Raglss Desg o Vbroeges Masostroee Legrad R Basevcs R Krlo A Bbls R Volceova Hg Freqec Vbrato Moto Trasorers Masostroee Legrad R Basevcs A Ivaov N Kasj A Kost L Lovov V Mceev K Raglss Idstral Robots or Matrsed Prodcts Masostroee Moscow K Raglss R Basevcs R Barasas G Klvets Vbrootors or Precso Mcrorobots Hespere New Yor Ltaa Scece Vbroegeerg Scece ad Arts o Ltaa Vls LMA K Raglss A Bbls Precso Vbrotecolog or Developet o Mecacal Sstes Blgara-85 Moscow 985 K Raglss Vbrato Based Trasorers o Moto Masostroee Legrad 984 K Raglss Wave Mecacal Sstes KUT Kaas 994 Vbroecaa Joral o Vbroegeerg 7 Jaar/Marc Vol 9 No ISSN

Noncommutative Solitons and Integrable Systems

Noncommutative Solitons and Integrable Systems Nocotatve Soltos ad Itegrable Systes Masash HAMANAKA Nagoya Uversty Dept. o Math. vstg Oord or oe year Based o EMPG Sear at Herot-Watt o Oct 7th MH JMP6 5 57 [hep-th/6] MH PB65 5 [hep-th/57] c. MH ``NC