CONTRIBUTIONS TO THE STUDY OF THE PASSING THROUGH THE RESONANCE OF THE LINEAR SYSTEMS HAVING A FINITE NUMBER OF DEGREES OF FREEDOM

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1 U.P.B.. Bull. eres D Vol. 69 No. 007 IN CONTRIBUTION TO THE TUDY OF THE PING THROUGH THE REONNCE OF THE LINER YTE HVING FINITE NUBER OF DEGREE OF FREEDO C- ION Ele Elvr ION G. C. ION Î esă lurre se v su roble regulu rzoru î oţle leg e vrţe lră freveţe forţelor erurbore uu sse ulsă. Euţle fereţle re esru şre sseulu els su vor f reuse l euţ e orul o. O sfel e reuere se oe relz r ouă our e oserre forţelor e rezseţă oresuzăore oezelor lu Fogo ş le lu E.. oro. e v ră ă î zul robleelor lre ulzre oulu e rezolvre lu oro ese ă eâ ulzre oulu e rerezere forţelor e frere forţe e vâsoze forţe e su roorţole u vez e eforre. Te usey ssg of ul-ss syse uer e lw of ler vro of surbe fores frequey wll be sue s er. Te fferel equos w esrbe e oo of e sue els syse wll be reue o e qur equos. u reuo y be eve by wo wys of oserg e resse fores orresog o e Fogo s E..oro s yoess. I e se of ler robles wll be sow e lo of oro s solvg wy s ore e e rereseo wy of fro fores s vsous fores w re roorol o e eforo see. Keywors: bsoro oeffe surbg oe vsous g resoe.. Irouo Te usey ssg of ul-ss syse wll be sue uer e lw of ler vro of surbe fores frequey. Te fferel equos w esrbe e oo of e sue els syse wll be reue o e qur equos lyze e er []. u reuo y be eve by wo wys of oserg e resse fores orresog o e Fogo s E..oro s yoess []. I e se of ler erls wll be sow e lo of oro s solvg wy s ore e e rereseo Prof. De. of es; Uversy Pole of Bures RONI Reer Dere G.D.G.I Uversy Pole of Bures RONI ss. De. of es; Uversy Pole of Bures RONI

2 0 C- Io Ele Elvr Io G. C. Io wy of fro fores s vsous fores w re roorol o e eforo see. Ts osero s exle o oly by es of goo orresoee of oro s yoess w exerel bu lso by es of reue lulo volue.. Fogo s yoeses eo Te geerl for of e fferel equos of e -freeo egrees syses osllos s ( q b q q ) Q ( ) () were b re os vlues wle q ( ) Q ( ) re e oores geerlze fores resevely. By eg e l oos: ere resuls fro () e syboll for x q ( ) ; F Q ( ) ( b ) x F olvg e syse () oe obs: x () F. ( ) () (4) were s e eer of e syse wle re e oleery eers orresog o e olu elees. s oly olex roos w egve rel r: If be broe u o sle fros e for ( ) ( ) (5)

3 Corbuos o e suy of e ss roug e resoe of e ler syses were: 4 γ δ ω ; γ ; δ ; [ ] If e relo (7) s roue ω. (6) ( γ δ ) F (7) by es of e relo (6) e relo (4) beoes x ω (8) or x Z (9) were every er be osere s ge of solvg e qur fferel equos ζ () ζ () ω ζ () () (0) s Z ζ ( ) ; ( ). () If e l oos re ull e equos (0) beoe ( τ ) ζ () ( τ ) e s ( τ ) τ () 0

4 C- Io Ele Elvr Io G. C. Io wle () e q 0 s τ τ τ τ. () If e syse s oere by geerlze fores vryg org o e lw () ex Q Q o (4) e egrl () s reue o robbly egrls w e olex rgue () 4 V V o u u o o o o e e W V e W u W V W u Q q π (5) were u V.. oro s eo Te fferel equos syse s esly obe fro e fore osllo equos () gorg e resse fores roug se of e E elsy oulus e olex oulus e for E ± π were Ψ s eergy bsoro oeffe. Tus s obe () Q q q ± π (6) or e syboll for F x ± π. (7)

5 Corbuos o e suy of e ss roug e resoe of e ler syses Ule e revous se e eer (s) of s syse s rel roos were ± ±... ±. (8) ± π Tus e lulo of lo of roos slfes. For e vlue wo vlues orreso. Fro (8) oe obe Reovg π π w s o erl eg oserg ± ± oe obe fro (7) π 4π were K ±. π u x ( ) e e roos equo 0 o sle fros wll be ( ) ( ) F K (9) re ± ±... ± s breg γ δ (0) w γ ; δ. ( ) ( ) e () s eve fuo ere resuls :

6 4 C- Io Ele Elvr Io G. C. Io ( ) ( ) γ 0 δ. ( ) Furer we oe δ ( ) F. () If (0) s roue o (9) ere resuls x () K or x Z ± π. () Fro () oe obs: ζ () ± ζ () (). (4) π Te fl soluo s q were (6) ws roue 0 () ( τ ) e ( τ ) 4π s ( τ ) ( ) δ ( ) F. ( ) τ (5) (6) I e se of e wo eos e slry bewee e soluos (6) soluos () s oe.

7 Corbuos o e suy of e ss roug e resoe of e ler syses 5 4. lo e se of e ssg roug e resoe of ree ss syse erfors ws osllos. If e se of e vsous g s osere e fferel equos syse wll be 0 0 b b b b (7) were er oes of e weels; elsy oeffe of e sf seor () surbg oe g o e ss ; b g oeffe e seor; roo gle (fgure ). Fg.. If we eoe ; fer severl lulos e syboll for beoes:. (8) Tere wll be roue l ull oos e oos

8 C- Io Ele Elvr Io G. C. Io 6. ; ; ; ; ; ; b b (9) ssoe olex roos of e equo 0 gve by e relos (5). Furer e oore s lule; us soluo (8) s e for (0) were ;. (0b) wle re ssoe w. Te exresso () () ex 0 ; () s roue o (0) vg e ew vrbles y x u y x v oe obe () [ ] ex 0 u W v W u W v W π.() Te vrbles x y x y re eere by es of e relos y x y x ξ λ ξ λ ξ λ ξ λ. ()

9 Corbuos o e suy of e ss roug e resoe of e ler syses 7 5. Nuerl lo Te lulo of e syse o-sey roess s erfore vg e uerl 9 6Ns 96Ns 9Ns b 4 7Ns b 9 6Ns N N 749s. Te followg equo s obe for eerg e roos (b) Te relos (5) re le oe obs 448 s 080 s 66 s. () 908 s λ Te exressos of e oeffes gve by (0b) re: Te y oeffe λ wll be eere usg e relo () ffere vlues of e ξ s were s 0 ( ). (e) I bles e W(Z) robbly egrls vlues re gve for ξ. Tble W u W ( v ) W ( u ) W ( v ) ().

10 8 C- Io Ele Elvr Io G. C. Io 8 Tble W u ξ Fg.. By vryg ξ fro o.046 e syse s e re of e seo resoe ( ξ s ofe orgly fro.8 o.5705). Durg e e fluee of e fuos W ( u ) W ( v ) W ( v ) s uor ey ve szes of egrees W ( u ) W ( v ) ( ) W ( v ) 0007 ( ). Beses fro e bles s well ere resuls e re (f) of e frs resoe W ( u ) s e os or vlue wle e re of e seo resoe s e gue of W ( u ) w s e os or vlue. I s wy s ofre e ess ofe use rl lulos e oe we e resoe s er url frequey e fors of e o-resoe vbros (w vlues of e url frequey fferg lo fro e resoe frequey) ve sll fluee o e gue of e lue of e syse resoe vbros. Te rs of e y oeffe λ we ssg roug e frs e seo resoe (fgure ) sow e exsee of e se rerss s se of e ssg roug e resoe of e ler syse vg oe freeo egree: overlyg reug e ges lue sroes euo se e. By oserg e fros org o oro e syse of equos wll ve e for ± ( ) 0 π ± ( ) ± ( ) ( ) (4) π π ± ( ) 0 π

11 Corbuos o e suy of e ss roug e resoe of e ler syses 9 or wll be e sybol for uer ull l oos (fer roug ) K K (5) were K π K ±. olvg e syse (5) w rese o oe obe K (6) or C C B B K (7) were ± re e roos of e equo (s) 0 wle C C B B re gve by B B C C. (8) Before obg e fl for (7) s wre uer e for K C K B or (9) were

12 40 C- Io Ele Elvr Io G. C. Io B; C; re ouge w (40) ( ) π Te resuls (0) (9) re e se. Tus e solvg wy of (0) wll be e se f we roue e oeffes (40) o (0). Ule e revous se se of four-ower equo ere s obe bqur equo 4 ( ) ( ) s 0. (g) s eve fuo beuse e eer of I e geerl se e equo ( ) 0 be wo-es reue. s oe bove ere s oe ore rers gve by e f e roos ± (...) re rel (R) o ssoe olex. ll ese slfy very u e lulos. Te bsoro oeffe of e vbros ower s ose usg e oo of oeffes oee 448s. Tere re fou e roos π s s e π Te oeffes (40) wll be () For e sue se fgure s sow oly urve () for ssg roug e seo resoe. s resul of oosg e oeffe e ssg urves roug e frs resoe re rlly overlg for bo ses. 6. Colusos s we exee e xu of e urve () s ger e oe of e urve (). Te exlo s e fro fores osere s suy o o ee o e vbro frequey. Tey ve ree sller e fores w re roorol w e soro see w vbros w ger frequey. I e se e e xu of e y oeffe of e frs resoe s ger e wo xu of e seo resoe. T es e se of e os lue 0 of e surbg oe. We 0 ees o frequey e e relo bewee e xu vlue of e oeffe λ s ge y resul e xu lue of vbros e seo resoe s ger e frs oe. Tus for exle

13 Corbuos o e suy of e ss roug e resoe of e ler syses 4 w () 0 ex w oer slr oos e xu oeffes wll be λ 4 for e frs resoe λ 9 5 for e seo oe. R E F E R E N C E []. E.G. Golosoov.P. Flov Nesorve oleb ees syse Kev 996. []. C. Io E.E. Io see Bezug uf e rse Drezle ulere Lger Welle Bule U.P.B. Buures 99. []. C. Io E.E. Io Ds su er rse Gerswgee fur Wele er se Uberrguge YRO Buures 99.

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